# bib

key: Diffusion
type: book
author: Stroock, D.W. and Varadhan, S.R.S.
title: Multidimensional Diffusion Processes
year: 1997
publisher: Springer Berlin Heidelberg
url: https://books.google.ca/books?id=DuDsmoyqCy4C
aux: {"isbn":"9783540903536","lccn":"96027093","series":"Grundlehren der mathematischen Wissenschaften"}
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@book{Diffusion,
    title={Multidimensional Diffusion Processes}
    author={Stroock, D.W. and Varadhan, S.R.S.}
    isbn={9783540903536}
    lccn={96027093}
    series={Grundlehren der mathematischen Wissenschaften}
    url={https://books.google.ca/books?id=DuDsmoyqCy4C}
    year={1997}
    publisher={Springer Berlin Heidelberg}
}


key: Baudoin
type: book
author: Baudoin, F.
title: Diffusion Processes and Stochastic Calculus
year: 2014
publisher: European Mathematical Society
url: https://books.google.ca/books?id=ov4kcKkzTP4C
aux: {"isbn":"9783037191330","lccn":"2014395958","series":"EMS textbooks in mathematics"}
bibtex:
@book{Baudoin,
    title={Diffusion Processes and Stochastic Calculus}
    author={Baudoin, F.}
    isbn={9783037191330}
    lccn={2014395958}
    series={EMS textbooks in mathematics}
    url={https://books.google.ca/books?id=ov4kcKkzTP4C}
    year={2014}
    publisher={European Mathematical Society}
}


key: Lang
type: book
author: Lang, Serge
title: Differential and Riemannian manifolds
year: 2012
publisher: Springer Science \& Business Media
aux: {}
bibtex:
@book{Lang,
    title={Differential and Riemannian manifolds}
    author={Lang, Serge}
    year={2012}
    publisher={Springer Science \& Business Media}
}


key: Rogers
type: article
author: Rogers, LCG and Williams, David
title: Book review: Diffusions, markov processes. and martingales, volume 2, it6 calculus
year: 1989
publisher: Taylor \& Francis
aux:
{"journal":"Stochastics: An International Journal of Probability and Stochastic Processes","volume":"27","number":"1","pages":"59--63"}

bibtex:
@article{Rogers,
    title={Book review: Diffusions, markov processes. and martingales, volume 2, it6 calculus}
    author={Rogers, LCG and Williams, David}
    journal={Stochastics: An International Journal of Probability and Stochastic Processes}
    volume={27}
    number={1}
    pages={59--63}
    year={1989}
    publisher={Taylor \& Francis}
}


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<div class="block-container"><p><a class="link-foreground" href="/t/5"><strong>Theorem 1.1.1.</strong></a><span> </span> Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space and $\bracs{B_t|t \ge 0}$ be a $\real^{d}$-valued, $\bracs{\cf_t}$-progressively measurable process such that $B_{0} = 0$ almost surely, then the following are equivalent:</p><ol class="enumerate"><li style="list-style-type: &#x27;(1) &#x27;"><p>$\bracs{B_t|t \ge 0}$ is a Brownian motion.</p></li><li style="list-style-type: &#x27;(2) &#x27;"><p>For each $\xi \in \real^{d}$, the process</p><div class="">\[X^{\theta}_{t} = \exp\braks{i \dpn{B_t, \xi}{\real^d} + \frac{1}{2}\norm{\xi}_{\real^d}^2t}\]</div><p>is a $\bracs{\mathcal{F}_t}$-martingale.</p></li><li style="list-style-type: &#x27;(3) &#x27;"><p>For each $\phi \in BC^{2}(\real^{d})$, the process</p><div class="">\[Y^{\phi}_{t} = \phi(B_{t}) - \frac{1}{2}\int_{0}^{t} \Delta f(B_{s})ds\]</div><p>is a $\bracs{\mathcal{F}_t}$-martingale.</p></li></ol></div><div class="proof"><p><i class="">Proof <span class="">[<span class="">Theorem 4.1.1</span>, <a class="link-primary" href="/b/1">SV97</a>]</span>. </i>(1) $\Rightarrow$ (2): For each $1 \le s \le t$ and $\xi \in \real^{d}$, </p><div class="" id="">\begin{align*}\ev\braks{e^{i \dpn{B_t, \xi}{\real^d}} \big| \cf_s}&#x26;= \ev\braks{e^{i \dpn{B_s, \xi}{\real^d}}e^{i \dpn{B_t - B_s, \xi}{\real^d}} \big| \cf_s}\\&#x26;= e^{i \dpn{B_s, \xi}{\real^d}}e^{-(t - s)\norm{\xi}_{\real^d}^2/2}\end{align*}</div><p>Therefore </p><div class="">\[\ev\braks{e^{i \dpn{B_t, \xi}{\real^d} + t\norm{\xi}^2/2} \big| \cf_s}= e^{i \dpn{B_s, \xi}{\real^d} + s\norm{\xi}_{\real^d}^2/2}\]</div><p>(2) $\Rightarrow$ (3): For each $0 \le s &#x3C; t$ and $\xi \in \real^{d}$, </p><div class="" id="">\begin{align*}\frac{d}{dt}\ev\braks{e^{i \dpn{B_t, \xi}{\real^d}} \big| \cf_s}&#x26;= -\frac{\norm{\xi}_{\real^d}^{2}}{2}e^{i \dpn{B_s, \xi}{\real^d}}e^{-(t - s)\norm{\xi}_{\real^d}^2/2}\end{align*}</div><p>Therefore for any $0 \le s &#x3C; t$, </p><div class="" id="">\begin{align*}\ev\braks{e^{i \dpn{B_t, \xi}{\real^d}}-e^{i \dpn{B_s, \xi}{\real^d}} \big| \cf_s}&#x26;= -\frac{\norm{\xi}_{\real^d}^{2}}{2}\int_{s}^{t} e^{i \dpn{B_s, \xi}{\real^d}}e^{-(r - s)\norm{\xi}_{\real^d}^2/2}dr \\&#x26;= -\frac{\norm{\xi}_{\real^d}^{2}}{2}\int_{s}^{t} \ev\braks{e^{i \dpn{B_r, \xi}{\real^d}} \big | \cf_{s}}dr\end{align*}</div><p>If $\phi_{\xi}(x) = e^{-i \dpn{x, \xi}{\real^d}}$, then </p><div class="">\[\ev\braks{e^{i \dpn{B_t, \xi}{\real^d}}-e^{i \dpn{B_s, \xi}{\real^d}} \big| \cf_s}= \ev\braks{\int_s^t \frac{1}{2}(\Delta \phi_\xi)(B_r)dd \bigg | \cf_s}\]</div><p>so the process </p><div class="">\[Z^{\xi}_{t} = e^{i \dpn{B_t, \xi}{\real^d}}+ \int_{0}^{t} \frac{1}{2}(\Delta \phi_{\xi})(B_{r})dr\]</div><p>is a $\bracs{\mathcal{F}_t}$-martingale.</p><p>Let $\phi \in \mathcal{S}(\real^{d})$, then by the Fourier Inversion Theorem, there exists $\psi \in \mathcal{S}(\real^{d})$ such that for any $x \in \real^{d}$, </p><div class="">\[\phi(x) = \int e^{i \dpn{\xi, x}{\real^d}}\psi(\xi)d\xi\]</div><p>In which case, for any $t \ge 0$, </p><div class="">\[\phi(B_{t}) + \int_{0}^{t} \frac{1}{2}(\Delta \phi)(B_{r})dr = \int_{\real^d}e^{i \dpn{\xi, B_t}{\real^d}}\psi(\xi)d\xi + \int_{\real^d}\int_{0}^{t} \frac{1}{2}(\Delta \phi_{\xi})(B_{r}) \psi(\xi) dr d\xi\]</div><p>By the Dominated Convergence Theorem, the above also holds for any $\phi \in BC^{2}(\real^{d})$. Therefore the process $\bracsn{Y^\phi_t|t \ge 0}$ is a $\bracs{\mathcal{F}_t}$-martingale as well.<span class="qed">$\square$</span></p></div>

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Brownian Motion Let ( , _t|t 0 , ) be a filtered probability space and B_t|t 0 be a d - valued , _t - progressively measurable process such that B 0 = 0 almost surely , then the following are equivalent : (1) B_t|t 0 is a Brownian motion . (2) For each d , the process X t = i B_t , ^d + 1 2 _ ^d ^2t is a F _t - martingale . (3) For each B C 2 ( d ) , the process Y t = ( B t ) - 1 2 0 t f ( B s ) d s is a F _t - martingale . ( 1 ) ( 2 ) : For each 1 s t and d , e^ i B_t , ^d | _s & = e^ i B_s , ^d e^ i B_t - B_s , ^d | _s & = e i B_s , ^d e - ( t - s ) _ ^d ^2 / 2 Therefore e^ i B_t , ^d + t ^2 / 2 | _s = e i B_s , ^d + s _ ^d ^2 / 2 ( 2 ) ( 3 ) : For each 0 s < t and d , d d t e^ i B_t , ^d | _s & = - ^d 2 2 e i B_s , ^d e - ( t - s ) _ ^d ^2 / 2 Therefore for any 0 s < t , e^ i B_t , ^d - e^ i B_s , ^d | _s & = - ^d 2 2 s t e i B_s , ^d e - ( r - s ) _ ^d ^2 / 2 d r & = - ^d 2 2 s t e^ i B_r , ^d | _ s d r If ( x ) = e - i x , ^d , then e^ i B_t , ^d - e^ i B_s , ^d | _s = _s^t 1 2 ( _ ) ( B_r ) dd | _s so the process Z t = e i B_t , ^d + 0 t 1 2 ( ) ( B r ) d r is a F _t - martingale . Let S ( d ) , then by the Fourier Inversion Theorem , there exists S ( d ) such that for any x d , ( x ) = e i , x ^d ( ) d In which case , for any t 0 , ( B t ) + 0 t 1 2 ( ) ( B r ) d r = ^d e i , B_t ^d ( ) d + ^d 0 t 1 2 ( ) ( B r ) ( ) d r d By the Dominated Convergence Theorem , the above also holds for any B C 2 ( d ) . Therefore the process Y^ _t|t 0 is a F _t - martingale as well . Proof Theorem 4.1 .1 Diffusion .

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<div class="block-container"><p><a class="link-foreground" href="/t/5"><strong>Theorem 1.1.1.</strong></a><span> </span> Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space and $\bracs{B_t|t \ge 0}$ be a $\real^{d}$-valued, $\bracs{\cf_t}$-progressively measurable process such that $B_{0} = 0$ almost surely, then the following are equivalent:</p><ol class="enumerate"><li style="list-style-type: &#x27;(1) &#x27;"><p>$\bracs{B_t|t \ge 0}$ is a Brownian motion.</p></li><li style="list-style-type: &#x27;(2) &#x27;"><p>For each $\xi \in \real^{d}$, the process</p><div class="">\[X^{\theta}_{t} = \exp\braks{i \dpn{B_t, \xi}{\real^d} + \frac{1}{2}\norm{\xi}_{\real^d}^2t}\]</div><p>is a $\bracs{\mathcal{F}_t}$-martingale.</p></li><li style="list-style-type: &#x27;(3) &#x27;"><p>For each $\phi \in BC^{2}(\real^{d})$, the process</p><div class="">\[Y^{\phi}_{t} = \phi(B_{t}) - \frac{1}{2}\int_{0}^{t} \Delta f(B_{s})ds\]</div><p>is a $\bracs{\mathcal{F}_t}$-martingale.</p></li></ol></div><div class="proof"><p><i class="">Proof <span class="">[<span class="">Theorem 4.1.1</span>, <a class="link-primary" href="/b/1">SV97</a>]</span>. </i>(1) $\Rightarrow$ (2): For each $1 \le s \le t$ and $\xi \in \real^{d}$, </p><div class="" id="">\begin{align*}\ev\braks{e^{i \dpn{B_t, \xi}{\real^d}} \big| \cf_s}&#x26;= \ev\braks{e^{i \dpn{B_s, \xi}{\real^d}}e^{i \dpn{B_t - B_s, \xi}{\real^d}} \big| \cf_s}\\&#x26;= e^{i \dpn{B_s, \xi}{\real^d}}e^{-(t - s)\norm{\xi}_{\real^d}^2/2}\end{align*}</div><p>Therefore </p><div class="">\[\ev\braks{e^{i \dpn{B_t, \xi}{\real^d} + t\norm{\xi}^2/2} \big| \cf_s}= e^{i \dpn{B_s, \xi}{\real^d} + s\norm{\xi}_{\real^d}^2/2}\]</div><p>(2) $\Rightarrow$ (3): For each $0 \le s &#x3C; t$ and $\xi \in \real^{d}$, </p><div class="" id="">\begin{align*}\frac{d}{dt}\ev\braks{e^{i \dpn{B_t, \xi}{\real^d}} \big| \cf_s}&#x26;= -\frac{\norm{\xi}_{\real^d}^{2}}{2}e^{i \dpn{B_s, \xi}{\real^d}}e^{-(t - s)\norm{\xi}_{\real^d}^2/2}\end{align*}</div><p>Therefore for any $0 \le s &#x3C; t$, </p><div class="" id="">\begin{align*}\ev\braks{e^{i \dpn{B_t, \xi}{\real^d}}-e^{i \dpn{B_s, \xi}{\real^d}} \big| \cf_s}&#x26;= -\frac{\norm{\xi}_{\real^d}^{2}}{2}\int_{s}^{t} e^{i \dpn{B_s, \xi}{\real^d}}e^{-(r - s)\norm{\xi}_{\real^d}^2/2}dr \\&#x26;= -\frac{\norm{\xi}_{\real^d}^{2}}{2}\int_{s}^{t} \ev\braks{e^{i \dpn{B_r, \xi}{\real^d}} \big | \cf_{s}}dr\end{align*}</div><p>If $\phi_{\xi}(x) = e^{-i \dpn{x, \xi}{\real^d}}$, then </p><div class="">\[\ev\braks{e^{i \dpn{B_t, \xi}{\real^d}}-e^{i \dpn{B_s, \xi}{\real^d}} \big| \cf_s}= \ev\braks{\int_s^t \frac{1}{2}(\Delta \phi_\xi)(B_r)dd \bigg | \cf_s}\]</div><p>so the process </p><div class="">\[Z^{\xi}_{t} = e^{i \dpn{B_t, \xi}{\real^d}}+ \int_{0}^{t} \frac{1}{2}(\Delta \phi_{\xi})(B_{r})dr\]</div><p>is a $\bracs{\mathcal{F}_t}$-martingale.</p><p>Let $\phi \in \mathcal{S}(\real^{d})$, then by the Fourier Inversion Theorem, there exists $\psi \in \mathcal{S}(\real^{d})$ such that for any $x \in \real^{d}$, </p><div class="">\[\phi(x) = \int e^{i \dpn{\xi, x}{\real^d}}\psi(\xi)d\xi\]</div><p>In which case, for any $t \ge 0$, </p><div class="">\[\phi(B_{t}) + \int_{0}^{t} \frac{1}{2}(\Delta \phi)(B_{r})dr = \int_{\real^d}e^{i \dpn{\xi, B_t}{\real^d}}\psi(\xi)d\xi + \int_{\real^d}\int_{0}^{t} \frac{1}{2}(\Delta \phi_{\xi})(B_{r}) \psi(\xi) dr d\xi\]</div><p>By the Dominated Convergence Theorem, the above also holds for any $\phi \in BC^{2}(\real^{d})$. Therefore the process $\bracsn{Y^\phi_t|t \ge 0}$ is a $\bracs{\mathcal{F}_t}$-martingale as well.<span class="qed">$\square$</span></p></div>

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Let ( , _t|t 0 , ) be a filtered probability space and B_t|t 0 be a d - valued , _t - progressively measurable process such that B 0 = 0 almost surely , then the following are equivalent : (1) B_t|t 0 is a Brownian motion . (2) For each d , the process X t = i B_t , ^d + 1 2 _ ^d ^2t is a F _t - martingale . (3) For each B C 2 ( d ) , the process Y t = ( B t ) - 1 2 0 t f ( B s ) d s is a F _t - martingale .

