diff --git a/.vscode/project.code-snippets b/.vscode/project.code-snippets index bcded9b..ed9ea72 100644 --- a/.vscode/project.code-snippets +++ b/.vscode/project.code-snippets @@ -126,6 +126,11 @@ "prefix": "cal", "body": ["\\mathcal{$1}$0"] }, + "Mathscr": { + "scope": "latex", + "prefix": "scr", + "body": ["\\mathscr{$1}$0"] + }, "Mathfrak": { "scope": "latex", "prefix": "fk", @@ -160,5 +165,10 @@ "scope": "latex", "prefix": "filt", "body": ["$\\bracs{\\mathcal{F}_t}$"] + }, + "Path Space": { + "scope": "latex", + "prefix": "path", + "body": ["C([0, \\infty); \\real^d)"] } } diff --git a/refs.bib b/refs.bib index 4cc872a..3a77691 100644 --- a/refs.bib +++ b/refs.bib @@ -8,3 +8,32 @@ year={1997}, publisher={Springer Berlin Heidelberg} } + +@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} +} + +@book{Lang, + title={Differential and Riemannian manifolds}, + author={Lang, Serge}, + year={2012}, + publisher={Springer Science \& Business Media} +} + +@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} +} diff --git a/spec.db b/spec.db index 7a09b4d..06e8d46 100644 Binary files a/spec.db and b/spec.db differ diff --git a/src/calculus/frobenius.tex b/src/calculus/frobenius.tex new file mode 100644 index 0000000..3db9433 --- /dev/null +++ b/src/calculus/frobenius.tex @@ -0,0 +1,77 @@ +\section{Frobenius' Theorem} +\label{section:frobenius} + +\begin{theorem}[Frobenius] +\label{theorem:frobenius} + Let $X$ be a $C^p$ ($p \ge 2$) manifold and $E \subset TX$ be a subbundle, then the following are equivalent: + \begin{enumerate} + \item For each pair of vector fields $\xi, \eta: X \to E$, $[\xi, \eta]$ lies in $E$. + \item For each $\omega \in \Lambda^1(TX)$ vanishing on $E$, $\xi, \eta: X \to E$, $d\omega(\xi, \eta) = 0$. + \item $E$ is integrable. + \end{enumerate} +\end{theorem} +\begin{proof}[Proof, {{\cite[Section VI.1]{Lang}}}. ] + Let $\omega \in \Lambda^1(TX)$, then + \[ + d\omega(\xi, \eta) = \omega([\xi, \eta]) - \eta\omega(\xi) - \xi\omega(\eta) + \] + + (1) $\Rightarrow$ (2): If $\omega$ vanishes on $E$ and $[\xi, \eta]$ lies in $E$, then $d\omega$ vanishes on $(\xi, \eta)$. + + (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). + + (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 + \[ + f: U \times V \times F \to U \times V \times (F \times G) + \] + + such that $\pi_1f(x, y)(z) = z$ for all $(x, y) \in U \times V$ and $z \in F$. Let + \[ + g: U \times V \to L(F; G) \quad g(x, y)(z) = \pi_2f(x, y, z) + \] + + then $g$ is also a $C^{p - 1}$-mapping. Let + \[ + \Xi: U \times V \to F \times G + \] + + be the local representation of a vector field, then the vector field lies in $E$ if and only if + \[ + \pi_2\Xi(x, y) = g(x, y)\pi_1\Xi(x, y) + \] + + 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 + \begin{align*} + [\xi, \eta] &= D\eta \cdot \xi - D\xi \cdot \eta \\ + g \cdot (D\eta \cdot \xi - D\xi \cdot \eta) &= (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 &= (Dg \cdot \eta) \cdot \xi + \end{align*} + + Let $(x_0, y_0) \in U \times V$, then by \autoref{theorem:local-frobenius}, 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 + \[ + \partial_x\alpha(x, y) = g(x, \alpha(x, y)) + \] + + Let + \[ + \varphi: U_0 \times V_0 \to U \times V \quad (x, y) \mapsto (x, \alpha(x, y)) + \] + + then + \[ + D\varphi(x_0, y_0) = \begin{bmatrix} Id & 0 \\ g(x, \alpha(x, y)) & Id \end{bmatrix} + \] + + so $\varphi$ is a local diffeomorphism at $(x_0, y_0)$. Since for any $(u, v) \in F \times G$, + \[ + \partial_x\varphi(x, y) \cdot (u, v) = (u, g(x, \alpha(x, y)) \cdot u) + \] + + the bundle $E$ is integrable. + + +\end{proof} + + + + diff --git a/src/calculus/index.tex b/src/calculus/index.tex new file mode 100644 index 0000000..acbd260 --- /dev/null +++ b/src/calculus/index.tex @@ -0,0 +1,5 @@ +\chapter{Integral Manifolds/Shenanigans} +\label{chap:integral} + +\input{./integrable.tex} +\input{./frobenius.tex} diff --git a/src/calculus/integrable.tex b/src/calculus/integrable.tex new file mode 100644 index 0000000..db98c57 --- /dev/null +++ b/src/calculus/integrable.tex @@ -0,0 +1,147 @@ +\section{Integrable Bundles} +\label{section:integrable} + +\begin{definition}[Integrable] +\label{definition:integrable} + Let $X$ be a $C^p$ ($p \ge 2$) manifold, $E \subset TX$ be a subbundle, and $x_0 \in X$, then $E$ is \textbf{integrable at $x_0$} if there exists a submanifold $x_0 \in Y \subset X$ such that + \[ + T\iota: TY \to E \subset TX + \] + + is an isomorphism. If $E$ is integrable at every $x_0 \in X$, then $E$ is \textbf{integrable}. +\end{definition} + +\begin{lemma} +\label{lemma:split-ode} + 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 + \[ + \beta: (-\delta, \delta) \times U_0 \times V_0 \to V + \] + + such that + \begin{enumerate} + \item For all $(x, y) \in U_0 \times V_0$, $\beta(0, x, y) = y$. + \item For all $(t, x, y) \in (-\delta, \delta) \times U_0 \times V_0$, + \[ + \frac{d}{dt}\beta(t, x, y) = g(t, x, \beta(t, x, y)) + \] + \item For each fixed $y \in V_0$, let $\beta(t, x) = \beta(t, x, y)$, then for each $(t, x) \in (-\delta, \delta) \times U_0$, + \[ + (\partial_t\partial_x\beta)(t, x) = (\partial_xg)(t, x, \beta(t, x)) + (\partial_yg)(t, x, \beta(t, x)) \circ (\partial_x\beta)(t, x) + \] + \end{enumerate} +\end{lemma} +\begin{proof}[Proof, {{\cite[Proposition VI.2.1]{Lang}}}.] + Let + \[ + G: (-\eps, \eps) \times U \times V \to E \times F \quad (t, x, y) \mapsto (0, g(t, x, y)) + \] + + 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 + \[ + B: (-\delta, \delta) \times U_0 \times V_0 \to E \times F + \] + + such that + \begin{enumerate} + \item[(a)] For each $(x, y) \in U_0 \times V_0$, $B(0, x, y) = (x, y)$. + \item[(b)] For each $(t, x, y) \in (-\delta, \delta) \times U_0 \times V_0$, + \[ + \frac{d}{dt}B(t, x, y) = (0, g(t, B(t, x, y))) + \] + \end{enumerate} + + Let $\beta(t, x, y) = \pi_2(t, x, y)$, then + \begin{enumerate} + \item For each $(x, y) \in U_0 \times V_0$, $\beta(0, x, y) = \pi_2(x, y) = y$. + \item For each $(t, x, y) \in (-\delta, \delta) \times U_0 \times V_0$, + \[ + \frac{d}{dt}\beta(t, x, y) = \pi_2(g(t, B(t, x, y))) = g(t, x, \beta(t, x, y)) + \] + \item 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, + \[ + (\partial_t\partial_x\beta)(t, x) = (\partial_xg)(t, x, \beta(t, x)) + (\partial_yg)(t, x, \beta(t, x)) \circ (\partial_x\beta)(t, x) + \] + + \end{enumerate} +\end{proof} + +\begin{theorem} +\label{theorem:local-frobenius} + Let $E, F$ be Banach spaces, $U \subset E$ and $V \subset F$ be open subsets, and $f \in C^k(