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titleText: Equivalence of Certain Martingales
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<div class="block-container"><p><a class="link-foreground" href="/t/7"><strong>Definition 1.2.1 (Random Differential Operator).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;definition:random-differential-operator&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $(\Omega, \cf, \bp)$ be a filtered probability space, $s \ge 0$, and $k \in \natp$. For each multi-index $\alpha \in \nat_{0}^{d}$ with $\abs{\alpha}\le k$, let $a_{\alpha}: [s, \infty) \times \Omega \to \real$ be measurable function. For each $f \in C^{k}(\real^{d})$ and $t \ge s$, let </p><div class="display-equation">\[[L_{t}(\omega)f](x) = \sum_{\abs{\alpha} \le k}a_{\alpha}(t, \omega) \partial^{\alpha} f(x)\]</div><p>then $L = \sum_{|\alpha| \le k}a_{\alpha} \partial^{\alpha}$ is a <b class="textbf">random differential operator</b> of order $k$ with coefficients $\bracs{a_\alpha}_{|\alpha| \le k}$.</p></div><div class="block-container"><p><a class="link-foreground" href="/t/8"><strong>Definition 1.2.2 (Progressively Measurable Random Differential Operator).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;definition:random-do-progressively-measurable&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $(\Omega, \cf, \bp)$ be a filtered probability space and $L = \sum_{|\alpha| \le k}a_{\alpha}$ be a random differential operator on $\Omega$, then $L$ is <b class="textbf">progressively measurable</b> with respect to $\bracs{\mathcal{F}_t}$ if for each $\alpha \in \nat_{0}$ with $|\alpha| \le k$, $a_{\alpha}: [s, \infty) \times \Omega \to \real$ is progressively measurable with respect to $\bracs{\mathcal{F}_t}$.</p></div><div class="block-container"><p><a class="link-foreground" href="/t/9"><strong>Lemma 1.2.3.</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;lemma:random-diff-process&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, $s \ge 0$, $k \in \natp$, $L = \sum_{|\alpha| \le k}a_{\alpha} \partial^{\alpha}$ be a $\bracs{\mathcal{F}_t}$-progressively measurable random differential operator, $f \in C^{k}(\real^{d})$, and $\bracs{X_t|t \ge 0}$ be a $\real^{d}$-valued progressively measurable process, then </p><div class="display-equation">\[Y_{t}(\omega) = (L_{t}(\omega)f)(X_{t}(\omega))\]</div><p>is a progressively measurable process.</p></div><div class="block-container"><p><a class="link-foreground" href="/t/D"><strong>Proposition 1.2.4.</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;proposition:martingale-quadratic&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, </p><div class="display-equation">\[L_{t}(\omega)u = \frac{1}{2}\dpn{A_t(\omega), u}{\real^{d \times d}}+ \dpn{B_t(\omega), u}{\real^d}\]</div><p>be a second-order $\bracs{\mathcal{F}_t}$-progressively measurable random differential operator, $\bracs{X_t|t \ge 0}$ be a $\bracs{\mathcal{F}_t}$-progressively measurable process with continuous sample paths, and $f \in C_{b}^{1, 2}([0, \infty) \times \real^{d})$. If the processes </p><div class="display-equation">\[Y_{t} = f(t, X_{t}) - \int_{0}^{t} [(\partial_{r} + L_{r})f](r, X_{r})dr\]</div><p>and </p><div class="display-equation">\[W_{t} = f(t, X_{t})^{2} - \int_{0}^{t} [(\partial_{r} + L_{r})(f^{2})](r, X_{r})dr\]</div><p>are martingales, then </p><div class="display-equation">\[[X]_{t} = \frac{1}{2}\int_{0}^{t} \dpn{Df, A(r)(Df)}{\real^d}(r, X_{r})dr\]</div></div><div class="proof"><p><i class="">Proof. </i>Firstly, since $f$ admits bounded derivatives, the integral term is of locally bounded variation. Therefore if $Z_{t} = f(t, X_{t})$, then $[X]_{t} = \angles{Z}_{t}$.</p><p>Let $\bracs{t_j}_{0}^{n} \subset [0, t]$ be a partition, then </p><div class="display-equation">\[Z_{t}^{2} - Z_{0}^{2} = \sum_{j = 1}^{n}Z_{t_j}^{2} - Z_{t_{j-1}}^{2} = \sum_{j = 1}^{n} 2Z_{t_{j-1}}(Z_{t_j}- Z_{t_{j-1}}) + \sum_{j = 1}^{n} (Z_{t_j}- Z_{t_{j-1}})^{2}\]</div><p>As the above holds for every partition, </p><div class="display-equation">\[Z_{t}^{2} - Z_{0}^{2} = 2\int_{0}^{t} Z_{r}Z(dr) + \angles{Z}_{t}\]</div><p>Since $Z_{t} = Y_{t} + \int_{0}^{t}(\partial_{r} + L_{r})fdr$, </p><div class="display-equation" id="">\begin{align*}\int_{0}^{t} Z_{r}Z(dr)&#x26;= \int_{0}^{t} Z_{r} Y(dr) + \int_{0}^{t} f(r, X_{r})(\partial_{r} + L_{r})f(r, X_{r})dr \\ Z_{t}^{2} - Z_{0}^{2}&#x26;= 2\int_{0}^{t} Z_{r} Y(dr) + 2\int_{0}^{t} f(r, X_{r})(\partial_{r} + L_{r})f(r, X_{r})dr + \angles{Z}_{t}\end{align*}</div><p>On the other hand, </p><div class="display-equation">\[Z_{t}^{2} - Z_{0}^{2} = W_{t} - W_{0} + \int_{0}^{t} [(\partial_{r} + L_{r})(f^{2})](r, X_{r})dr\]</div><p>Therefore </p><div class="display-equation" id="">\begin{align*}\angles{Z}_{t}&#x26;= W_{t} - W_{0} + \int_{0}^{t} [(\partial_{r} + L_{r})(f^{2})](r, X_{r})dr \\&#x26;- 2\int_{0}^{t} Z_{r} Y(dr) - 2\int_{0}^{t} f(r, X_{r})(\partial_{r} + L_{r})f(r, X_{r})dr \\&#x26;= W_{t} - W_{0} - 2 \int_{0}^{t} f(r, X_{r}) Y(dr) \\&#x26;+ \int_{0}^{t} [L_{r}(f^{2})(r, X_{r}) - 2f(r, X_{r})(L_{r}f)(r, X_{r})]dr \\&#x26;= W_{t} - W_{0} - 2\int_{0}^{t} f(r, X_{r})Y(dr) + \frac{1}{2}\int_{0}^{t} \dpn{Df, A(r)(Df)}{\real^d}(r, X_{r})dr\end{align*}</div><p>where the process $t \mapsto W_{t} - W_{0} - 2\int_{0}^{t} f(r, X_{r})Y(dr)$ is of finite variation, so it must be $0$ by <span class="">[<span class="">Lemma 5.13</span>, <a class="link-primary" href="/b/E">Bau14</a>]</span>.<span class="qed">$\square$</span></p></div><div class="block-container"><p><a class="link-foreground" href="/t/C"><strong>Theorem 1.2.5 (Integration by Parts).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;theorem:martingale-ibp&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, $\bracs{X_t}$ be a $\bracs{\mathcal{F}_t}$-martingale, and $\phi: [0, \infty) \times \Omega \to \complex$ be a continuous, progressively measurable function. If:</p><ol class="enumerate"><li style="list-style-type: &#x27;(1) &#x27;"><p>For every $\omega \in \Omega$, $\phi(\cdot, \omega) \in BV_{\text{loc}}([0, \infty))$.</p></li><li style="list-style-type: &#x27;(2) &#x27;"><p>For all $t \ge 0$,</p><div class="display-equation">\[\ev\braks{\sup_{0 \le s \le t}|X_s|(|\phi|(s) + |\phi|(t))}&#x3C; \infty\]</div></li></ol><p>then the process </p><div class="display-equation">\[Y_{t} = X_{t} \phi(t) - \int_{0}^{t} X_{r} \phi(dr)\]</div><p>is a $\bracs{\mathcal{F}_t}$-martingale.</p></div><div class="block-container"><p><a class="link-foreground" href="/t/A"><strong>Theorem 1.2.6 (Equivalence of Martingales).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;theorem:equivalence-of-martingales&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, </p><div class="display-equation">\[L_{t}(\omega)u = \frac{1}{2}\dpn{A_t(\omega), u}{\real^{d \times d}}+ \dpn{B_t(\omega), u}{\real^d}\]</div><p>be a second-order $\bracs{\mathcal{F}_t}$-progressively measurable random differential operator, and $\bracs{X_t|t \ge 0}$ be a $\bracs{\mathcal{F}_t}$-progressively measurable process with continuous sample paths, then the following are equivalent:</p><ol class="enumerate"><li style="list-style-type: &#x27;(1) &#x27;"><p>For every $f \in C_{c}^{\infty}(\real^{d})$, the process</p><div class="display-equation">\[E_{t} = f(X_{t}) - \int_{0}^{t} (L_{r}f)(X_{r})dr\]</div><p>is a $\bracs{\mathcal{F}_t}$-martingale.</p></li><li style="list-style-type: &#x27;(2) &#x27;"><p>For every $f \in C_{b}^{1, 2}([0, \infty) \times \real^{d})$,</p><div class="display-equation">\[H_{t} = f(t, X_{t}) - \int_{0}^{t} [(\partial_{r} + L_{r})f](r, X_{r})dr\]</div><p>is a $\bracs{\mathcal{F}_t}$-martingale.</p></li><li style="list-style-type: &#x27;(3) &#x27;"><p>For each uniformly positive $f \in C_{b}^{1, 2}([0, \infty) \times \real^{d})$,</p><div class="display-equation">\[X_{t} = f(t, X_{t})\exp\braks{-\int_0^t \frac{(\partial_{r} + L_{r})f}{f}(r, X_r)dr}\]</div><p>is a $\bracs{\mathcal{F}_t}$-martingale.</p></li><li style="list-style-type: &#x27;(4) &#x27;"><p>For each $x \in \real^{d}$ and $g \in C_{b}^{1, 2}([0, \infty) \times \real^{d})$, the process</p><div class="display-equation" id="">\begin{align*}Y_{t}^{x, g}&#x26;= \exp\braks{\dpn{x, X_t - X_0 - \int_0^t b(r)dr}{\real^d} + g(t, X_t)}\\&#x26;\cdot \exp\braks{-\frac{1}{2}\int_0^t \dpn{x + Dg, A(r)(x + Dg)}{\real^d}(r, X_r)dr}\\&#x26;\cdot \exp\braks{-\int_0^t [(\partial_r + L_r)g](r, X_r)}dr\end{align*}</div><p>is a $\bracs{\mathcal{F}_t}$-martingale.</p></li><li style="list-style-type: &#x27;(5) &#x27;"><p>For each $\theta \in \real^{d}$,</p></li></ol></div><div class="proof"><p><i class="">Proof <span class="">[<span class="">Theorem 4.2.1</span>, <a class="link-primary" href="/b/1">SV97</a>]</span>. </i>(1) $\Rightarrow$ (2): Let $0 \le s \le t$ and $f \in C_{c}^{\infty}([0, \infty) \times \real^{d})$, then by the Fundamental Theorem of Calculus, </p><div class="display-equation" id="">\begin{align*}f(t, X_{t}) - f(s, X_{s})&#x26;= f(t, X_{t}) - f(s, X_{t}) + f(s, X_{t}) - f(s, X_{s}) \\&#x26;= \int_{s}^{t} \partial_{r}f(r, X_{t})dr + f(s, X_{t}) - f(s, X_{s})\end{align*}</div><p>where by (1), </p><div class="display-equation">\[\ev\braks{f(s, X_t) - f(s, X_s) |\cf_s}= \ev\braks{\int_s^t (L_rf)(s, X_r)dr \bigg | \cf_s}\]</div><p>By the Fundamental Theorem of Calculus, </p><div class="display-equation" id="">\begin{align*}\int_{s}^{t} (\partial_{r}f)(r, X_{t})dr&#x26;= \int_{s}^{t} (\partial_{r} f)(r, X_{r})dr + \int_{s}^{t} (\partial_{r} f)(r, X_{t}) - (\partial_{r} f)(r, X_{r})dr \\&#x26;= \int_{s}^{t} (\partial_{r} f)(r, X_{r})dr + \int_{s}^{t} (\partial_{r} f)(r, X_{t}) - (\partial_{r} f)(r, X_{r})dr \\\end{align*}</div><p>By (1) applied to $(\partial_{r} f)$, </p><div class="display-equation">\[\ev\braks{\int_s^t (\partial_rf)(r, X_t)dr \bigg | \cf_s}= \braks{\int_s^t (\partial_r f)(r, X_r)dr + \int_s^t \int_r^t (\partial_r f)(r, X_u)du dr \bigg | \cf_s}\]</div><p>Similarly, by the Fundamental Theorem of Calculus, </p><div class="display-equation" id="">\begin{align*}\int_{s}^{t} (L_{r}f)(s, X_{r})dr&#x26;= \int_{s}^{t} (L_{r}f)(r, X_{r})dr + \int_{s}^{t} (L_{r}f)(s, X_{r}) - (L_{r}f)(r, X_{r})dr \\&#x26;= \int_{s}^{t}(L_{r}f)(r, X_{r})dr - \int_{s}^{t} \int_{s}^{r} (\partial_{r}L_{r}f)(u, X_{r})dudr\end{align*}</div><p>Since the two iterated integrals are over the same region, </p><div class="display-equation">\[\ev\braks{f(t, X_t) - f(s, X_s) | \cf_s}= \ev\braks{\int_s^t (\partial_r + L_rf)(r, X_r)dr \bigg | \cf_s}\]</div><p>Therefore $\bracs{E_t}$ is a $\bracs{\mathcal{F}_t}$-martingale.</p><p>(2) $\Rightarrow$ (3): Let $f \in C_{b}^{1, 2}([0, \infty) \times \real^{d})$ be uniformly positive. Define </p><div class="display-equation">\[Z_{t} = f(t, X_{t}) - \int_{0}^{t} (\partial_{r}f + L_{r}f)(r, X_{r})dr\]</div><p>and </p><div class="display-equation">\[\phi_{t} = \exp\braks{-\int_0^t \frac{\partial_{r}f + L_{r}f}{f}(r, X_r)dr}\]</div><p>then using the by parts formula for Riemann-Stieltjes integrals, </p><div class="display-equation" id="">\begin{align*}Z_{t}\phi_{t} - \int_{0}^{t} Z_{s} \phi(ds)&#x26;= Z_{0}\phi_{0} + \int_{0}^{t} \phi_{s} Z(ds) \\&#x26;= Z_{0}\phi_{0} + \int_{0}^{t} \phi_{s} f(ds, X_{ds}) + \int_{0}^{t} \phi_{s} \cdot (\partial_{s} f L_{s}f)(s, X_{s})ds\end{align*}</div><p>Since </p><div class="display-equation">\[\partial_{s} \phi_{s} = -\phi_{s} \frac{\partial_{s} f + L_{s}f}{f}(s, X_{s})\]</div><p>Using the by parts formula again,</p><div class="display-equation" id="">\begin{align*}Z_{t}\phi_{t} - \int_{0}^{t} Z_{s} \phi(ds)&#x26;= Z_{0}\phi_{0} + \int_{0}^{t} \phi_{s} f(ds, X_{ds}) + \int_{0}^{t} f(s, X_{s})\phi(ds) \\&#x26;= Z_{0}\phi_{0} + f(t, X_{t})\phi_{t} - f(0, X_{0})\phi_{0} = f(t, X_{t})\phi_{t} \\&#x26;= f(t, X_{t})\exp\braks{-\int_0^t \frac{\partial_{r}f + L_{r}f}{f}(r, X_r)dr}\end{align*}</div><p>Therefore <a href="/t/C" class="link-primary">Theorem 1.2.5</a> implies that the above process is a $\bracs{\mathcal{F}_t}$-martingale.</p><p>(3) $\Rightarrow$ (4): Let $x \in \real^{d}$ and $g \in C_{b}^{1, 2}([0, \infty) \times \real^{d})$. Define </p><div class="display-equation">\[f: [0, \infty) \times \real^{d} \to \real \quad (t, y) \mapsto \exp(\dpn{x, y}{\real^d}+ g(t, x))\]</div><p>then (3) applied to $f$ yields that $\bracs{Y_t|t \ge 0}$ is a $\bracs{\mathcal{F}_t}$-martingale. However, since $f$ is unbounded and not uniformly positive, a direct application is ineffective.</p><p>Let $\seq{f_n}\subset C_{b}^{1, 2}([0, \infty) \times \real^{d})$ be uniformly positive such that $f_{n}|_{\bracs{|y| \le n}}= f$. Define </p><div class="display-equation">\[\tau_{n} = \inf\bracs{t \ge 0 \bigg | \sup_{0 \le s \le t}|X_t(u)| \ge n}\wedge n\]</div><p>then $\bracsn{Y_{t \wedge \tau_n}^{x, g}}$ is a $\bracs{\mathcal{F}_t}$-martingale with $Y_{t \wedge \tau_n}^{x, g}\to Y_{t}^{x, g}$ almost surely as $n \to \infty$. In addition, </p><div class="display-equation">\[(Y_{t \wedge \tau_n}^{x, g})^{2} \le Y_{t \wedge \tau_n}^{2x, 2g}\cdot \exp\braks{\int_0^{t \wedge \tau_n} \dpn{x + Dg, A_s(x + Dg) }{\real^d}(s, X_s)ds}\le CY_{t \wedge \tau_n}^{2x, 2g}\]</div><p>where $\ev\braks{Y_{t \wedge \tau_n}^{2x, 2g}}= f(s, X_{s})^{2} \le \exp(2\norm{g}_{u})$. Therefore $\bracsn{Y_{t \wedge \tau_n}^{x, g}}$ is bounded in $L^{2}$, uniformly integrable in $L^{1}$, and converges to $Y_{t}^{x, g}$ in $L^{1}$.<span class="qed">$\square$</span></p></div>

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Equivalence of Certain Martingales Let ( , , ) be a filtered probability space , s 0 , and k . For each multi - index 0 d with k , let a : [ s , ) be measurable function . For each f C k ( d ) and t s , let [ L t ( ) f ] ( x ) = k a ( t , ) f ( x ) then L = | | k a is a random differential operator of order k with coefficients a_ | | k . Random Differential Operator Let ( , , ) be a filtered probability space and L = | | k a be a random differential operator on , then L is progressively measurable with respect to F _t if for each 0 with | | k , a : [ s , ) is progressively measurable with respect to F _t . Progressively Measurable Random Differential Operator Let ( , _t|t 0 , ) be a filtered probability space , s 0 , k , L = | | k a be a F _t - progressively measurable random differential operator , f C k ( d ) , and X_t|t 0 be a d - valued progressively measurable process , then Y t ( ) = ( L t ( ) f ) ( X t ( ) ) is a progressively measurable process . Let ( , _t|t 0 , ) be a filtered probability space , L t ( ) u = 1 2 A_t ( ) , u ^ d d + B_t ( ) , u ^d be a second - order F _t - progressively measurable random differential operator , X_t|t 0 be a F _t - progressively measurable process with continuous sample paths , and f C b 1 , 2 ( [ 0 , ) d ) . If the processes Y t = f ( t , X t ) - 0 t [ ( r + L r ) f ] ( r , X r ) d r and W t = f ( t , X t ) 2 - 0 t [ ( r + L r ) ( f 2 ) ] ( r , X r ) d r are martingales , then [ X ] t = 1 2 0 t Df , A ( r ) ( Df ) ^d ( r , X r ) d r Firstly , since f admits bounded derivatives , the integral term is of locally bounded variation . Therefore if Z t = f ( t , X t ) , then [ X ] t = Z t . Let t_j 0 n [ 0 , t ] be a partition , then Z t 2 - Z 0 2 = j = 1 n Z t_j 2 - Z t_ j - 1 2 = j = 1 n 2 Z t_ j - 1 ( Z t_j - Z t_ j - 1 ) + j = 1 n ( Z t_j - Z t_ j - 1 ) 2 As the above holds for every partition , Z t 2 - Z 0 2 = 2 0 t Z r Z ( d r ) + Z t Since Z t = Y t + 0 t ( r + L r ) f d r , 0 t Z r Z ( d r ) & = 0 t Z r Y ( d r ) + 0 t f ( r , X r ) ( r + L r ) f ( r , X r ) d r Z t 2 - Z 0 2 & = 2 0 t Z r Y ( d r ) + 2 0 t f ( r , X r ) ( r + L r ) f ( r , X r ) d r + Z t On the other hand , Z t 2 - Z 0 2 = W t - W 0 + 0 t [ ( r + L r ) ( f 2 ) ] ( r , X r ) d r Therefore Z t & = W t - W 0 + 0 t [ ( r + L r ) ( f 2 ) ] ( r , X r ) d r & - 2 0 t Z r Y ( d r ) - 2 0 t f ( r , X r ) ( r + L r ) f ( r , X r ) d r & = W t - W 0 - 2 0 t f ( r , X r ) Y ( d r ) & + 0 t [ L r ( f 2 ) ( r , X r ) - 2 f ( r , X r ) ( L r f ) ( r , X r ) ] d r & = W t - W 0 - 2 0 t f ( r , X r ) Y ( d r ) + 1 2 0 t Df , A ( r ) ( Df ) ^d ( r , X r ) d r where the process t W t - W 0 - 2 0 t f ( r , X r ) Y ( d r ) is of finite variation , so it must be 0 by Lemma 5.13 Baudoin . Let ( , _t|t 0 , ) be a filtered probability space , X_t be a F _t - martingale , and : [ 0 , ) be a continuous , progressively measurable function . If : (1) For every , ( , ) B V loc ( [ 0 , ) ) . (2) For all t 0 , _ 0 s t |X_s| ( | | ( s ) + | | ( t ) ) < then the process Y t = X t ( t ) - 0 t X r ( d r ) is a F _t - martingale . Integration by Parts Let ( , _t|t 0 , ) be a filtered probability space , L t ( ) u = 1 2 A_t ( ) , u ^ d d + B_t ( ) , u ^d be a second - order F _t - progressively measurable random differential operator , and X_t|t 0 be a F _t - progressively measurable process with continuous sample paths , then the following are equivalent : (1) For every f C c ( d ) , the process E t = f ( X t ) - 0 t ( L r f ) ( X r ) d r is a F _t - martingale . (2) For every f C b 1 , 2 ( [ 0 , ) d ) , H t = f ( t , X t ) - 0 t [ ( r + L r ) f ] ( r , X r ) d r is a F _t - martingale . (3) For each uniformly positive f C b 1 , 2 ( [ 0 , ) d ) , X t = f ( t , X t ) - _0^t ( r + L r ) f f ( r , X_r ) dr is a F _t - martingale . (4) For each x d and g C b 1 , 2 ( [ 0 , ) d ) , the process Y t x , g & = x , X_t - X_0 - _0^t b ( r ) dr ^d + g ( t , X_t ) & - 1 2 _0^t x + Dg , A ( r ) ( x + Dg ) ^d ( r , X_r ) dr & - _0^t [ ( _r + L_r ) g ] ( r , X_r ) d r is a F _t - martingale . (5) For each d , Equivalence of Martingales ( 1 ) ( 2 ) : Let 0 s t and f C c ( [ 0 , ) d ) , then by the Fundamental Theorem of Calculus , f ( t , X t ) - f ( s , X s ) & = f ( t , X t ) - f ( s , X t ) + f ( s , X t ) - f ( s , X s ) & = s t r f ( r , X t ) d r + f ( s , X t ) - f ( s , X s ) where by ( 1 ) , f ( s , X_t ) - f ( s , X_s ) | _s = _s^t ( L_rf ) ( s , X_r ) dr | _s By the Fundamental Theorem of Calculus , s t ( r f ) ( r , X t ) d r & = s t ( r f ) ( r , X r ) d r + s t ( r f ) ( r , X t ) - ( r f ) ( r , X r ) d r & = s t ( r f ) ( r , X r ) d r + s t ( r f ) ( r , X t ) - ( r f ) ( r , X r ) d r By ( 1 ) applied to ( r f ) , _s^t ( _rf ) ( r , X_t ) dr | _s = _s^t ( _r f ) ( r , X_r ) dr + _s^t _r^t ( _r f ) ( r , X_u ) du dr | _s Similarly , by the Fundamental Theorem of Calculus , s t ( L r f ) ( s , X r ) d r & = s t ( L r f ) ( r , X r ) d r + s t ( L r f ) ( s , X r ) - ( L r f ) ( r , X r ) d r & = s t ( L r f ) ( r , X r ) d r - s t s r ( r L r f ) ( u , X r ) d u d r Since the two iterated integrals are over the same region , f ( t , X_t ) - f ( s , X_s ) | _s = _s^t ( _r + L_rf ) ( r , X_r ) dr | _s Therefore E_t is a F _t - martingale . ( 2 ) ( 3 ) : Let f C b 1 , 2 ( [ 0 , ) d ) be uniformly positive . Define Z t = f ( t , X t ) - 0 t ( r f + L r f ) ( r , X r ) d r and t = - _0^t r f + L r f f ( r , X_r ) dr then using the by parts formula for Riemann - Stieltjes integrals , Z t t - 0 t Z s ( d s ) & = Z 0 0 + 0 t s Z ( d s ) & = Z 0 0 + 0 t s f ( d s , X ds ) + 0 t s ( s f L s f ) ( s , X s ) d s Since s s = - s s f + L s f f ( s , X s ) Using the by parts formula again , Z t t - 0 t Z s ( d s ) & = Z 0 0 + 0 t s f ( d s , X ds ) + 0 t f ( s , X s ) ( d s ) & = Z 0 0 + f ( t , X t ) t - f ( 0 , X 0 ) 0 = f ( t , X t ) t & = f ( t , X t ) - _0^t r f + L r f f ( r , X_r ) dr Therefore theorem : martingale - ibp implies that the above process is a F _t - martingale . ( 3 ) ( 4 ) : Let x d and g C b 1 , 2 ( [ 0 , ) d ) . Define f : [ 0 , ) d ( t , y ) ( x , y ^d + g ( t , x ) ) then ( 3 ) applied to f yields that Y_t|t 0 is a F _t - martingale . However , since f is unbounded and not uniformly positive , a direct application is ineffective . Let f_n C b 1 , 2 ( [ 0 , ) d ) be uniformly positive such that f n | |y| n = f . Define n = t 0 | _ 0 s t |X_t ( u ) | n n then Y_ t _n ^ x , g is a F _t - martingale with Y t _n x , g Y t x , g almost surely as n . In addition , ( Y t _n x , g ) 2 Y t _n 2x , 2g _0^ t _n x + Dg , A_s ( x + Dg ) ^d ( s , X_s ) ds C Y t _n 2x , 2g where Y_ t _n ^ 2x , 2g = f ( s , X s ) 2 ( 2 g u ) . Therefore Y_ t _n ^ x , g is bounded in L 2 , uniformly integrable in L 1 , and converges to Y t x , g in L 1 . Proof Theorem 4.2 .1 Diffusion .

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hash: b6bd36f490fb78f5ae65a09ca478aeaceecc79cdeefc8e949080a90c4aba6668
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<div class="block-container"><p><a class="link-foreground" href="/t/7"><strong>Definition 1.2.1 (Random Differential Operator).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;definition:random-differential-operator&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $(\Omega, \cf, \bp)$ be a filtered probability space, $s \ge 0$, and $k \in \natp$. For each multi-index $\alpha \in \nat_{0}^{d}$ with $\abs{\alpha}\le k$, let $a_{\alpha}: [s, \infty) \times \Omega \to \real$ be measurable function. For each $f \in C^{k}(\real^{d})$ and $t \ge s$, let </p><div class="display-equation">\[[L_{t}(\omega)f](x) = \sum_{\abs{\alpha} \le k}a_{\alpha}(t, \omega) \partial^{\alpha} f(x)\]</div><p>then $L = \sum_{|\alpha| \le k}a_{\alpha} \partial^{\alpha}$ is a <b class="textbf">random differential operator</b> of order $k$ with coefficients $\bracs{a_\alpha}_{|\alpha| \le k}$.</p></div>

contentText:
Random Differential Operator Let ( , , ) be a filtered probability space , s 0 , and k . For each multi - index 0 d with k , let a : [ s , ) be measurable function . For each f C k ( d ) and t s , let [ L t ( ) f ] ( x ) = k a ( t , ) f ( x ) then L = | | k a is a random differential operator of order k with coefficients a_ | | k . Random Differential Operator

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titleText: Progressively Measurable Random Differential Operator
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<div class="block-container"><p><a class="link-foreground" href="/t/8"><strong>Definition 1.2.2 (Progressively Measurable Random Differential Operator).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;definition:random-do-progressively-measurable&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $(\Omega, \cf, \bp)$ be a filtered probability space and $L = \sum_{|\alpha| \le k}a_{\alpha}$ be a random differential operator on $\Omega$, then $L$ is <b class="textbf">progressively measurable</b> with respect to $\bracs{\mathcal{F}_t}$ if for each $\alpha \in \nat_{0}$ with $|\alpha| \le k$, $a_{\alpha}: [s, \infty) \times \Omega \to \real$ is progressively measurable with respect to $\bracs{\mathcal{F}_t}$.</p></div>

contentText:
Progressively Measurable Random Differential Operator Let ( , , ) be a filtered probability space and L = | | k a be a random differential operator on , then L is progressively measurable with respect to F _t if for each 0 with | | k , a : [ s , ) is progressively measurable with respect to F _t . Progressively Measurable Random Differential Operator

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<div class="block-container"><p><a class="link-foreground" href="/t/9"><strong>Lemma 1.2.3.</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;lemma:random-diff-process&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, $s \ge 0$, $k \in \natp$, $L = \sum_{|\alpha| \le k}a_{\alpha} \partial^{\alpha}$ be a $\bracs{\mathcal{F}_t}$-progressively measurable random differential operator, $f \in C^{k}(\real^{d})$, and $\bracs{X_t|t \ge 0}$ be a $\real^{d}$-valued progressively measurable process, then </p><div class="display-equation">\[Y_{t}(\omega) = (L_{t}(\omega)f)(X_{t}(\omega))\]</div><p>is a progressively measurable process.</p></div>

contentText:
Let ( , _t|t 0 , ) be a filtered probability space , s 0 , k , L = | | k a be a F _t - progressively measurable random differential operator , f C k ( d ) , and X_t|t 0 be a d - valued progressively measurable process , then Y t ( ) = ( L t ( ) f ) ( X t ( ) ) is a progressively measurable process .

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label: theorem:equivalence-of-martingales
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<div class="block-container"><p><a class="link-foreground" href="/t/A"><strong>Theorem 1.2.6 (Equivalence of Martingales).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;theorem:equivalence-of-martingales&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, </p><div class="display-equation">\[L_{t}(\omega)u = \frac{1}{2}\dpn{A_t(\omega), u}{\real^{d \times d}}+ \dpn{B_t(\omega), u}{\real^d}\]</div><p>be a second-order $\bracs{\mathcal{F}_t}$-progressively measurable random differential operator, and $\bracs{X_t|t \ge 0}$ be a $\bracs{\mathcal{F}_t}$-progressively measurable process with continuous sample paths, then the following are equivalent:</p><ol class="enumerate"><li style="list-style-type: &#x27;(1) &#x27;"><p>For every $f \in C_{c}^{\infty}(\real^{d})$, the process</p><div class="display-equation">\[E_{t} = f(X_{t}) - \int_{0}^{t} (L_{r}f)(X_{r})dr\]</div><p>is a $\bracs{\mathcal{F}_t}$-martingale.</p></li><li style="list-style-type: &#x27;(2) &#x27;"><p>For every $f \in C_{b}^{1, 2}([0, \infty) \times \real^{d})$,</p><div class="display-equation">\[H_{t} = f(t, X_{t}) - \int_{0}^{t} [(\partial_{r} + L_{r})f](r, X_{r})dr\]</div><p>is a $\bracs{\mathcal{F}_t}$-martingale.</p></li><li style="list-style-type: &#x27;(3) &#x27;"><p>For each uniformly positive $f \in C_{b}^{1, 2}([0, \infty) \times \real^{d})$,</p><div class="display-equation">\[X_{t} = f(t, X_{t})\exp\braks{-\int_0^t \frac{(\partial_{r} + L_{r})f}{f}(r, X_r)dr}\]</div><p>is a $\bracs{\mathcal{F}_t}$-martingale.</p></li><li style="list-style-type: &#x27;(4) &#x27;"><p>For each $x \in \real^{d}$ and $g \in C_{b}^{1, 2}([0, \infty) \times \real^{d})$, the process</p><div class="display-equation" id="">\begin{align*}Y_{t}^{x, g}&#x26;= \exp\braks{\dpn{x, X_t - X_0 - \int_0^t b(r)dr}{\real^d} + g(t, X_t)}\\&#x26;\cdot \exp\braks{-\frac{1}{2}\int_0^t \dpn{x + Dg, A(r)(x + Dg)}{\real^d}(r, X_r)dr}\\&#x26;\cdot \exp\braks{-\int_0^t [(\partial_r + L_r)g](r, X_r)}dr\end{align*}</div><p>is a $\bracs{\mathcal{F}_t}$-martingale.</p></li><li style="list-style-type: &#x27;(5) &#x27;"><p>For each $\theta \in \real^{d}$,</p></li></ol></div><div class="proof"><p><i class="">Proof <span class="">[<span class="">Theorem 4.2.1</span>, <a class="link-primary" href="/b/1">SV97</a>]</span>. </i>(1) $\Rightarrow$ (2): Let $0 \le s \le t$ and $f \in C_{c}^{\infty}([0, \infty) \times \real^{d})$, then by the Fundamental Theorem of Calculus, </p><div class="display-equation" id="">\begin{align*}f(t, X_{t}) - f(s, X_{s})&#x26;= f(t, X_{t}) - f(s, X_{t}) + f(s, X_{t}) - f(s, X_{s}) \\&#x26;= \int_{s}^{t} \partial_{r}f(r, X_{t})dr + f(s, X_{t}) - f(s, X_{s})\end{align*}</div><p>where by (1), </p><div class="display-equation">\[\ev\braks{f(s, X_t) - f(s, X_s) |\cf_s}= \ev\braks{\int_s^t (L_rf)(s, X_r)dr \bigg | \cf_s}\]</div><p>By the Fundamental Theorem of Calculus, </p><div class="display-equation" id="">\begin{align*}\int_{s}^{t} (\partial_{r}f)(r, X_{t})dr&#x26;= \int_{s}^{t} (\partial_{r} f)(r, X_{r})dr + \int_{s}^{t} (\partial_{r} f)(r, X_{t}) - (\partial_{r} f)(r, X_{r})dr \\&#x26;= \int_{s}^{t} (\partial_{r} f)(r, X_{r})dr + \int_{s}^{t} (\partial_{r} f)(r, X_{t}) - (\partial_{r} f)(r, X_{r})dr \\\end{align*}</div><p>By (1) applied to $(\partial_{r} f)$, </p><div class="display-equation">\[\ev\braks{\int_s^t (\partial_rf)(r, X_t)dr \bigg | \cf_s}= \braks{\int_s^t (\partial_r f)(r, X_r)dr + \int_s^t \int_r^t (\partial_r f)(r, X_u)du dr \bigg | \cf_s}\]</div><p>Similarly, by the Fundamental Theorem of Calculus, </p><div class="display-equation" id="">\begin{align*}\int_{s}^{t} (L_{r}f)(s, X_{r})dr&#x26;= \int_{s}^{t} (L_{r}f)(r, X_{r})dr + \int_{s}^{t} (L_{r}f)(s, X_{r}) - (L_{r}f)(r, X_{r})dr \\&#x26;= \int_{s}^{t}(L_{r}f)(r, X_{r})dr - \int_{s}^{t} \int_{s}^{r} (\partial_{r}L_{r}f)(u, X_{r})dudr\end{align*}</div><p>Since the two iterated integrals are over the same region, </p><div class="display-equation">\[\ev\braks{f(t, X_t) - f(s, X_s) | \cf_s}= \ev\braks{\int_s^t (\partial_r + L_rf)(r, X_r)dr \bigg | \cf_s}\]</div><p>Therefore $\bracs{E_t}$ is a $\bracs{\mathcal{F}_t}$-martingale.</p><p>(2) $\Rightarrow$ (3): Let $f \in C_{b}^{1, 2}([0, \infty) \times \real^{d})$ be uniformly positive. Define </p><div class="display-equation">\[Z_{t} = f(t, X_{t}) - \int_{0}^{t} (\partial_{r}f + L_{r}f)(r, X_{r})dr\]</div><p>and </p><div class="display-equation">\[\phi_{t} = \exp\braks{-\int_0^t \frac{\partial_{r}f + L_{r}f}{f}(r, X_r)dr}\]</div><p>then using the by parts formula for Riemann-Stieltjes integrals, </p><div class="display-equation" id="">\begin{align*}Z_{t}\phi_{t} - \int_{0}^{t} Z_{s} \phi(ds)&#x26;= Z_{0}\phi_{0} + \int_{0}^{t} \phi_{s} Z(ds) \\&#x26;= Z_{0}\phi_{0} + \int_{0}^{t} \phi_{s} f(ds, X_{ds}) + \int_{0}^{t} \phi_{s} \cdot (\partial_{s} f L_{s}f)(s, X_{s})ds\end{align*}</div><p>Since </p><div class="display-equation">\[\partial_{s} \phi_{s} = -\phi_{s} \frac{\partial_{s} f + L_{s}f}{f}(s, X_{s})\]</div><p>Using the by parts formula again,</p><div class="display-equation" id="">\begin{align*}Z_{t}\phi_{t} - \int_{0}^{t} Z_{s} \phi(ds)&#x26;= Z_{0}\phi_{0} + \int_{0}^{t} \phi_{s} f(ds, X_{ds}) + \int_{0}^{t} f(s, X_{s})\phi(ds) \\&#x26;= Z_{0}\phi_{0} + f(t, X_{t})\phi_{t} - f(0, X_{0})\phi_{0} = f(t, X_{t})\phi_{t} \\&#x26;= f(t, X_{t})\exp\braks{-\int_0^t \frac{\partial_{r}f + L_{r}f}{f}(r, X_r)dr}\end{align*}</div><p>Therefore <a href="/t/C" class="link-primary">Theorem 1.2.5</a> implies that the above process is a $\bracs{\mathcal{F}_t}$-martingale.</p><p>(3) $\Rightarrow$ (4): Let $x \in \real^{d}$ and $g \in C_{b}^{1, 2}([0, \infty) \times \real^{d})$. Define </p><div class="display-equation">\[f: [0, \infty) \times \real^{d} \to \real \quad (t, y) \mapsto \exp(\dpn{x, y}{\real^d}+ g(t, x))\]</div><p>then (3) applied to $f$ yields that $\bracs{Y_t|t \ge 0}$ is a $\bracs{\mathcal{F}_t}$-martingale. However, since $f$ is unbounded and not uniformly positive, a direct application is ineffective.</p><p>Let $\seq{f_n}\subset C_{b}^{1, 2}([0, \infty) \times \real^{d})$ be uniformly positive such that $f_{n}|_{\bracs{|y| \le n}}= f$. Define </p><div class="display-equation">\[\tau_{n} = \inf\bracs{t \ge 0 \bigg | \sup_{0 \le s \le t}|X_t(u)| \ge n}\wedge n\]</div><p>then $\bracsn{Y_{t \wedge \tau_n}^{x, g}}$ is a $\bracs{\mathcal{F}_t}$-martingale with $Y_{t \wedge \tau_n}^{x, g}\to Y_{t}^{x, g}$ almost surely as $n \to \infty$. In addition, </p><div class="display-equation">\[(Y_{t \wedge \tau_n}^{x, g})^{2} \le Y_{t \wedge \tau_n}^{2x, 2g}\cdot \exp\braks{\int_0^{t \wedge \tau_n} \dpn{x + Dg, A_s(x + Dg) }{\real^d}(s, X_s)ds}\le CY_{t \wedge \tau_n}^{2x, 2g}\]</div><p>where $\ev\braks{Y_{t \wedge \tau_n}^{2x, 2g}}= f(s, X_{s})^{2} \le \exp(2\norm{g}_{u})$. Therefore $\bracsn{Y_{t \wedge \tau_n}^{x, g}}$ is bounded in $L^{2}$, uniformly integrable in $L^{1}$, and converges to $Y_{t}^{x, g}$ in $L^{1}$.<span class="qed">$\square$</span></p></div>

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Equivalence of Martingales Let ( , _t|t 0 , ) be a filtered probability space , L t ( ) u = 1 2 A_t ( ) , u ^ d d + B_t ( ) , u ^d be a second - order F _t - progressively measurable random differential operator , and X_t|t 0 be a F _t - progressively measurable process with continuous sample paths , then the following are equivalent : (1) For every f C c ( d ) , the process E t = f ( X t ) - 0 t ( L r f ) ( X r ) d r is a F _t - martingale . (2) For every f C b 1 , 2 ( [ 0 , ) d ) , H t = f ( t , X t ) - 0 t [ ( r + L r ) f ] ( r , X r ) d r is a F _t - martingale . (3) For each uniformly positive f C b 1 , 2 ( [ 0 , ) d ) , X t = f ( t , X t ) - _0^t ( r + L r ) f f ( r , X_r ) dr is a F _t - martingale . (4) For each x d and g C b 1 , 2 ( [ 0 , ) d ) , the process Y t x , g & = x , X_t - X_0 - _0^t b ( r ) dr ^d + g ( t , X_t ) & - 1 2 _0^t x + Dg , A ( r ) ( x + Dg ) ^d ( r , X_r ) dr & - _0^t [ ( _r + L_r ) g ] ( r , X_r ) d r is a F _t - martingale . (5) For each d , Equivalence of Martingales ( 1 ) ( 2 ) : Let 0 s t and f C c ( [ 0 , ) d ) , then by the Fundamental Theorem of Calculus , f ( t , X t ) - f ( s , X s ) & = f ( t , X t ) - f ( s , X t ) + f ( s , X t ) - f ( s , X s ) & = s t r f ( r , X t ) d r + f ( s , X t ) - f ( s , X s ) where by ( 1 ) , f ( s , X_t ) - f ( s , X_s ) | _s = _s^t ( L_rf ) ( s , X_r ) dr | _s By the Fundamental Theorem of Calculus , s t ( r f ) ( r , X t ) d r & = s t ( r f ) ( r , X r ) d r + s t ( r f ) ( r , X t ) - ( r f ) ( r , X r ) d r & = s t ( r f ) ( r , X r ) d r + s t ( r f ) ( r , X t ) - ( r f ) ( r , X r ) d r By ( 1 ) applied to ( r f ) , _s^t ( _rf ) ( r , X_t ) dr | _s = _s^t ( _r f ) ( r , X_r ) dr + _s^t _r^t ( _r f ) ( r , X_u ) du dr | _s Similarly , by the Fundamental Theorem of Calculus , s t ( L r f ) ( s , X r ) d r & = s t ( L r f ) ( r , X r ) d r + s t ( L r f ) ( s , X r ) - ( L r f ) ( r , X r ) d r & = s t ( L r f ) ( r , X r ) d r - s t s r ( r L r f ) ( u , X r ) d u d r Since the two iterated integrals are over the same region , f ( t , X_t ) - f ( s , X_s ) | _s = _s^t ( _r + L_rf ) ( r , X_r ) dr | _s Therefore E_t is a F _t - martingale . ( 2 ) ( 3 ) : Let f C b 1 , 2 ( [ 0 , ) d ) be uniformly positive . Define Z t = f ( t , X t ) - 0 t ( r f + L r f ) ( r , X r ) d r and t = - _0^t r f + L r f f ( r , X_r ) dr then using the by parts formula for Riemann - Stieltjes integrals , Z t t - 0 t Z s ( d s ) & = Z 0 0 + 0 t s Z ( d s ) & = Z 0 0 + 0 t s f ( d s , X ds ) + 0 t s ( s f L s f ) ( s , X s ) d s Since s s = - s s f + L s f f ( s , X s ) Using the by parts formula again , Z t t - 0 t Z s ( d s ) & = Z 0 0 + 0 t s f ( d s , X ds ) + 0 t f ( s , X s ) ( d s ) & = Z 0 0 + f ( t , X t ) t - f ( 0 , X 0 ) 0 = f ( t , X t ) t & = f ( t , X t ) - _0^t r f + L r f f ( r , X_r ) dr Therefore theorem : martingale - ibp implies that the above process is a F _t - martingale . ( 3 ) ( 4 ) : Let x d and g C b 1 , 2 ( [ 0 , ) d ) . Define f : [ 0 , ) d ( t , y ) ( x , y ^d + g ( t , x ) ) then ( 3 ) applied to f yields that Y_t|t 0 is a F _t - martingale . However , since f is unbounded and not uniformly positive , a direct application is ineffective . Let f_n C b 1 , 2 ( [ 0 , ) d ) be uniformly positive such that f n | |y| n = f . Define n = t 0 | _ 0 s t |X_t ( u ) | n n then Y_ t _n ^ x , g is a F _t - martingale with Y t _n x , g Y t x , g almost surely as n . In addition , ( Y t _n x , g ) 2 Y t _n 2x , 2g _0^ t _n x + Dg , A_s ( x + Dg ) ^d ( s , X_s ) ds C Y t _n 2x , 2g where Y_ t _n ^ 2x , 2g = f ( s , X s ) 2 ( 2 g u ) . Therefore Y_ t _n ^ x , g is bounded in L 2 , uniformly integrable in L 1 , and converges to Y t x , g in L 1 . Proof Theorem 4.2 .1 Diffusion .

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<div class="block-container"><p><a class="link-foreground" href="/t/C"><strong>Theorem 1.2.5 (Integration by Parts).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;theorem:martingale-ibp&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, $\bracs{X_t}$ be a $\bracs{\mathcal{F}_t}$-martingale, and $\phi: [0, \infty) \times \Omega \to \complex$ be a continuous, progressively measurable function. If:</p><ol class="enumerate"><li style="list-style-type: &#x27;(1) &#x27;"><p>For every $\omega \in \Omega$, $\phi(\cdot, \omega) \in BV_{\text{loc}}([0, \infty))$.</p></li><li style="list-style-type: &#x27;(2) &#x27;"><p>For all $t \ge 0$,</p><div class="display-equation">\[\ev\braks{\sup_{0 \le s \le t}|X_s|(|\phi|(s) + |\phi|(t))}&#x3C; \infty\]</div></li></ol><p>then the process </p><div class="display-equation">\[Y_{t} = X_{t} \phi(t) - \int_{0}^{t} X_{r} \phi(dr)\]</div><p>is a $\bracs{\mathcal{F}_t}$-martingale.</p></div>

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Integration by Parts Let ( , _t|t 0 , ) be a filtered probability space , X_t be a F _t - martingale , and : [ 0 , ) be a continuous , progressively measurable function . If : (1) For every , ( , ) B V loc ( [ 0 , ) ) . (2) For all t 0 , _ 0 s t |X_s| ( | | ( s ) + | | ( t ) ) < then the process Y t = X t ( t ) - 0 t X r ( d r ) is a F _t - martingale . Integration by Parts

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<div class="block-container"><p><a class="link-foreground" href="/t/D"><strong>Proposition 1.2.4.</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;proposition:martingale-quadratic&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, </p><div class="display-equation">\[L_{t}(\omega)u = \frac{1}{2}\dpn{A_t(\omega), u}{\real^{d \times d}}+ \dpn{B_t(\omega), u}{\real^d}\]</div><p>be a second-order $\bracs{\mathcal{F}_t}$-progressively measurable random differential operator, $\bracs{X_t|t \ge 0}$ be a $\bracs{\mathcal{F}_t}$-progressively measurable process with continuous sample paths, and $f \in C_{b}^{1, 2}([0, \infty) \times \real^{d})$. If the processes </p><div class="display-equation">\[Y_{t} = f(t, X_{t}) - \int_{0}^{t} [(\partial_{r} + L_{r})f](r, X_{r})dr\]</div><p>and </p><div class="display-equation">\[W_{t} = f(t, X_{t})^{2} - \int_{0}^{t} [(\partial_{r} + L_{r})(f^{2})](r, X_{r})dr\]</div><p>are martingales, then </p><div class="display-equation">\[[X]_{t} = \frac{1}{2}\int_{0}^{t} \dpn{Df, A(r)(Df)}{\real^d}(r, X_{r})dr\]</div></div><div class="proof"><p><i class="">Proof. </i>Firstly, since $f$ admits bounded derivatives, the integral term is of locally bounded variation. Therefore if $Z_{t} = f(t, X_{t})$, then $[X]_{t} = \angles{Z}_{t}$.</p><p>Let $\bracs{t_j}_{0}^{n} \subset [0, t]$ be a partition, then </p><div class="display-equation">\[Z_{t}^{2} - Z_{0}^{2} = \sum_{j = 1}^{n}Z_{t_j}^{2} - Z_{t_{j-1}}^{2} = \sum_{j = 1}^{n} 2Z_{t_{j-1}}(Z_{t_j}- Z_{t_{j-1}}) + \sum_{j = 1}^{n} (Z_{t_j}- Z_{t_{j-1}})^{2}\]</div><p>As the above holds for every partition, </p><div class="display-equation">\[Z_{t}^{2} - Z_{0}^{2} = 2\int_{0}^{t} Z_{r}Z(dr) + \angles{Z}_{t}\]</div><p>Since $Z_{t} = Y_{t} + \int_{0}^{t}(\partial_{r} + L_{r})fdr$, </p><div class="display-equation" id="">\begin{align*}\int_{0}^{t} Z_{r}Z(dr)&#x26;= \int_{0}^{t} Z_{r} Y(dr) + \int_{0}^{t} f(r, X_{r})(\partial_{r} + L_{r})f(r, X_{r})dr \\ Z_{t}^{2} - Z_{0}^{2}&#x26;= 2\int_{0}^{t} Z_{r} Y(dr) + 2\int_{0}^{t} f(r, X_{r})(\partial_{r} + L_{r})f(r, X_{r})dr + \angles{Z}_{t}\end{align*}</div><p>On the other hand, </p><div class="display-equation">\[Z_{t}^{2} - Z_{0}^{2} = W_{t} - W_{0} + \int_{0}^{t} [(\partial_{r} + L_{r})(f^{2})](r, X_{r})dr\]</div><p>Therefore </p><div class="display-equation" id="">\begin{align*}\angles{Z}_{t}&#x26;= W_{t} - W_{0} + \int_{0}^{t} [(\partial_{r} + L_{r})(f^{2})](r, X_{r})dr \\&#x26;- 2\int_{0}^{t} Z_{r} Y(dr) - 2\int_{0}^{t} f(r, X_{r})(\partial_{r} + L_{r})f(r, X_{r})dr \\&#x26;= W_{t} - W_{0} - 2 \int_{0}^{t} f(r, X_{r}) Y(dr) \\&#x26;+ \int_{0}^{t} [L_{r}(f^{2})(r, X_{r}) - 2f(r, X_{r})(L_{r}f)(r, X_{r})]dr \\&#x26;= W_{t} - W_{0} - 2\int_{0}^{t} f(r, X_{r})Y(dr) + \frac{1}{2}\int_{0}^{t} \dpn{Df, A(r)(Df)}{\real^d}(r, X_{r})dr\end{align*}</div><p>where the process $t \mapsto W_{t} - W_{0} - 2\int_{0}^{t} f(r, X_{r})Y(dr)$ is of finite variation, so it must be $0$ by <span class="">[<span class="">Lemma 5.13</span>, <a class="link-primary" href="/b/E">Bau14</a>]</span>.<span class="qed">$\square$</span></p></div>

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Let ( , _t|t 0 , ) be a filtered probability space , L t ( ) u = 1 2 A_t ( ) , u ^ d d + B_t ( ) , u ^d be a second - order F _t - progressively measurable random differential operator , X_t|t 0 be a F _t - progressively measurable process with continuous sample paths , and f C b 1 , 2 ( [ 0 , ) d ) . If the processes Y t = f ( t , X t ) - 0 t [ ( r + L r ) f ] ( r , X r ) d r and W t = f ( t , X t ) 2 - 0 t [ ( r + L r ) ( f 2 ) ] ( r , X r ) d r are martingales , then [ X ] t = 1 2 0 t Df , A ( r ) ( Df ) ^d ( r , X r ) d r Firstly , since f admits bounded derivatives , the integral term is of locally bounded variation . Therefore if Z t = f ( t , X t ) , then [ X ] t = Z t . Let t_j 0 n [ 0 , t ] be a partition , then Z t 2 - Z 0 2 = j = 1 n Z t_j 2 - Z t_ j - 1 2 = j = 1 n 2 Z t_ j - 1 ( Z t_j - Z t_ j - 1 ) + j = 1 n ( Z t_j - Z t_ j - 1 ) 2 As the above holds for every partition , Z t 2 - Z 0 2 = 2 0 t Z r Z ( d r ) + Z t Since Z t = Y t + 0 t ( r + L r ) f d r , 0 t Z r Z ( d r ) & = 0 t Z r Y ( d r ) + 0 t f ( r , X r ) ( r + L r ) f ( r , X r ) d r Z t 2 - Z 0 2 & = 2 0 t Z r Y ( d r ) + 2 0 t f ( r , X r ) ( r + L r ) f ( r , X r ) d r + Z t On the other hand , Z t 2 - Z 0 2 = W t - W 0 + 0 t [ ( r + L r ) ( f 2 ) ] ( r , X r ) d r Therefore Z t & = W t - W 0 + 0 t [ ( r + L r ) ( f 2 ) ] ( r , X r ) d r & - 2 0 t Z r Y ( d r ) - 2 0 t f ( r , X r ) ( r + L r ) f ( r , X r ) d r & = W t - W 0 - 2 0 t f ( r , X r ) Y ( d r ) & + 0 t [ L r ( f 2 ) ( r , X r ) - 2 f ( r , X r ) ( L r f ) ( r , X r ) ] d r & = W t - W 0 - 2 0 t f ( r , X r ) Y ( d r ) + 1 2 0 t Df , A ( r ) ( Df ) ^d ( r , X r ) d r where the process t W t - W 0 - 2 0 t f ( r , X r ) Y ( d r ) is of finite variation , so it must be 0 by Lemma 5.13 Baudoin .


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<div class="block-container"><p><a class="link-foreground" href="/t/J"><strong>Definition 2.1.1 (Integrable).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;definition:integrable&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $X$ be a $C^{p}$ ($p \ge 2$) manifold, $E \subset TX$ be a subbundle, and $x_{0} \in X$, then $E$ is <b class="textbf">integrable at $x_{0}$</b> if there exists a submanifold $x_{0} \in Y \subset X$ such that </p><div class="display-equation">\[T\iota: TY \to E \subset TX\]</div><p>is an isomorphism. If $E$ is integrable at every $x_{0} \in X$, then $E$ is <b class="textbf">integrable</b>.</p></div><div class="block-container"><p><a class="link-foreground" href="/t/K"><strong>Lemma 2.1.2.</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;lemma:split-ode&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $E, F$ be Banach spaces, $U \subset E$, $V \subset F$ be open sets, $\eps > 0$, $g \in C^{k}((-\eps, \eps) \times U \times V; F)$ ($k \ge 2$), and $(x_{0}, y_{0}) \in U \times V$, then there exists $\delta > 0$, $U_{0} \in \cn_{E}(x_{0})$, $V_{0} \in \cn_{F}(x_{0})$ and a unique $C^{k}$-mapping </p><div class="display-equation">\[\beta: (-\delta, \delta) \times U_{0} \times V_{0} \to V\]</div><p>such that</p><ol class="enumerate"><li style="list-style-type: &#x27;(1) &#x27;"><p>For all $(x, y) \in U_{0} \times V_{0}$, $\beta(0, x, y) = y$.</p></li><li style="list-style-type: &#x27;(2) &#x27;"><p>For all $(t, x, y) \in (-\delta, \delta) \times U_{0} \times V_{0}$,</p><div class="display-equation">\[\frac{d}{dt}\beta(t, x, y) = g(t, x, \beta(t, x, y))\]</div></li><li style="list-style-type: &#x27;(3) &#x27;"><p>For each fixed $y \in V_{0}$, let $\beta(t, x) = \beta(t, x, y)$, then for each $(t, x) \in (-\delta, \delta) \times U_{0}$,</p><div class="display-equation">\[(\partial_{t}\partial_{x}\beta)(t, x) = (\partial_{x}g)(t, x, \beta(t, x)) + (\partial_{y}g)(t, x, \beta(t, x)) \circ (\partial_{x}\beta)(t, x)\]</div></li></ol></div><div class="proof"><p><i class="">Proof, <span class="">[<span class="">Proposition VI.2.1</span>, <a class="link-primary" href="/b/F">Lan12</a>]</span>.</i>Let </p><div class="display-equation">\[G: (-\eps, \eps) \times U \times V \to E \times F \quad (t, x, y) \mapsto (0, g(t, x, y))\]</div><p>then by the existence and uniqueness of ODEs, there exists $\delta > 0$, $U_{0} \in \cn_{E}(x_{0})$, $V_{0} \in \cn_{F}(y_{0})$, and a unique $C^{k}$-mapping </p><div class="display-equation">\[B: (-\delta, \delta) \times U_{0} \times V_{0} \to E \times F\]</div><p>such that</p><ol class="enumerate"><li style="list-style-type: &#x27;(a) &#x27;"><p>For each $(x, y) \in U_{0} \times V_{0}$, $B(0, x, y) = (x, y)$.</p></li><li style="list-style-type: &#x27;(b) &#x27;"><p>For each $(t, x, y) \in (-\delta, \delta) \times U_{0} \times V_{0}$,</p><div class="display-equation">\[\frac{d}{dt}B(t, x, y) = (0, g(t, B(t, x, y)))\]</div></li></ol><p>Let $\beta(t, x, y) = \pi_{2}(t, x, y)$, then</p><ol class="enumerate"><li style="list-style-type: &#x27;(1) &#x27;"><p>For each $(x, y) \in U_{0} \times V_{0}$, $\beta(0, x, y) = \pi_{2}(x, y) = y$.</p></li><li style="list-style-type: &#x27;(2) &#x27;"><p>For each $(t, x, y) \in (-\delta, \delta) \times U_{0} \times V_{0}$,</p><div class="display-equation">\[\frac{d}{dt}\beta(t, x, y) = \pi_{2}(g(t, B(t, x, y))) = g(t, x, \beta(t, x, y))\]</div></li><li style="list-style-type: &#x27;(3) &#x27;"><p>For each fixed $y \in V_{0}$, let $\beta(t, x) = \beta(t, x, y)$, then for each $(t, x) \in (-\delta, \delta) \times U_{0}$, by the chain rule,</p><div class="display-equation">\[(\partial_{t}\partial_{x}\beta)(t, x) = (\partial_{x}g)(t, x, \beta(t, x)) + (\partial_{y}g)(t, x, \beta(t, x)) \circ (\partial_{x}\beta)(t, x)\]</div></li></ol><p><span class="qed">$\square$</span></p></div><div class="block-container"><p><a class="link-foreground" href="/t/N"><strong>Theorem 2.1.3.</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;theorem:local-frobenius&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $E, F$ be Banach spaces, $U \subset E$ and $V \subset F$ be open subsets, and $f \in C^{k}(U \times V; L(E; F))$ ($k \ge 2$).</p><p>For any $\xi, \eta \in C^{k}(U \times V; E)$, denote </p><div class="display-equation">\[\Xi(x, y) = (\xi(x, y), f(x, y) \cdot \xi(x, y)) \quad H(x, y) = (\eta(x, y), f(x, y) \cdot \eta(x, y))\]</div><p>If for every $\xi, \eta \in C^{k}(U \times V; E)$ and $(x, y) \in U \times V$, </p><div class="display-equation">\[[Df(x, y) \cdot \Xi(x, y)] \cdot \eta(x, y) = [Df(x, y) \cdot H(x, y)] \cdot \xi(x, y)\]</div><p>then for every $(x_{0}, y_{0}) \in U \times V$, there exists $U_{0} \in \cn_{E}(x_{0})$, $V_{0} \in \cn_{F}(y_{0})$, and a unique $\alpha \in C^{k}(U_{0} \times V_{0}; V)$ such that</p><ol class="enumerate"><li style="list-style-type: &#x27;(1) &#x27;"><p>For every $y \in V_{0}$, $\alpha(x_{0}, y) = y$.</p></li><li style="list-style-type: &#x27;(2) &#x27;"><p>For every $(x, y) \in U_{0} \times V_{0}$,</p><div class="display-equation">\[(\partial_{x} \alpha)(x, y) = f(x, \alpha(x, y))\]</div></li></ol></div><div class="proof"><p><i class="">Proof. </i>Using translation, assume without loss of generality that $x_{0} = 0$ and $y_{0} = 0$. Let $B \in \cn_{E}(0)$ and </p><div class="display-equation">\[g: (-\eps, \eps) \times B \times V \to F \quad (t, z, y) \mapsto f(tz, y) \cdot z\]</div><p>then by <a href="/t/K" class="link-primary">Lemma 2.1.2</a>, there exists $\delta > 0$, $B_{0} \in \cn_{E}(0)$, and $\beta: (-\delta, \delta) \times B_{0} \times V_{0}$ such that</p><ol class="enumerate"><li style="list-style-type: &#x27;(a) &#x27;"><p>For each $(z, y) \in B_{0} \times V_{0}$, $\beta(0, z, y) = y$.</p></li><li style="list-style-type: &#x27;(b) &#x27;"><p>For each $(t, z, y) \in (-\delta, \delta) \times B_{0} \times V_{0}$,</p><div class="display-equation">\[\frac{d}{dt}\beta(t, z, y) = f(tz, \beta(t, z, y)) \cdot z\]</div></li><li style="list-style-type: &#x27;(c) &#x27;"><p>For each fixed $y \in V_{0}$, if $\beta(t, z) = \beta(t, z, y)$, then for each $(t, x) \in (-\delta, \delta) \times B_{0}$ and $h \in E$,</p><div class="display-equation" id="">\begin{align*}(\partial_{t}\partial_{z}\beta)(t, z)\cdot h&#x26;= t(\partial_{x}f)(tz, \beta(t, z))\cdot h \cdot z + f(tz, \beta(t, z))\cdot h \\&#x26;+ (\partial_{y}f)(tz, \beta(t, z)) \circ (\partial_{z} \beta)(t, z)\cdot h \cdot z\end{align*}</div></li></ol><p>Following (c), let $k(t, z) = (\partial_{z}\beta)(t, z)\cdot h - tf(tz, \beta(t, z))\cdot h$, then </p><div class="display-equation" id="">\begin{align*}\frac{d}{dt}k(t)&#x26;= (\partial_{t}\partial_{z}\beta)(t, z)\cdot h - f(tz, \beta)\cdot h \\&#x26;- t [(\partial_{x}f)(tz, \beta)]\cdot z \cdot h - t[(\partial_{y} f)(tz, \beta)]f(tz, \beta) \cdot z\cdot h\end{align*}</div><p>where by assumption, </p><div class="display-equation" id="">\begin{align*}&#x26;[(\partial_{x}f)(tz, \beta)]\cdot z \cdot h - [(\partial_{y} f)(tz, \beta)]f(tz, \beta) \cdot z \cdot h \\&#x26;= Df(tz, \beta) \cdot (z, f(tz, \beta) \cdot z) \cdot h \\&#x26;= Df(tz, \beta) \cdot (h, f(tz, \beta) \cdot h) \cdot z \\&#x26;= [(\partial_{x}f)(tz, \beta)]\cdot h \cdot z - [(\partial_{y} f)(tz, \beta)]f(tz, \beta) \cdot h\cdot z\end{align*}</div><p>Therefore by (c), </p><div class="display-equation" id="">\begin{align*}\frac{d}{dt}k(t)&#x26;= (\partial_{y}f)(tz, \beta) \circ (\partial_{z} \beta)(t, z)\cdot h \cdot z - t[(\partial_{y} f)(tz, \beta)]f(tz, \beta) \cdot h\cdot z \\&#x26;= (\partial_{y} f)(tz, \beta)[\partial_{z} \beta(t, z) - tf(tz, \beta)] \cdot h \cdot z \\&#x26;= (\partial_{y} f)(tz, \beta) \cdot k(t) \cdot z\end{align*}</div><p>Since $0$ is a solution to the above equation, $k(t) = 0$ by the uniqueness of solutions to ODEs. Hence for every $t \in (-\delta, \delta)$ and $z \in B_{0}$, </p><div class="display-equation">\[\partial_{z} \beta(t, z) = tf(tz, \beta(t, z))\]</div><p>By adjusting $\delta$ and $B_{0}$, assume without loss of generality that $\delta > 1$. In which case, if $\alpha(x) = \beta(1, x, y)$, then </p><div class="display-equation">\[\partial_{x}\alpha(x, y) = \partial_{z}\beta(1, x, y) = f(z, \beta(1, x, y)) = f(x, \alpha(x, y))\]</div><p><span class="qed">$\square$</span></p></div>

contentText:
Integrable Bundles Let X be a C p ( p 2 ) manifold , E T X be a subbundle , and x 0 X , then E is integrable at x 0 if there exists a submanifold x 0 Y X such that T : T Y E T X is an isomorphism . If E is integrable at every x 0 X , then E is integrable . Integrable Let E , F be Banach spaces , U E , V F be open sets , > 0 , g C k ( ( - , ) U V ; F ) ( k 2 ) , and ( x 0 , y 0 ) U V , then there exists > 0 , U 0 E ( x 0 ) , V 0 F ( x 0 ) and a unique C k - mapping : ( - , ) U 0 V 0 V such that (1) For all ( x , y ) U 0 V 0 , ( 0 , x , y ) = y . (2) For all ( t , x , y ) ( - , ) U 0 V 0 , d d t ( t , x , y ) = g ( t , x , ( t , x , y ) ) (3) For each fixed y V 0 , let ( t , x ) = ( t , x , y ) , then for each ( t , x ) ( - , ) U 0 , ( t x ) ( t , x ) = ( x g ) ( t , x , ( t , x ) ) + ( y g ) ( t , x , ( t , x ) ) ( x ) ( t , x ) Let G : ( - , ) U V E F ( t , x , y ) ( 0 , g ( t , x , y ) ) then by the existence and uniqueness of ODEs , there exists > 0 , U 0 E ( x 0 ) , V 0 F ( y 0 ) , and a unique C k - mapping B : ( - , ) U 0 V 0 E F such that ( a ) For each ( x , y ) U 0 V 0 , B ( 0 , x , y ) = ( x , y ) . ( b ) For each ( t , x , y ) ( - , ) U 0 V 0 , d d t B ( t , x , y ) = ( 0 , g ( t , B ( t , x , y ) ) ) Let ( t , x , y ) = 2 ( t , x , y ) , then (1) For each ( x , y ) U 0 V 0 , ( 0 , x , y ) = 2 ( x , y ) = y . (2) For each ( t , x , y ) ( - , ) U 0 V 0 , d d t ( t , x , y ) = 2 ( g ( t , B ( t , x , y ) ) ) = g ( t , x , ( t , x , y ) ) (3) For each fixed y V 0 , let ( t , x ) = ( t , x , y ) , then for each ( t , x ) ( - , ) U 0 , by the chain rule , ( t x ) ( t , x ) = ( x g ) ( t , x , ( t , x ) ) + ( y g ) ( t , x , ( t , x ) ) ( x ) ( t , x ) Proof , Proposition VI .2 .1 Lang . Let E , F be Banach spaces , U E and V F be open subsets , and f C k ( U V ; L ( E ; F ) ) ( k 2 ) . For any , C k ( U V ; E ) , denote ( x , y ) = ( ( x , y ) , f ( x , y ) ( x , y ) ) H ( x , y ) = ( ( x , y ) , f ( x , y ) ( x , y ) ) If for every , C k ( U V ; E ) and ( x , y ) U V , [ D f ( x , y ) ( x , y ) ] ( x , y ) = [ D f ( x , y ) H ( x , y ) ] ( x , y ) then for every ( x 0 , y 0 ) U V , there exists U 0 E ( x 0 ) , V 0 F ( y 0 ) , and a unique C k ( U 0 V 0 ; V ) such that (1) For every y V 0 , ( x 0 , y ) = y . (2) For every ( x , y ) U 0 V 0 , ( x ) ( x , y ) = f ( x , ( x , y ) ) Using translation , assume without loss of generality that x 0 = 0 and y 0 = 0 . Let B E ( 0 ) and g : ( - , ) B V F ( t , z , y ) f ( t z , y ) z then by lemma : split - ode , there exists > 0 , B 0 E ( 0 ) , and : ( - , ) B 0 V 0 such that ( a ) For each ( z , y ) B 0 V 0 , ( 0 , z , y ) = y . ( b ) For each ( t , z , y ) ( - , ) B 0 V 0 , d d t ( t , z , y ) = f ( t z , ( t , z , y ) ) z ( c ) For each fixed y V 0 , if ( t , z ) = ( t , z , y ) , then for each ( t , x ) ( - , ) B 0 and h E , ( t z ) ( t , z ) h & = t ( x f ) ( t z , ( t , z ) ) h z + f ( t z , ( t , z ) ) h & + ( y f ) ( t z , ( t , z ) ) ( z ) ( t , z ) h z Following ( c ) , let k ( t , z ) = ( z ) ( t , z ) h - t f ( t z , ( t , z ) ) h , then d d t k ( t ) & = ( t z ) ( t , z ) h - f ( t z , ) h & - t [ ( x f ) ( t z , ) ] z h - t [ ( y f ) ( t z , ) ] f ( t z , ) z h where by assumption , & [ ( x f ) ( t z , ) ] z h - [ ( y f ) ( t z , ) ] f ( t z , ) z h & = D f ( t z , ) ( z , f ( t z , ) z ) h & = D f ( t z , ) ( h , f ( t z , ) h ) z & = [ ( x f ) ( t z , ) ] h z - [ ( y f ) ( t z , ) ] f ( t z , ) h z Therefore by ( c ) , d d t k ( t ) & = ( y f ) ( t z , ) ( z ) ( t , z ) h z - t [ ( y f ) ( t z , ) ] f ( t z , ) h z & = ( y f ) ( t z , ) [ z ( t , z ) - t f ( t z , ) ] h z & = ( y f ) ( t z , ) k ( t ) z Since 0 is a solution to the above equation , k ( t ) = 0 by the uniqueness of solutions to ODEs . Hence for every t ( - , ) and z B 0 , z ( t , z ) = t f ( t z , ( t , z ) ) By adjusting and B 0 , assume without loss of generality that > 1 . In which case , if ( x ) = ( 1 , x , y ) , then x ( x , y ) = z ( 1 , x , y ) = f ( z , ( 1 , x , y ) ) = f ( x , ( x , y ) )

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<div class="block-container"><p><a class="link-foreground" href="/t/J"><strong>Definition 2.1.1 (Integrable).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;definition:integrable&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $X$ be a $C^{p}$ ($p \ge 2$) manifold, $E \subset TX$ be a subbundle, and $x_{0} \in X$, then $E$ is <b class="textbf">integrable at $x_{0}$</b> if there exists a submanifold $x_{0} \in Y \subset X$ such that </p><div class="display-equation">\[T\iota: TY \to E \subset TX\]</div><p>is an isomorphism. If $E$ is integrable at every $x_{0} \in X$, then $E$ is <b class="textbf">integrable</b>.</p></div>

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Integrable Let X be a C p ( p 2 ) manifold , E T X be a subbundle , and x 0 X , then E is integrable at x 0 if there exists a submanifold x 0 Y X such that T : T Y E T X is an isomorphism . If E is integrable at every x 0 X , then E is integrable . Integrable

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<div class="block-container"><p><a class="link-foreground" href="/t/K"><strong>Lemma 2.1.2.</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;lemma:split-ode&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $E, F$ be Banach spaces, $U \subset E$, $V \subset F$ be open sets, $\eps > 0$, $g \in C^{k}((-\eps, \eps) \times U \times V; F)$ ($k \ge 2$), and $(x_{0}, y_{0}) \in U \times V$, then there exists $\delta > 0$, $U_{0} \in \cn_{E}(x_{0})$, $V_{0} \in \cn_{F}(x_{0})$ and a unique $C^{k}$-mapping </p><div class="display-equation">\[\beta: (-\delta, \delta) \times U_{0} \times V_{0} \to V\]</div><p>such that</p><ol class="enumerate"><li style="list-style-type: &#x27;(1) &#x27;"><p>For all $(x, y) \in U_{0} \times V_{0}$, $\beta(0, x, y) = y$.</p></li><li style="list-style-type: &#x27;(2) &#x27;"><p>For all $(t, x, y) \in (-\delta, \delta) \times U_{0} \times V_{0}$,</p><div class="display-equation">\[\frac{d}{dt}\beta(t, x, y) = g(t, x, \beta(t, x, y))\]</div></li><li style="list-style-type: &#x27;(3) &#x27;"><p>For each fixed $y \in V_{0}$, let $\beta(t, x) = \beta(t, x, y)$, then for each $(t, x) \in (-\delta, \delta) \times U_{0}$,</p><div class="display-equation">\[(\partial_{t}\partial_{x}\beta)(t, x) = (\partial_{x}g)(t, x, \beta(t, x)) + (\partial_{y}g)(t, x, \beta(t, x)) \circ (\partial_{x}\beta)(t, x)\]</div></li></ol></div><div class="proof"><p><i class="">Proof, <span class="">[<span class="">Proposition VI.2.1</span>, <a class="link-primary" href="/b/F">Lan12</a>]</span>.</i>Let </p><div class="display-equation">\[G: (-\eps, \eps) \times U \times V \to E \times F \quad (t, x, y) \mapsto (0, g(t, x, y))\]</div><p>then by the existence and uniqueness of ODEs, there exists $\delta > 0$, $U_{0} \in \cn_{E}(x_{0})$, $V_{0} \in \cn_{F}(y_{0})$, and a unique $C^{k}$-mapping </p><div class="display-equation">\[B: (-\delta, \delta) \times U_{0} \times V_{0} \to E \times F\]</div><p>such that</p><ol class="enumerate"><li style="list-style-type: &#x27;(a) &#x27;"><p>For each $(x, y) \in U_{0} \times V_{0}$, $B(0, x, y) = (x, y)$.</p></li><li style="list-style-type: &#x27;(b) &#x27;"><p>For each $(t, x, y) \in (-\delta, \delta) \times U_{0} \times V_{0}$,</p><div class="display-equation">\[\frac{d}{dt}B(t, x, y) = (0, g(t, B(t, x, y)))\]</div></li></ol><p>Let $\beta(t, x, y) = \pi_{2}(t, x, y)$, then</p><ol class="enumerate"><li style="list-style-type: &#x27;(1) &#x27;"><p>For each $(x, y) \in U_{0} \times V_{0}$, $\beta(0, x, y) = \pi_{2}(x, y) = y$.</p></li><li style="list-style-type: &#x27;(2) &#x27;"><p>For each $(t, x, y) \in (-\delta, \delta) \times U_{0} \times V_{0}$,</p><div class="display-equation">\[\frac{d}{dt}\beta(t, x, y) = \pi_{2}(g(t, B(t, x, y))) = g(t, x, \beta(t, x, y))\]</div></li><li style="list-style-type: &#x27;(3) &#x27;"><p>For each fixed $y \in V_{0}$, let $\beta(t, x) = \beta(t, x, y)$, then for each $(t, x) \in (-\delta, \delta) \times U_{0}$, by the chain rule,</p><div class="display-equation">\[(\partial_{t}\partial_{x}\beta)(t, x) = (\partial_{x}g)(t, x, \beta(t, x)) + (\partial_{y}g)(t, x, \beta(t, x)) \circ (\partial_{x}\beta)(t, x)\]</div></li></ol><p><span class="qed">$\square$</span></p></div>

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Let E , F be Banach spaces , U E , V F be open sets , > 0 , g C k ( ( - , ) U V ; F ) ( k 2 ) , and ( x 0 , y 0 ) U V , then there exists > 0 , U 0 E ( x 0 ) , V 0 F ( x 0 ) and a unique C k - mapping : ( - , ) U 0 V 0 V such that (1) For all ( x , y ) U 0 V 0 , ( 0 , x , y ) = y . (2) For all ( t , x , y ) ( - , ) U 0 V 0 , d d t ( t , x , y ) = g ( t , x , ( t , x , y ) ) (3) For each fixed y V 0 , let ( t , x ) = ( t , x , y ) , then for each ( t , x ) ( - , ) U 0 , ( t x ) ( t , x ) = ( x g ) ( t , x , ( t , x ) ) + ( y g ) ( t , x , ( t , x ) ) ( x ) ( t , x ) Let G : ( - , ) U V E F ( t , x , y ) ( 0 , g ( t , x , y ) ) then by the existence and uniqueness of ODEs , there exists > 0 , U 0 E ( x 0 ) , V 0 F ( y 0 ) , and a unique C k - mapping B : ( - , ) U 0 V 0 E F such that ( a ) For each ( x , y ) U 0 V 0 , B ( 0 , x , y ) = ( x , y ) . ( b ) For each ( t , x , y ) ( - , ) U 0 V 0 , d d t B ( t , x , y ) = ( 0 , g ( t , B ( t , x , y ) ) ) Let ( t , x , y ) = 2 ( t , x , y ) , then (1) For each ( x , y ) U 0 V 0 , ( 0 , x , y ) = 2 ( x , y ) = y . (2) For each ( t , x , y ) ( - , ) U 0 V 0 , d d t ( t , x , y ) = 2 ( g ( t , B ( t , x , y ) ) ) = g ( t , x , ( t , x , y ) ) (3) For each fixed y V 0 , let ( t , x ) = ( t , x , y ) , then for each ( t , x ) ( - , ) U 0 , by the chain rule , ( t x ) ( t , x ) = ( x g ) ( t , x , ( t , x ) ) + ( y g ) ( t , x , ( t , x ) ) ( x ) ( t , x ) Proof , Proposition VI .2 .1 Lang .

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<div class="block-container"><p><a class="link-foreground" href="/t/M"><strong>Theorem 2.2.1 (Frobenius).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;theorem:frobenius&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $X$ be a $C^{p}$ ($p \ge 2$) manifold and $E \subset TX$ be a subbundle, then the following are equivalent:</p><ol class="enumerate"><li style="list-style-type: &#x27;(1) &#x27;"><p>For each pair of vector fields $\xi, \eta: X \to E$, $[\xi, \eta]$ lies in $E$.</p></li><li style="list-style-type: &#x27;(2) &#x27;"><p>For each $\omega \in \Lambda^{1}(TX)$ vanishing on $E$, $\xi, \eta: X \to E$, $d\omega(\xi, \eta) = 0$.</p></li><li style="list-style-type: &#x27;(3) &#x27;"><p>$E$ is integrable.</p></li></ol></div><div class="proof"><p><i class="">Proof, <span class="">[<span class="">Section VI.1</span>, <a class="link-primary" href="/b/F">Lan12</a>]</span>. </i>Let $\omega \in \Lambda^{1}(TX)$, then </p><div class="display-equation">\[d\omega(\xi, \eta) = \omega([\xi, \eta]) - \eta\omega(\xi) - \xi\omega(\eta)\]</div><p>(1) $\Rightarrow$ (2): If $\omega$ vanishes on $E$ and $[\xi, \eta]$ lies in $E$, then $d\omega$ vanishes on $(\xi, \eta)$.</p><p>(2) $\Rightarrow$ (1): If $[\xi, \eta]$ does not lie in $E$, then there exists $\omega \in \Lambda^{1}(TX)$ that vanishes on $E$ but not $[\xi, \eta]$, which contradicts (2).</p><p>(1) $\Leftrightarrow$ (3): For any point on $X$, since $E$ is a subbundle, there exists Banach spaces $F, G$ and a coordinate neighbourhood $U \times V$ such that $E$ is given by a $C^{p-1}$-mapping </p><div class="display-equation">\[f: U \times V \times F \to U \times V \times (F \times G)\]</div><p>such that $\pi_{1}f(x, y)(z) = z$ for all $(x, y) \in U \times V$ and $z \in F$. Let </p><div class="display-equation">\[g: U \times V \to L(F; G) \quad g(x, y)(z) = \pi_{2}f(x, y, z)\]</div><p>then $g$ is also a $C^{p - 1}$-mapping. Let </p><div class="display-equation">\[\Xi: U \times V \to F \times G\]</div><p>be the local representation of a vector field, then the vector field lies in $E$ if and only if </p><div class="display-equation">\[\pi_{2}\Xi(x, y) = g(x, y)\pi_{1}\Xi(x, y)\]</div><p>for all $(x, y) \im U \times V$. Let $H: U \times V \to F \times G$ be another vector field, then by assumption (1), $[\Xi, H]$ lies in $E$. Denote $\xi = \pi_{1}\Xi$ and $\eta = \pi_{1} H$, then </p><div class="display-equation" id="">\begin{align*}[\xi, \eta]&#x26;= D\eta \cdot \xi - D\xi \cdot \eta \\ g \cdot (D\eta \cdot \xi - D\xi \cdot \eta)&#x26;= (Dg \cdot \xi) \cdot \eta + g \cdot D\eta \cdot \xi - (Dg \cdot \eta) \cdot \xi - g \cdot D\eta \cdot \xi \\ (Dg \cdot \xi) \cdot \eta&#x26;= (Dg \cdot \eta) \cdot \xi\end{align*}</div><p>Let $(x_{0}, y_{0}) \in U \times V$, then by <a href="/t/N" class="link-primary">Theorem 2.1.3</a>, there exists $U_{0} \in \cn_{F}(x_{0})$ and $V_{0} \in \cn_{G}(y_{0})$ and $\alpha \in C^{p-1}(U_{0} \times V_{0}; V)$ such that </p><div class="display-equation">\[\partial_{x}\alpha(x, y) = g(x, \alpha(x, y))\]</div><p>Let </p><div class="display-equation">\[\varphi: U_{0} \times V_{0} \to U \times V \quad (x, y) \mapsto (x, \alpha(x, y))\]</div><p>then </p><div class="display-equation">\[D\varphi(x_{0}, y_{0}) = \begin{bmatrix}Id&#x26;0 \\ g(x, \alpha(x, y))&#x26;Id\end{bmatrix}\]</div><p>so $\varphi$ is a local diffeomorphism at $(x_{0}, y_{0})$. Since for any $(u, v) \in F \times G$, </p><div class="display-equation">\[\partial_{x}\varphi(x, y) \cdot (u, v) = (u, g(x, \alpha(x, y)) \cdot u)\]</div><p>the bundle $E$ is integrable.<span class="qed">$\square$</span></p></div>

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Frobenius ' Theorem Let X be a C p ( p 2 ) manifold and E T X be a subbundle , then the following are equivalent : (1) For each pair of vector fields , : X E , [ , ] lies in E . (2) For each 1 ( T X ) vanishing on E , , : X E , d ( , ) = 0 . (3) E is integrable . Frobenius Let 1 ( T X ) , then d ( , ) = ( [ , ] ) - ( ) - ( ) ( 1 ) ( 2 ) : If vanishes on E and [ , ] lies in E , then d vanishes on ( , ) . ( 2 ) ( 1 ) : If [ , ] does not lie in E , then there exists 1 ( T X ) that vanishes on E but not [ , ] , which contradicts ( 2 ) . ( 1 ) ( 3 ) : For any point on X , since E is a subbundle , there exists Banach spaces F , G and a coordinate neighbourhood U V such that E is given by a C p - 1 - mapping f : U V F U V ( F G ) such that 1 f ( x , y ) ( z ) = z for all ( x , y ) U V and z F . Let g : U V L ( F ; G ) g ( x , y ) ( z ) = 2 f ( x , y , z ) then g is also a C p - 1 - mapping . Let : U V F G be the local representation of a vector field , then the vector field lies in E if and only if 2 ( x , y ) = g ( x , y ) 1 ( x , y ) for all ( x , y ) U V . Let H : U V F G be another vector field , then by assumption ( 1 ) , [ , H ] lies in E . Denote = 1 and = 1 H , then [ , ] & = D - D g ( D - D ) & = ( D g ) + g D - ( D g ) - g D ( D g ) & = ( D g ) Let ( x 0 , y 0 ) U V , then by theorem : local - frobenius , there exists U 0 F ( x 0 ) and V 0 G ( y 0 ) and C p - 1 ( U 0 V 0 ; V ) such that x ( x , y ) = g ( x , ( x , y ) ) Let : U 0 V 0 U V ( x , y ) ( x , ( x , y ) ) then D ( x 0 , y 0 ) = I d & 0 g ( x , ( x , y ) ) & I d so is a local diffeomorphism at ( x 0 , y 0 ) . Since for any ( u , v ) F G , x ( x , y ) ( u , v ) = ( u , g ( x , ( x , y ) ) u ) the bundle E is integrable . Proof , Section VI .1 Lang .

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<div class="block-container"><p><a class="link-foreground" href="/t/M"><strong>Theorem 2.2.1 (Frobenius).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;theorem:frobenius&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $X$ be a $C^{p}$ ($p \ge 2$) manifold and $E \subset TX$ be a subbundle, then the following are equivalent:</p><ol class="enumerate"><li style="list-style-type: &#x27;(1) &#x27;"><p>For each pair of vector fields $\xi, \eta: X \to E$, $[\xi, \eta]$ lies in $E$.</p></li><li style="list-style-type: &#x27;(2) &#x27;"><p>For each $\omega \in \Lambda^{1}(TX)$ vanishing on $E$, $\xi, \eta: X \to E$, $d\omega(\xi, \eta) = 0$.</p></li><li style="list-style-type: &#x27;(3) &#x27;"><p>$E$ is integrable.</p></li></ol></div><div class="proof"><p><i class="">Proof, <span class="">[<span class="">Section VI.1</span>, <a class="link-primary" href="/b/F">Lan12</a>]</span>. </i>Let $\omega \in \Lambda^{1}(TX)$, then </p><div class="display-equation">\[d\omega(\xi, \eta) = \omega([\xi, \eta]) - \eta\omega(\xi) - \xi\omega(\eta)\]</div><p>(1) $\Rightarrow$ (2): If $\omega$ vanishes on $E$ and $[\xi, \eta]$ lies in $E$, then $d\omega$ vanishes on $(\xi, \eta)$.</p><p>(2) $\Rightarrow$ (1): If $[\xi, \eta]$ does not lie in $E$, then there exists $\omega \in \Lambda^{1}(TX)$ that vanishes on $E$ but not $[\xi, \eta]$, which contradicts (2).</p><p>(1) $\Leftrightarrow$ (3): For any point on $X$, since $E$ is a subbundle, there exists Banach spaces $F, G$ and a coordinate neighbourhood $U \times V$ such that $E$ is given by a $C^{p-1}$-mapping </p><div class="display-equation">\[f: U \times V \times F \to U \times V \times (F \times G)\]</div><p>such that $\pi_{1}f(x, y)(z) = z$ for all $(x, y) \in U \times V$ and $z \in F$. Let </p><div class="display-equation">\[g: U \times V \to L(F; G) \quad g(x, y)(z) = \pi_{2}f(x, y, z)\]</div><p>then $g$ is also a $C^{p - 1}$-mapping. Let </p><div class="display-equation">\[\Xi: U \times V \to F \times G\]</div><p>be the local representation of a vector field, then the vector field lies in $E$ if and only if </p><div class="display-equation">\[\pi_{2}\Xi(x, y) = g(x, y)\pi_{1}\Xi(x, y)\]</div><p>for all $(x, y) \im U \times V$. Let $H: U \times V \to F \times G$ be another vector field, then by assumption (1), $[\Xi, H]$ lies in $E$. Denote $\xi = \pi_{1}\Xi$ and $\eta = \pi_{1} H$, then </p><div class="display-equation" id="">\begin{align*}[\xi, \eta]&#x26;= D\eta \cdot \xi - D\xi \cdot \eta \\ g \cdot (D\eta \cdot \xi - D\xi \cdot \eta)&#x26;= (Dg \cdot \xi) \cdot \eta + g \cdot D\eta \cdot \xi - (Dg \cdot \eta) \cdot \xi - g \cdot D\eta \cdot \xi \\ (Dg \cdot \xi) \cdot \eta&#x26;= (Dg \cdot \eta) \cdot \xi\end{align*}</div><p>Let $(x_{0}, y_{0}) \in U \times V$, then by <a href="/t/N" class="link-primary">Theorem 2.1.3</a>, there exists $U_{0} \in \cn_{F}(x_{0})$ and $V_{0} \in \cn_{G}(y_{0})$ and $\alpha \in C^{p-1}(U_{0} \times V_{0}; V)$ such that </p><div class="display-equation">\[\partial_{x}\alpha(x, y) = g(x, \alpha(x, y))\]</div><p>Let </p><div class="display-equation">\[\varphi: U_{0} \times V_{0} \to U \times V \quad (x, y) \mapsto (x, \alpha(x, y))\]</div><p>then </p><div class="display-equation">\[D\varphi(x_{0}, y_{0}) = \begin{bmatrix}Id&#x26;0 \\ g(x, \alpha(x, y))&#x26;Id\end{bmatrix}\]</div><p>so $\varphi$ is a local diffeomorphism at $(x_{0}, y_{0})$. Since for any $(u, v) \in F \times G$, </p><div class="display-equation">\[\partial_{x}\varphi(x, y) \cdot (u, v) = (u, g(x, \alpha(x, y)) \cdot u)\]</div><p>the bundle $E$ is integrable.<span class="qed">$\square$</span></p></div>

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Frobenius Let X be a C p ( p 2 ) manifold and E T X be a subbundle , then the following are equivalent : (1) For each pair of vector fields , : X E , [ , ] lies in E . (2) For each 1 ( T X ) vanishing on E , , : X E , d ( , ) = 0 . (3) E is integrable . Frobenius Let 1 ( T X ) , then d ( , ) = ( [ , ] ) - ( ) - ( ) ( 1 ) ( 2 ) : If vanishes on E and [ , ] lies in E , then d vanishes on ( , ) . ( 2 ) ( 1 ) : If [ , ] does not lie in E , then there exists 1 ( T X ) that vanishes on E but not [ , ] , which contradicts ( 2 ) . ( 1 ) ( 3 ) : For any point on X , since E is a subbundle , there exists Banach spaces F , G and a coordinate neighbourhood U V such that E is given by a C p - 1 - mapping f : U V F U V ( F G ) such that 1 f ( x , y ) ( z ) = z for all ( x , y ) U V and z F . Let g : U V L ( F ; G ) g ( x , y ) ( z ) = 2 f ( x , y , z ) then g is also a C p - 1 - mapping . Let : U V F G be the local representation of a vector field , then the vector field lies in E if and only if 2 ( x , y ) = g ( x , y ) 1 ( x , y ) for all ( x , y ) U V . Let H : U V F G be another vector field , then by assumption ( 1 ) , [ , H ] lies in E . Denote = 1 and = 1 H , then [ , ] & = D - D g ( D - D ) & = ( D g ) + g D - ( D g ) - g D ( D g ) & = ( D g ) Let ( x 0 , y 0 ) U V , then by theorem : local - frobenius , there exists U 0 F ( x 0 ) and V 0 G ( y 0 ) and C p - 1 ( U 0 V 0 ; V ) such that x ( x , y ) = g ( x , ( x , y ) ) Let : U 0 V 0 U V ( x , y ) ( x , ( x , y ) ) then D ( x 0 , y 0 ) = I d & 0 g ( x , ( x , y ) ) & I d so is a local diffeomorphism at ( x 0 , y 0 ) . Since for any ( u , v ) F G , x ( x , y ) ( u , v ) = ( u , g ( x , ( x , y ) ) u ) the bundle E is integrable . Proof , Section VI .1 Lang .

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<div class="block-container"><p><a class="link-foreground" href="/t/N"><strong>Theorem 2.1.3.</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;theorem:local-frobenius&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $E, F$ be Banach spaces, $U \subset E$ and $V \subset F$ be open subsets, and $f \in C^{k}(U \times V; L(E; F))$ ($k \ge 2$).</p><p>For any $\xi, \eta \in C^{k}(U \times V; E)$, denote </p><div class="display-equation">\[\Xi(x, y) = (\xi(x, y), f(x, y) \cdot \xi(x, y)) \quad H(x, y) = (\eta(x, y), f(x, y) \cdot \eta(x, y))\]</div><p>If for every $\xi, \eta \in C^{k}(U \times V; E)$ and $(x, y) \in U \times V$, </p><div class="display-equation">\[[Df(x, y) \cdot \Xi(x, y)] \cdot \eta(x, y) = [Df(x, y) \cdot H(x, y)] \cdot \xi(x, y)\]</div><p>then for every $(x_{0}, y_{0}) \in U \times V$, there exists $U_{0} \in \cn_{E}(x_{0})$, $V_{0} \in \cn_{F}(y_{0})$, and a unique $\alpha \in C^{k}(U_{0} \times V_{0}; V)$ such that</p><ol class="enumerate"><li style="list-style-type: &#x27;(1) &#x27;"><p>For every $y \in V_{0}$, $\alpha(x_{0}, y) = y$.</p></li><li style="list-style-type: &#x27;(2) &#x27;"><p>For every $(x, y) \in U_{0} \times V_{0}$,</p><div class="display-equation">\[(\partial_{x} \alpha)(x, y) = f(x, \alpha(x, y))\]</div></li></ol></div><div class="proof"><p><i class="">Proof. </i>Using translation, assume without loss of generality that $x_{0} = 0$ and $y_{0} = 0$. Let $B \in \cn_{E}(0)$ and </p><div class="display-equation">\[g: (-\eps, \eps) \times B \times V \to F \quad (t, z, y) \mapsto f(tz, y) \cdot z\]</div><p>then by <a href="/t/K" class="link-primary">Lemma 2.1.2</a>, there exists $\delta > 0$, $B_{0} \in \cn_{E}(0)$, and $\beta: (-\delta, \delta) \times B_{0} \times V_{0}$ such that</p><ol class="enumerate"><li style="list-style-type: &#x27;(a) &#x27;"><p>For each $(z, y) \in B_{0} \times V_{0}$, $\beta(0, z, y) = y$.</p></li><li style="list-style-type: &#x27;(b) &#x27;"><p>For each $(t, z, y) \in (-\delta, \delta) \times B_{0} \times V_{0}$,</p><div class="display-equation">\[\frac{d}{dt}\beta(t, z, y) = f(tz, \beta(t, z, y)) \cdot z\]</div></li><li style="list-style-type: &#x27;(c) &#x27;"><p>For each fixed $y \in V_{0}$, if $\beta(t, z) = \beta(t, z, y)$, then for each $(t, x) \in (-\delta, \delta) \times B_{0}$ and $h \in E$,</p><div class="display-equation" id="">\begin{align*}(\partial_{t}\partial_{z}\beta)(t, z)\cdot h&#x26;= t(\partial_{x}f)(tz, \beta(t, z))\cdot h \cdot z + f(tz, \beta(t, z))\cdot h \\&#x26;+ (\partial_{y}f)(tz, \beta(t, z)) \circ (\partial_{z} \beta)(t, z)\cdot h \cdot z\end{align*}</div></li></ol><p>Following (c), let $k(t, z) = (\partial_{z}\beta)(t, z)\cdot h - tf(tz, \beta(t, z))\cdot h$, then </p><div class="display-equation" id="">\begin{align*}\frac{d}{dt}k(t)&#x26;= (\partial_{t}\partial_{z}\beta)(t, z)\cdot h - f(tz, \beta)\cdot h \\&#x26;- t [(\partial_{x}f)(tz, \beta)]\cdot z \cdot h - t[(\partial_{y} f)(tz, \beta)]f(tz, \beta) \cdot z\cdot h\end{align*}</div><p>where by assumption, </p><div class="display-equation" id="">\begin{align*}&#x26;[(\partial_{x}f)(tz, \beta)]\cdot z \cdot h - [(\partial_{y} f)(tz, \beta)]f(tz, \beta) \cdot z \cdot h \\&#x26;= Df(tz, \beta) \cdot (z, f(tz, \beta) \cdot z) \cdot h \\&#x26;= Df(tz, \beta) \cdot (h, f(tz, \beta) \cdot h) \cdot z \\&#x26;= [(\partial_{x}f)(tz, \beta)]\cdot h \cdot z - [(\partial_{y} f)(tz, \beta)]f(tz, \beta) \cdot h\cdot z\end{align*}</div><p>Therefore by (c), </p><div class="display-equation" id="">\begin{align*}\frac{d}{dt}k(t)&#x26;= (\partial_{y}f)(tz, \beta) \circ (\partial_{z} \beta)(t, z)\cdot h \cdot z - t[(\partial_{y} f)(tz, \beta)]f(tz, \beta) \cdot h\cdot z \\&#x26;= (\partial_{y} f)(tz, \beta)[\partial_{z} \beta(t, z) - tf(tz, \beta)] \cdot h \cdot z \\&#x26;= (\partial_{y} f)(tz, \beta) \cdot k(t) \cdot z\end{align*}</div><p>Since $0$ is a solution to the above equation, $k(t) = 0$ by the uniqueness of solutions to ODEs. Hence for every $t \in (-\delta, \delta)$ and $z \in B_{0}$, </p><div class="display-equation">\[\partial_{z} \beta(t, z) = tf(tz, \beta(t, z))\]</div><p>By adjusting $\delta$ and $B_{0}$, assume without loss of generality that $\delta > 1$. In which case, if $\alpha(x) = \beta(1, x, y)$, then </p><div class="display-equation">\[\partial_{x}\alpha(x, y) = \partial_{z}\beta(1, x, y) = f(z, \beta(1, x, y)) = f(x, \alpha(x, y))\]</div><p><span class="qed">$\square$</span></p></div>

contentText:
Let E , F be Banach spaces , U E and V F be open subsets , and f C k ( U V ; L ( E ; F ) ) ( k 2 ) . For any , C k ( U V ; E ) , denote ( x , y ) = ( ( x , y ) , f ( x , y ) ( x , y ) ) H ( x , y ) = ( ( x , y ) , f ( x , y ) ( x , y ) ) If for every , C k ( U V ; E ) and ( x , y ) U V , [ D f ( x , y ) ( x , y ) ] ( x , y ) = [ D f ( x , y ) H ( x , y ) ] ( x , y ) then for every ( x 0 , y 0 ) U V , there exists U 0 E ( x 0 ) , V 0 F ( y 0 ) , and a unique C k ( U 0 V 0 ; V ) such that (1) For every y V 0 , ( x 0 , y ) = y . (2) For every ( x , y ) U 0 V 0 , ( x ) ( x , y ) = f ( x , ( x , y ) ) Using translation , assume without loss of generality that x 0 = 0 and y 0 = 0 . Let B E ( 0 ) and g : ( - , ) B V F ( t , z , y ) f ( t z , y ) z then by lemma : split - ode , there exists > 0 , B 0 E ( 0 ) , and : ( - , ) B 0 V 0 such that ( a ) For each ( z , y ) B 0 V 0 , ( 0 , z , y ) = y . ( b ) For each ( t , z , y ) ( - , ) B 0 V 0 , d d t ( t , z , y ) = f ( t z , ( t , z , y ) ) z ( c ) For each fixed y V 0 , if ( t , z ) = ( t , z , y ) , then for each ( t , x ) ( - , ) B 0 and h E , ( t z ) ( t , z ) h & = t ( x f ) ( t z , ( t , z ) ) h z + f ( t z , ( t , z ) ) h & + ( y f ) ( t z , ( t , z ) ) ( z ) ( t , z ) h z Following ( c ) , let k ( t , z ) = ( z ) ( t , z ) h - t f ( t z , ( t , z ) ) h , then d d t k ( t ) & = ( t z ) ( t , z ) h - f ( t z , ) h & - t [ ( x f ) ( t z , ) ] z h - t [ ( y f ) ( t z , ) ] f ( t z , ) z h where by assumption , & [ ( x f ) ( t z , ) ] z h - [ ( y f ) ( t z , ) ] f ( t z , ) z h & = D f ( t z , ) ( z , f ( t z , ) z ) h & = D f ( t z , ) ( h , f ( t z , ) h ) z & = [ ( x f ) ( t z , ) ] h z - [ ( y f ) ( t z , ) ] f ( t z , ) h z Therefore by ( c ) , d d t k ( t ) & = ( y f ) ( t z , ) ( z ) ( t , z ) h z - t [ ( y f ) ( t z , ) ] f ( t z , ) h z & = ( y f ) ( t z , ) [ z ( t , z ) - t f ( t z , ) ] h z & = ( y f ) ( t z , ) k ( t ) z Since 0 is a solution to the above equation , k ( t ) = 0 by the uniqueness of solutions to ODEs . Hence for every t ( - , ) and z B 0 , z ( t , z ) = t f ( t z , ( t , z ) ) By adjusting and B 0 , assume without loss of generality that > 1 . In which case , if ( x ) = ( 1 , x , y ) , then x ( x , y ) = z ( 1 , x , y ) = f ( z , ( 1 , x , y ) ) = f ( x , ( x , y ) )

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<div class="block-container"><p><a class="link-foreground" href="/t/R"><strong>Definition 3.1.1 (Previsible $\sigma$-Algebra).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;definition:previsible-sigma-algebra&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $(\Omega, \cf)$ be a measurable space and $\bracs{\cf_t|t \ge 0}$ be a filtration on $\Omega$, then the <b class="textbf">previsible $\sigma$-algebra</b> $\mathscr{J}_{\omega}$ on $(0, \infty) \times \Omega$ associated with $\bracs{\mathcal{F}_t}$ is the $\sigma$-algebra generated by </p><div class="display-equation">\[\bracs{(s, t] \times A| 0 \le s &#x3C; t &#x3C; \infty, A \in \cf_s}\]</div><p>In other words, it is the smallest $\sigma$-algebra on $(0, \infty) \times \Omega$ on which every $\bracs{\mathcal{F}_t}$-adapted process with left-continuous sample paths is measurable.</p></div><div class="block-container"><p><a class="link-foreground" href="/t/T"><strong>Definition 3.1.2 (Previsible Path Functional).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;definition:previsible-path-functional&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $C([0, \infty); \real^{d})$ be the space of $\real^{d}$-valued continuous functions on $[0, \infty)$, equipped with the topology of uniform convergence. For each $t \ge 0$, let </p><div class="display-equation">\[\mathscr{X}_{t} = \sigma(\bracs{\pi_s|s \le t})\]</div><p>and $\mathscr{J}_{C([0, \infty); \real^d)}$ be the previsible $\sigma$-algebra on $(0, \infty) \times C([0, \infty); \real^{d})$, then a <b class="textbf">previsible path functional</b> is a $\mathscr{J}$-measurable mapping on $(0, \infty) \times C([0, \infty); \real^{d})$.</p></div><div class="block-container"><p><a class="link-foreground" href="/t/U"><strong>Lemma 3.1.3.</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;lemma:adapted-composition&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $(\Omega, \bracs{\cf_t})$ be a filtered space and $X: \Omega \to C([0, \infty); \real^{d})$ be a $\bracs{\mathcal{F}_t}$-adapted process with continuous sample paths, then</p><ol class="enumerate"><li style="list-style-type: &#x27;(1) &#x27;"><p>$X$ is $(\mathscr{J}_{\Omega}, \mathscr{J}_{C([0, \infty); \real^d)})$-measurable.</p></li><li style="list-style-type: &#x27;(2) &#x27;"><p>For any previsible path functional $\alpha: (0, \infty) \times C([0, \infty); \real^{d})$, $\alpha(t, X)$ is $\bracs{\mathcal{F}_t}$-previsible.</p></li></ol></div><div class="block-container"><p><a class="link-foreground" href="/t/V"><strong>Definition 3.1.4 (Diffusion Type SDE).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;definition:diffusion-sde&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $\sigma: \real^{d} \to L(\real^{d}; \real^{n})$ and $b: \real^{n} \to \real^{n}$ be measurable functions, then a <b class="textbf">SDE of diffusion type</b> is the equation </p><div class="display-equation">\[X_{t} = \xi + \int_{0}^{t} \sigma(X_{s}) dB_{s} + \int_{0}^{t} b(X_{s})ds\]</div><p>under the constraint </p><div class="display-equation" id="X">\begin{equation}\int_{0}^{t} \norm{\sigma(X_s)}_{\real^n}^{2} + \norm{b(X_s)}_{\real^n}ds &#x3C; \infty \tag{3.1}\end{equation}</div><p>for all $t > 0$, where</p><ul class="itemize"><li><p>$B$ is a standard Brownian motion on a filtered probability space $(\Omega, \bracs{\cf_t}, \bp)$.</p></li><li><p>$\xi$ is a $\cf_{0}$-measurable random variable.</p></li></ul></div><div class="block-container"><p><a class="link-foreground" href="/t/W"><strong>Definition 3.1.5 (Pathwise Uniqueness).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;definition:pathwise-uniqueness&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $\sigma: \real^{d} \to L(\real^{d}; \real^{n})$ and $b: \real^{n} \to \real^{n}$ be measurable functions, then the diffusion SDE </p><div class="display-equation" id="Y">\begin{equation}X_{t} = \xi + \int_{0}^{t} \sigma(X_{s}) dB_{s} + \int_{0}^{t} b(X_{s})ds \tag{3.2}\end{equation}</div><p>has <b class="textbf">pathwise uniqueness</b> if given</p><ul class="itemize"><li><p>A filtered probability space $(\Omega, \bracs{\cf_t}, \bp)$,</p></li><li><p>$\bracs{\mathcal{F}_t}$-adapted standard Brownian motion $B$,</p></li><li><p>Continuous $\bracs{\mathcal{F}_t}$-semimartingales $X, Y: \Omega \to C([0, \infty); \real^{d})$ satisfying <a href="/t/W#Y" class="link-primary" onclick="
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        " targetTag="Y">Equation 3.2</a> and <a href="/t/V#X" class="link-primary" onclick="
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        " targetTag="X">Equation 3.1</a>,</p></li></ul><p>then $X = Y$ almost surely.</p></div><div class="block-container"><p><a class="link-foreground" href="/t/10"><strong>Definition 3.1.6 (Pathwise Exact).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;definition:pathwise-exact&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $\sigma: \real^{d} \to L(\real^{d}; \real^{n})$ and $b: \real^{n} \to \real^{n}$ be measurable functions, then the diffusion SDE </p><div class="display-equation" id="">\begin{equation}X_{t} = \xi + \int_{0}^{t} \sigma(X_{s}) dB_{s} + \int_{0}^{t} b(X_{s})ds \end{equation}</div><p>is <b class="textbf">pathwise exact</b> if given</p><ul class="itemize"><li><p>A filtered probability space $(\Omega, \bracs{\cf_t}, \bp)$,</p></li><li><p>$\bracs{\mathcal{F}_t}$-adapted standard Brownian motion $B$,</p></li><li><p>$\bracs{\mathcal{F}_t}$-semimartingales $X, Y: \Omega \to C([0, \infty); \real^{d})$ satisfying <a href="/t/W#Y" class="link-primary" onclick="
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Definitions Let ( , ) be a measurable space and _t|t 0 be a filtration on , then the previsible - algebra J on ( 0 , ) associated with F _t is the - algebra generated by ( s , t ] A| 0 s < t < , A _s In other words , it is the smallest - algebra on ( 0 , ) on which every F _t - adapted process with left - continuous sample paths is measurable . Previsible - Algebra Let C ( [ 0 , ) ; d ) be the space of d - valued continuous functions on [ 0 , ) , equipped with the topology of uniform convergence . For each t 0 , let X t = ( _s|s t ) and J C ( [ 0 , ) ; ^d ) be the previsible - algebra on ( 0 , ) C ( [ 0 , ) ; d ) , then a previsible path functional is a J - measurable mapping on ( 0 , ) C ( [ 0 , ) ; d ) . Previsible Path Functional Let ( , _t ) be a filtered space and X : C ( [ 0 , ) ; d ) be a F _t - adapted process with continuous sample paths , then (1) X is ( J , J C ( [ 0 , ) ; ^d ) ) - measurable . (2) For any previsible path functional : ( 0 , ) C ( [ 0 , ) ; d ) , ( t , X ) is F _t - previsible . Let : d L ( d ; n ) and b : n n be measurable functions , then a SDE of diffusion type is the equation X t = + 0 t ( X s ) d B s + 0 t b ( X s ) d s under the constraint 0 t ( X_s ) ^n 2 + b ( X_s ) ^n d s < for all t > 0 , where B is a standard Brownian motion on a filtered probability space ( , _t , ) . is a 0 - measurable random variable . Diffusion Type SDE Let : d L ( d ; n ) and b : n n be measurable functions , then the diffusion SDE X t = + 0 t ( X s ) d B s + 0 t b ( X s ) d s has pathwise uniqueness if given A filtered probability space ( , _t , ) , F _t - adapted standard Brownian motion B , Continuous F _t - semimartingales X , Y : C ( [ 0 , ) ; d ) satisfying equation : diffusion - sde and equation : diffusion - constraint , then X = Y almost surely . Pathwise Uniqueness Let : d L ( d ; n ) and b : n n be measurable functions , then the diffusion SDE X t = + 0 t ( X s ) d B s + 0 t b ( X s ) d s is pathwise exact if given A filtered probability space ( , _t , ) , F _t - adapted standard Brownian motion B , F _t - semimartingales X , Y : C ( [ 0 , ) ; d ) satisfying equation : diffusion - sde and equation : diffusion - constraint , then for every t 0 , X t = Y t almost surely . Pathwise Exact

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<div class="block-container"><p><a class="link-foreground" href="/t/R"><strong>Definition 3.1.1 (Previsible $\sigma$-Algebra).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;definition:previsible-sigma-algebra&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $(\Omega, \cf)$ be a measurable space and $\bracs{\cf_t|t \ge 0}$ be a filtration on $\Omega$, then the <b class="textbf">previsible $\sigma$-algebra</b> $\mathscr{J}_{\omega}$ on $(0, \infty) \times \Omega$ associated with $\bracs{\mathcal{F}_t}$ is the $\sigma$-algebra generated by </p><div class="display-equation">\[\bracs{(s, t] \times A| 0 \le s &#x3C; t &#x3C; \infty, A \in \cf_s}\]</div><p>In other words, it is the smallest $\sigma$-algebra on $(0, \infty) \times \Omega$ on which every $\bracs{\mathcal{F}_t}$-adapted process with left-continuous sample paths is measurable.</p></div>

contentText:
Previsible - Algebra Let ( , ) be a measurable space and _t|t 0 be a filtration on , then the previsible - algebra J on ( 0 , ) associated with F _t is the - algebra generated by ( s , t ] A| 0 s < t < , A _s In other words , it is the smallest - algebra on ( 0 , ) on which every F _t - adapted process with left - continuous sample paths is measurable . Previsible - Algebra

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<div class="block-container"><p><a class="link-foreground" href="/t/T"><strong>Definition 3.1.2 (Previsible Path Functional).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;definition:previsible-path-functional&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $C([0, \infty); \real^{d})$ be the space of $\real^{d}$-valued continuous functions on $[0, \infty)$, equipped with the topology of uniform convergence. For each $t \ge 0$, let </p><div class="display-equation">\[\mathscr{X}_{t} = \sigma(\bracs{\pi_s|s \le t})\]</div><p>and $\mathscr{J}_{C([0, \infty); \real^d)}$ be the previsible $\sigma$-algebra on $(0, \infty) \times C([0, \infty); \real^{d})$, then a <b class="textbf">previsible path functional</b> is a $\mathscr{J}$-measurable mapping on $(0, \infty) \times C([0, \infty); \real^{d})$.</p></div>

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Previsible Path Functional Let C ( [ 0 , ) ; d ) be the space of d - valued continuous functions on [ 0 , ) , equipped with the topology of uniform convergence . For each t 0 , let X t = ( _s|s t ) and J C ( [ 0 , ) ; ^d ) be the previsible - algebra on ( 0 , ) C ( [ 0 , ) ; d ) , then a previsible path functional is a J - measurable mapping on ( 0 , ) C ( [ 0 , ) ; d ) . Previsible Path Functional

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<div class="block-container"><p><a class="link-foreground" href="/t/U"><strong>Lemma 3.1.3.</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;lemma:adapted-composition&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $(\Omega, \bracs{\cf_t})$ be a filtered space and $X: \Omega \to C([0, \infty); \real^{d})$ be a $\bracs{\mathcal{F}_t}$-adapted process with continuous sample paths, then</p><ol class="enumerate"><li style="list-style-type: &#x27;(1) &#x27;"><p>$X$ is $(\mathscr{J}_{\Omega}, \mathscr{J}_{C([0, \infty); \real^d)})$-measurable.</p></li><li style="list-style-type: &#x27;(2) &#x27;"><p>For any previsible path functional $\alpha: (0, \infty) \times C([0, \infty); \real^{d})$, $\alpha(t, X)$ is $\bracs{\mathcal{F}_t}$-previsible.</p></li></ol></div>

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Let ( , _t ) be a filtered space and X : C ( [ 0 , ) ; d ) be a F _t - adapted process with continuous sample paths , then (1) X is ( J , J C ( [ 0 , ) ; ^d ) ) - measurable . (2) For any previsible path functional : ( 0 , ) C ( [ 0 , ) ; d ) , ( t , X ) is F _t - previsible .

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<div class="block-container"><p><a class="link-foreground" href="/t/V"><strong>Definition 3.1.4 (Diffusion Type SDE).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;definition:diffusion-sde&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $\sigma: \real^{d} \to L(\real^{d}; \real^{n})$ and $b: \real^{n} \to \real^{n}$ be measurable functions, then a <b class="textbf">SDE of diffusion type</b> is the equation </p><div class="display-equation">\[X_{t} = \xi + \int_{0}^{t} \sigma(X_{s}) dB_{s} + \int_{0}^{t} b(X_{s})ds\]</div><p>under the constraint </p><div class="display-equation" id="X">\begin{equation}\int_{0}^{t} \norm{\sigma(X_s)}_{\real^n}^{2} + \norm{b(X_s)}_{\real^n}ds &#x3C; \infty \tag{3.1}\end{equation}</div><p>for all $t > 0$, where</p><ul class="itemize"><li><p>$B$ is a standard Brownian motion on a filtered probability space $(\Omega, \bracs{\cf_t}, \bp)$.</p></li><li><p>$\xi$ is a $\cf_{0}$-measurable random variable.</p></li></ul></div>

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Diffusion Type SDE Let : d L ( d ; n ) and b : n n be measurable functions , then a SDE of diffusion type is the equation X t = + 0 t ( X s ) d B s + 0 t b ( X s ) d s under the constraint 0 t ( X_s ) ^n 2 + b ( X_s ) ^n d s < for all t > 0 , where B is a standard Brownian motion on a filtered probability space ( , _t , ) . is a 0 - measurable random variable . Diffusion Type SDE


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<div class="block-container"><p><a class="link-foreground" href="/t/W"><strong>Definition 3.1.5 (Pathwise Uniqueness).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;definition:pathwise-uniqueness&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $\sigma: \real^{d} \to L(\real^{d}; \real^{n})$ and $b: \real^{n} \to \real^{n}$ be measurable functions, then the diffusion SDE </p><div class="display-equation" id="Y">\begin{equation}X_{t} = \xi + \int_{0}^{t} \sigma(X_{s}) dB_{s} + \int_{0}^{t} b(X_{s})ds \tag{3.2}\end{equation}</div><p>has <b class="textbf">pathwise uniqueness</b> if given</p><ul class="itemize"><li><p>A filtered probability space $(\Omega, \bracs{\cf_t}, \bp)$,</p></li><li><p>$\bracs{\mathcal{F}_t}$-adapted standard Brownian motion $B$,</p></li><li><p>Continuous $\bracs{\mathcal{F}_t}$-semimartingales $X, Y: \Omega \to C([0, \infty); \real^{d})$ satisfying <a href="/t/W#Y" class="link-primary" onclick="
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Pathwise Uniqueness Let : d L ( d ; n ) and b : n n be measurable functions , then the diffusion SDE X t = + 0 t ( X s ) d B s + 0 t b ( X s ) d s has pathwise uniqueness if given A filtered probability space ( , _t , ) , F _t - adapted standard Brownian motion B , Continuous F _t - semimartingales X , Y : C ( [ 0 , ) ; d ) satisfying equation : diffusion - sde and equation : diffusion - constraint , then X = Y almost surely . Pathwise Uniqueness

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<div class="display-equation" id="X">\begin{equation}\int_{0}^{t} \norm{\sigma(X_s)}_{\real^n}^{2} + \norm{b(X_s)}_{\real^n}ds &#x3C; \infty \tag{3.1}\end{equation}</div>

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<div class="display-equation" id="Y">\begin{equation}X_{t} = \xi + \int_{0}^{t} \sigma(X_{s}) dB_{s} + \int_{0}^{t} b(X_{s})ds \tag{3.2}\end{equation}</div>

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<div class="block-container"><p><a class="link-foreground" href="/t/10"><strong>Definition 3.1.6 (Pathwise Exact).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;definition:pathwise-exact&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $\sigma: \real^{d} \to L(\real^{d}; \real^{n})$ and $b: \real^{n} \to \real^{n}$ be measurable functions, then the diffusion SDE </p><div class="display-equation" id="">\begin{equation}X_{t} = \xi + \int_{0}^{t} \sigma(X_{s}) dB_{s} + \int_{0}^{t} b(X_{s})ds \end{equation}</div><p>is <b class="textbf">pathwise exact</b> if given</p><ul class="itemize"><li><p>A filtered probability space $(\Omega, \bracs{\cf_t}, \bp)$,</p></li><li><p>$\bracs{\mathcal{F}_t}$-adapted standard Brownian motion $B$,</p></li><li><p>$\bracs{\mathcal{F}_t}$-semimartingales $X, Y: \Omega \to C([0, \infty); \real^{d})$ satisfying <a href="/t/W#Y" class="link-primary" onclick="
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Pathwise Exact Let : d L ( d ; n ) and b : n n be measurable functions , then the diffusion SDE X t = + 0 t ( X s ) d B s + 0 t b ( X s ) d s is pathwise exact if given A filtered probability space ( , _t , ) , F _t - adapted standard Brownian motion B , F _t - semimartingales X , Y : C ( [ 0 , ) ; d ) satisfying equation : diffusion - sde and equation : diffusion - constraint , then for every t 0 , X t = Y t almost surely . Pathwise Exact

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<div class="block-container"><p><a class="link-foreground" href="/t/12"><strong>Definition 3.2.1 (Lipschitz Coefficient).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;definition:lipschitz-coefficient&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $\sigma: [0, \infty) \times C([0, \infty); \real^{d}) \to \real^{n}$, then $\sigma$ is <b class="textbf">Lipschitz</b> if there exists $C \ge 0$ such that for any $\theta, \eta \in C([0, \infty); \real^{d})$, </p><div class="display-equation">\[\norm{\sigma(t, \theta) - \sigma(t, \eta)}_{\real^n}\le C\norm{\theta - \eta}_{u, [0, t]}\]</div></div><div class="block-container"><p><a class="link-foreground" href="/t/13"><strong>Lemma 3.2.2.</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;lemma:exact-uniform-bound&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, $B$ be a $\bracs{\mathcal{F}_t}$-adapted standard Brownian motion, $\sigma$ be a $\bracs{\mathcal{F}_t}$-previsible $L(\real^{d}; \real^{n})$-valued process, $b$ be a $\bracs{\mathcal{F}_t}$-previsible $\real^{n}$-valued process, $\xi \in L^{p}(\Omega, \cf_{0}; \real^{n})$, and </p><div class="display-equation">\[X_{t} = \xi + \int_{0}^{t} \sigma_{s} dB_{s} + \int_{0}^{t} b_{s} ds\]</div><p>then for any $T > 0$ and $p \ge 2$, there exists $C_{T, p, n}\ge 0$ such that for all $0 \le t \le T$, </p><div class="display-equation">\[\ev\braks{\norm{X}_{u, [0, t]}^p}\le C_{T, p, n}\braks{\ev(\norm{\xi}_{\real^n}^p) + \ev\paren{\int_0^t \norm{\sigma_s}_{\real^n}^p + \norm{b_s}_{\real^n}^p ds }}\]</div></div><div class="proof"><p><i class="">Proof, <span class="">[<span class="">Theorem 11.5</span>, <a class="link-primary" href="/b/Z">RW89</a>]</span>. </i>Assume without loss of generality that $\xi = 0$, then </p><div class="display-equation">\[\norm{X}_{u, [0, t]}^{p} \le C_{p}\paren{\sup_{0 \le s \le t}\int_0^s \sigma_rdB_r}^{p} + \paren{\int_0^t \norm{b_s}_{\real^n}ds}^{p}\]</div><p>where by Jensen’s inequality, </p><div class="display-equation">\[\paren{\int_0^t \norm{b_s}ds}^{p} \le C_{t, p}\int_{0}^{t} \norm{b_s}_{\real^n}^{p} ds\]</div><p>and by the BDG inequality and Jensen’s inequality, </p><div class="display-equation">\[\ev\braks{\paren{\sup_{0 \le s \le t}\int_0^s \sigma_rdB_r}^p}\le C_{p} \ev\braks{\paren{\int_0^t \norm{\sigma_s}_{\real^d}^2 ds}^{p/2}}\le C_{p} \int_{0}^{t} \norm{\sigma_s}_{\real^d}^{p} ds\]</div><p><span class="qed">$\square$</span></p></div><div class="block-container"><p><a class="link-foreground" href="/t/14"><strong>Lemma 3.2.3.</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;lemma:lipschitz-picard&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $\sigma: [0, \infty) \times C([0, \infty); \real^{d}) \to L(\real^{d}; \real^{n})$ and $b: [0, \infty) \times C([0, \infty); \real^{d}) \to \real^{n}$ be previsible path functionals satisfying the <a href="/t/12" class="link-primary">Lipschitz condition</a> with constant $K$.</p><p>Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space and $B$ be a $\bracs{\mathcal{F}_t}$-adapted standard Brownian motion. For any $\bracs{\mathcal{F}_t}$-adapted process $X: \Omega \to C([0, \infty); \real^{d})$ with continuous sample paths and $\xi \in L^{p}(\Omega, \cf_{0}; \real^{n})$, let </p><div class="display-equation">\[P(X, \xi)_{t} = \xi + \int_{0}^{t} \sigma(s, X)dB_{s} + \int_{0}^{t} b(s, X)ds\]</div><p>then for any $\bracs{\mathcal{F}_t}$-adapted process $Y: \Omega \to C([0, \infty); \real^{d})$, $\eta \in L^{p}(\Omega, \cf_{0}; \real^{n})$, $T > 0$, and $0 \le t \le T$, </p><div class="display-equation" id="">\begin{align*}&#x26;\ev\braks{\norm{P(X, \xi) - P(Y, \eta)}_{u, [0, t]}^p}\\&#x26;\le C_{K, n, T, p}\braks{\norm{\xi - \eta}_{L^p(\Omega; \real^n)}^p + \ev\paren{\int_0^t \norm{X - Y}_{u, [0, s]}^p ds}}\end{align*}</div></div><div class="block-container"><p><a class="link-foreground" href="/t/15"><strong>Theorem 3.2.4 (Itô).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;theorem:ito-existence-uniqueness&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $\sigma: [0, \infty) \times C([0, \infty); \real^{d}) \to L(\real^{d}; \real^{n})$ and $b: [0, \infty) \times C([0, \infty); \real^{d}) \to \real^{n}$ be previsible path functionals satisfying the <a href="/t/12" class="link-primary">Lipschitz condition</a>. If for each $T \ge 0$, </p><div class="display-equation">\[\sup_{0 \le s \le T}\norm{\sigma(s, 0)}_{L(\real^d; \real^n)}+ \norm{b(s, 0)}_{\real^n}&#x3C; \infty\]</div><p>then the SDE </p><div class="display-equation">\[X_{t} = \xi + \int_{0}^{t} \sigma(s, X)dB_{s} + \int_{0}^{t} b(s, X)ds\]</div><p>is exact and pathwise unique.</p></div><div class="proof"><p><i class="">Proof of existence, <span class="">[<span class="">Theorem 11.2</span>, <a class="link-primary" href="/b/Z">RW89</a>]</span>. </i>For each $t \ge 0$, let $\mathcal{G}_{t}^{\circ} = \sigma(\bracs{B_s|1 \le s \le t}\cup \bracs{\xi})$, $\mathcal{G}^{\circ} = \sigma(\bracsn{\mathcal{G}_t^\circ|t \ge 0})$, $\mathcal{N}$ be the collection of $\bp$-null sets in the completion of $\sigma(\bracsn{\mathcal{G}_t^\circ|t \ge 0})$, and $\mathcal{G}_{t} = \sigma(\mathcal{G}_{t}^{\circ} \cup \mathcal{N})$.</p><p>Let $\xi \in L^{\infty}(\Omega, \cf_{0}; \real^{n})$ and $X^{(0)}= \xi$. For each $m \in \natz$, define inductively </p><div class="display-equation">\[X^{(m+1)}= P(X^{(m)}, \xi)_{t} = \xi + \int_{0}^{t} \sigma(s, X^{(m)})dB_{s} + \int_{0}^{t} b(s, X^{(m)})ds\]</div><p>then $X^{(m)}$ has continuous sample paths, and for each $T > 0$, $\ev[{\normn{X^{(m)}}_{u, [0, s]}^2}] &#x3C; \infty$. By <a href="/t/14" class="link-primary">Lemma 3.2.3</a>, </p><div class="display-equation">\[\ev\braks{\norm{X^{(m+2)} - X^{(m+1)}}_{u, [0, T]}^2}\le C \int_{0}^{t} \ev\braks{\norm{X^{(m+1)} - X^{(m)}}_{u, [0, t]}^2}dt\]</div><p>Therefore </p><div class="display-equation">\[\ev\braks{\norm{X^{(m+1)} - X^{(m)}}_{u, [0, T]}^2}\le C_{0} C \frac{T^{n}}{n!}\]</div><p>and $X^{(m)}$ converges uniformly in $L^{2}$ to a limiting process $X$, which satisfies the given equation.<span class="qed">$\square$</span></p></div><div class="proof"><p><i class="">Proof of uniqueness. </i>Let $X$ and $Y$ be solutions of the SDE with the same setup, then by <a href="/t/14" class="link-primary">Lemma 3.2.3</a>, </p><div class="display-equation">\[\ev\braks{\norm{X - Y}_{u, [0, T]}^2}\le C \int_{0}^{t} \ev\braks{\norm{X - Y}_{u, [0, t]}^2}dt\]</div><p>which implies that $X|_{[0, T]}= Y|_{[0, T]}$ almost surely.<span class="qed">$\square$</span></p></div>

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Itô Existence and Uniqueness Let : [ 0 , ) C ( [ 0 , ) ; d ) n , then is Lipschitz if there exists C 0 such that for any , C ( [ 0 , ) ; d ) , ( t , ) - ( t , ) ^n C - u , [ 0 , t ] Lipschitz Coefficient Let ( , _t|t 0 , ) be a filtered probability space , B be a F _t - adapted standard Brownian motion , be a F _t - previsible L ( d ; n ) - valued process , b be a F _t - previsible n - valued process , L p ( , 0 ; n ) , and X t = + 0 t s d B s + 0 t b s d s then for any T > 0 and p 2 , there exists C T , p , n 0 such that for all 0 t T , X _ u , [ 0 , t ] ^p C T , p , n ( _ ^n ^p ) + _0^t _s _ ^n ^p + b_s _ ^n ^p ds Assume without loss of generality that = 0 , then X u , [ 0 , t ] p C p _ 0 s t _0^s _rdB_r p + _0^t b_s _ ^n ds p where by Jensen ' s inequality , _0^t b_s ds p C t , p 0 t b_s ^n p d s and by the BDG inequality and Jensen ' s inequality , _ 0 s t _0^s _rdB_r ^p C p _0^t _s _ ^d ^2 ds ^ p / 2 C p 0 t _s ^d p d s Proof , Theorem 11.5 Rogers . Let : [ 0 , ) C ( [ 0 , ) ; d ) L ( d ; n ) and b : [ 0 , ) C ( [ 0 , ) ; d ) n be previsible path functionals satisfying the Lipschitz condition definition : lipschitz - coefficient with constant K . Let ( , _t|t 0 , ) be a filtered probability space and B be a F _t - adapted standard Brownian motion . For any F _t - adapted process X : C ( [ 0 , ) ; d ) with continuous sample paths and L p ( , 0 ; n ) , let P ( X , ) t = + 0 t ( s , X ) d B s + 0 t b ( s , X ) d s then for any F _t - adapted process Y : C ( [ 0 , ) ; d ) , L p ( , 0 ; n ) , T > 0 , and 0 t T , & P ( X , ) - P ( Y , ) _ u , [ 0 , t ] ^p & C K , n , T , p - _ L^p ( ; ^n ) ^p + _0^t X - Y _ u , [ 0 , s ] ^p ds Let : [ 0 , ) C ( [ 0 , ) ; d ) L ( d ; n ) and b : [ 0 , ) C ( [ 0 , ) ; d ) n be previsible path functionals satisfying the Lipschitz condition definition : lipschitz - coefficient . If for each T 0 , 0 s T ( s , 0 ) L ( ^d ; ^n ) + b ( s , 0 ) ^n < then the SDE X t = + 0 t ( s , X ) d B s + 0 t b ( s , X ) d s is exact and pathwise unique . Itô For each t 0 , let G t = ( B_s|1 s t ) , G = ( G _t^ |t 0 ) , N be the collection of - null sets in the completion of ( G _t^ |t 0 ) , and G t = ( G t N ) . Let L ( , 0 ; n ) and X ( 0 ) = . For each m , define inductively X ( m + 1 ) = P ( X ( m ) , ) t = + 0 t ( s , X ( m ) ) d B s + 0 t b ( s , X ( m ) ) d s then X ( m ) has continuous sample paths , and for each T > 0 , [ X^ ( m ) _ u , [ 0 , s ] ^2 ] < . By lemma : lipschitz - picard , X^ ( m + 2 ) - X^ ( m + 1 ) _ u , [ 0 , T ] ^2 C 0 t X^ ( m + 1 ) - X^ ( m ) _ u , [ 0 , t ] ^2 d t Therefore X^ ( m + 1 ) - X^ ( m ) _ u , [ 0 , T ] ^2 C 0 C T n n ! and X ( m ) converges uniformly in L 2 to a limiting process X , which satisfies the given equation . Proof of existence , Theorem 11.2 Rogers . Let X and Y be solutions of the SDE with the same setup , then by lemma : lipschitz - picard , X - Y _ u , [ 0 , T ] ^2 C 0 t X - Y _ u , [ 0 , t ] ^2 d t which implies that X | [ 0 , T ] = Y | [ 0 , T ] almost surely . Proof of uniqueness .

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<div class="block-container"><p><a class="link-foreground" href="/t/12"><strong>Definition 3.2.1 (Lipschitz Coefficient).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;definition:lipschitz-coefficient&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $\sigma: [0, \infty) \times C([0, \infty); \real^{d}) \to \real^{n}$, then $\sigma$ is <b class="textbf">Lipschitz</b> if there exists $C \ge 0$ such that for any $\theta, \eta \in C([0, \infty); \real^{d})$, </p><div class="display-equation">\[\norm{\sigma(t, \theta) - \sigma(t, \eta)}_{\real^n}\le C\norm{\theta - \eta}_{u, [0, t]}\]</div></div>

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Lipschitz Coefficient Let : [ 0 , ) C ( [ 0 , ) ; d ) n , then is Lipschitz if there exists C 0 such that for any , C ( [ 0 , ) ; d ) , ( t , ) - ( t , ) ^n C - u , [ 0 , t ] Lipschitz Coefficient

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<div class="block-container"><p><a class="link-foreground" href="/t/13"><strong>Lemma 3.2.2.</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;lemma:exact-uniform-bound&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, $B$ be a $\bracs{\mathcal{F}_t}$-adapted standard Brownian motion, $\sigma$ be a $\bracs{\mathcal{F}_t}$-previsible $L(\real^{d}; \real^{n})$-valued process, $b$ be a $\bracs{\mathcal{F}_t}$-previsible $\real^{n}$-valued process, $\xi \in L^{p}(\Omega, \cf_{0}; \real^{n})$, and </p><div class="display-equation">\[X_{t} = \xi + \int_{0}^{t} \sigma_{s} dB_{s} + \int_{0}^{t} b_{s} ds\]</div><p>then for any $T > 0$ and $p \ge 2$, there exists $C_{T, p, n}\ge 0$ such that for all $0 \le t \le T$, </p><div class="display-equation">\[\ev\braks{\norm{X}_{u, [0, t]}^p}\le C_{T, p, n}\braks{\ev(\norm{\xi}_{\real^n}^p) + \ev\paren{\int_0^t \norm{\sigma_s}_{\real^n}^p + \norm{b_s}_{\real^n}^p ds }}\]</div></div><div class="proof"><p><i class="">Proof, <span class="">[<span class="">Theorem 11.5</span>, <a class="link-primary" href="/b/Z">RW89</a>]</span>. </i>Assume without loss of generality that $\xi = 0$, then </p><div class="display-equation">\[\norm{X}_{u, [0, t]}^{p} \le C_{p}\paren{\sup_{0 \le s \le t}\int_0^s \sigma_rdB_r}^{p} + \paren{\int_0^t \norm{b_s}_{\real^n}ds}^{p}\]</div><p>where by Jensen’s inequality, </p><div class="display-equation">\[\paren{\int_0^t \norm{b_s}ds}^{p} \le C_{t, p}\int_{0}^{t} \norm{b_s}_{\real^n}^{p} ds\]</div><p>and by the BDG inequality and Jensen’s inequality, </p><div class="display-equation">\[\ev\braks{\paren{\sup_{0 \le s \le t}\int_0^s \sigma_rdB_r}^p}\le C_{p} \ev\braks{\paren{\int_0^t \norm{\sigma_s}_{\real^d}^2 ds}^{p/2}}\le C_{p} \int_{0}^{t} \norm{\sigma_s}_{\real^d}^{p} ds\]</div><p><span class="qed">$\square$</span></p></div>

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Let ( , _t|t 0 , ) be a filtered probability space , B be a F _t - adapted standard Brownian motion , be a F _t - previsible L ( d ; n ) - valued process , b be a F _t - previsible n - valued process , L p ( , 0 ; n ) , and X t = + 0 t s d B s + 0 t b s d s then for any T > 0 and p 2 , there exists C T , p , n 0 such that for all 0 t T , X _ u , [ 0 , t ] ^p C T , p , n ( _ ^n ^p ) + _0^t _s _ ^n ^p + b_s _ ^n ^p ds Assume without loss of generality that = 0 , then X u , [ 0 , t ] p C p _ 0 s t _0^s _rdB_r p + _0^t b_s _ ^n ds p where by Jensen ' s inequality , _0^t b_s ds p C t , p 0 t b_s ^n p d s and by the BDG inequality and Jensen ' s inequality , _ 0 s t _0^s _rdB_r ^p C p _0^t _s _ ^d ^2 ds ^ p / 2 C p 0 t _s ^d p d s Proof , Theorem 11.5 Rogers .

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<div class="block-container"><p><a class="link-foreground" href="/t/14"><strong>Lemma 3.2.3.</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;lemma:lipschitz-picard&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $\sigma: [0, \infty) \times C([0, \infty); \real^{d}) \to L(\real^{d}; \real^{n})$ and $b: [0, \infty) \times C([0, \infty); \real^{d}) \to \real^{n}$ be previsible path functionals satisfying the <a href="/t/12" class="link-primary">Lipschitz condition</a> with constant $K$.</p><p>Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space and $B$ be a $\bracs{\mathcal{F}_t}$-adapted standard Brownian motion. For any $\bracs{\mathcal{F}_t}$-adapted process $X: \Omega \to C([0, \infty); \real^{d})$ with continuous sample paths and $\xi \in L^{p}(\Omega, \cf_{0}; \real^{n})$, let </p><div class="display-equation">\[P(X, \xi)_{t} = \xi + \int_{0}^{t} \sigma(s, X)dB_{s} + \int_{0}^{t} b(s, X)ds\]</div><p>then for any $\bracs{\mathcal{F}_t}$-adapted process $Y: \Omega \to C([0, \infty); \real^{d})$, $\eta \in L^{p}(\Omega, \cf_{0}; \real^{n})$, $T > 0$, and $0 \le t \le T$, </p><div class="display-equation" id="">\begin{align*}&#x26;\ev\braks{\norm{P(X, \xi) - P(Y, \eta)}_{u, [0, t]}^p}\\&#x26;\le C_{K, n, T, p}\braks{\norm{\xi - \eta}_{L^p(\Omega; \real^n)}^p + \ev\paren{\int_0^t \norm{X - Y}_{u, [0, s]}^p ds}}\end{align*}</div></div>

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Let : [ 0 , ) C ( [ 0 , ) ; d ) L ( d ; n ) and b : [ 0 , ) C ( [ 0 , ) ; d ) n be previsible path functionals satisfying the Lipschitz condition definition : lipschitz - coefficient with constant K . Let ( , _t|t 0 , ) be a filtered probability space and B be a F _t - adapted standard Brownian motion . For any F _t - adapted process X : C ( [ 0 , ) ; d ) with continuous sample paths and L p ( , 0 ; n ) , let P ( X , ) t = + 0 t ( s , X ) d B s + 0 t b ( s , X ) d s then for any F _t - adapted process Y : C ( [ 0 , ) ; d ) , L p ( , 0 ; n ) , T > 0 , and 0 t T , & P ( X , ) - P ( Y , ) _ u , [ 0 , t ] ^p & C K , n , T , p - _ L^p ( ; ^n ) ^p + _0^t X - Y _ u , [ 0 , s ] ^p ds

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<div class="block-container"><p><a class="link-foreground" href="/t/15"><strong>Theorem 3.2.4 (Itô).</strong></a><a class="text-action copy-label" onclick="navigator.clipboard.writeText(&#x22;theorem:ito-existence-uniqueness&#x22;)" title="Copy Label" href="#">label</a><span> </span> Let $\sigma: [0, \infty) \times C([0, \infty); \real^{d}) \to L(\real^{d}; \real^{n})$ and $b: [0, \infty) \times C([0, \infty); \real^{d}) \to \real^{n}$ be previsible path functionals satisfying the <a href="/t/12" class="link-primary">Lipschitz condition</a>. If for each $T \ge 0$, </p><div class="display-equation">\[\sup_{0 \le s \le T}\norm{\sigma(s, 0)}_{L(\real^d; \real^n)}+ \norm{b(s, 0)}_{\real^n}&#x3C; \infty\]</div><p>then the SDE </p><div class="display-equation">\[X_{t} = \xi + \int_{0}^{t} \sigma(s, X)dB_{s} + \int_{0}^{t} b(s, X)ds\]</div><p>is exact and pathwise unique.</p></div><div class="proof"><p><i class="">Proof of existence, <span class="">[<span class="">Theorem 11.2</span>, <a class="link-primary" href="/b/Z">RW89</a>]</span>. </i>For each $t \ge 0$, let $\mathcal{G}_{t}^{\circ} = \sigma(\bracs{B_s|1 \le s \le t}\cup \bracs{\xi})$, $\mathcal{G}^{\circ} = \sigma(\bracsn{\mathcal{G}_t^\circ|t \ge 0})$, $\mathcal{N}$ be the collection of $\bp$-null sets in the completion of $\sigma(\bracsn{\mathcal{G}_t^\circ|t \ge 0})$, and $\mathcal{G}_{t} = \sigma(\mathcal{G}_{t}^{\circ} \cup \mathcal{N})$.</p><p>Let $\xi \in L^{\infty}(\Omega, \cf_{0}; \real^{n})$ and $X^{(0)}= \xi$. For each $m \in \natz$, define inductively </p><div class="display-equation">\[X^{(m+1)}= P(X^{(m)}, \xi)_{t} = \xi + \int_{0}^{t} \sigma(s, X^{(m)})dB_{s} + \int_{0}^{t} b(s, X^{(m)})ds\]</div><p>then $X^{(m)}$ has continuous sample paths, and for each $T > 0$, $\ev[{\normn{X^{(m)}}_{u, [0, s]}^2}] &#x3C; \infty$. By <a href="/t/14" class="link-primary">Lemma 3.2.3</a>, </p><div class="display-equation">\[\ev\braks{\norm{X^{(m+2)} - X^{(m+1)}}_{u, [0, T]}^2}\le C \int_{0}^{t} \ev\braks{\norm{X^{(m+1)} - X^{(m)}}_{u, [0, t]}^2}dt\]</div><p>Therefore </p><div class="display-equation">\[\ev\braks{\norm{X^{(m+1)} - X^{(m)}}_{u, [0, T]}^2}\le C_{0} C \frac{T^{n}}{n!}\]</div><p>and $X^{(m)}$ converges uniformly in $L^{2}$ to a limiting process $X$, which satisfies the given equation.<span class="qed">$\square$</span></p></div><div class="proof"><p><i class="">Proof of uniqueness. </i>Let $X$ and $Y$ be solutions of the SDE with the same setup, then by <a href="/t/14" class="link-primary">Lemma 3.2.3</a>, </p><div class="display-equation">\[\ev\braks{\norm{X - Y}_{u, [0, T]}^2}\le C \int_{0}^{t} \ev\braks{\norm{X - Y}_{u, [0, t]}^2}dt\]</div><p>which implies that $X|_{[0, T]}= Y|_{[0, T]}$ almost surely.<span class="qed">$\square$</span></p></div>

contentText:
Itô Let : [ 0 , ) C ( [ 0 , ) ; d ) L ( d ; n ) and b : [ 0 , ) C ( [ 0 , ) ; d ) n be previsible path functionals satisfying the Lipschitz condition definition : lipschitz - coefficient . If for each T 0 , 0 s T ( s , 0 ) L ( ^d ; ^n ) + b ( s , 0 ) ^n < then the SDE X t = + 0 t ( s , X ) d B s + 0 t b ( s , X ) d s is exact and pathwise unique . Itô For each t 0 , let G t = ( B_s|1 s t ) , G = ( G _t^ |t 0 ) , N be the collection of - null sets in the completion of ( G _t^ |t 0 ) , and G t = ( G t N ) . Let L ( , 0 ; n ) and X ( 0 ) = . For each m , define inductively X ( m + 1 ) = P ( X ( m ) , ) t = + 0 t ( s , X ( m ) ) d B s + 0 t b ( s , X ( m ) ) d s then X ( m ) has continuous sample paths , and for each T > 0 , [ X^ ( m ) _ u , [ 0 , s ] ^2 ] < . By lemma : lipschitz - picard , X^ ( m + 2 ) - X^ ( m + 1 ) _ u , [ 0 , T ] ^2 C 0 t X^ ( m + 1 ) - X^ ( m ) _ u , [ 0 , t ] ^2 d t Therefore X^ ( m + 1 ) - X^ ( m ) _ u , [ 0 , T ] ^2 C 0 C T n n ! and X ( m ) converges uniformly in L 2 to a limiting process X , which satisfies the given equation . Proof of existence , Theorem 11.2 Rogers . Let X and Y be solutions of the SDE with the same setup , then by lemma : lipschitz - picard , X - Y _ u , [ 0 , T ] ^2 C 0 t X - Y _ u , [ 0 , t ] ^2 d t which implies that X | [ 0 , T ] = Y | [ 0 , T ] almost surely . Proof of uniqueness .

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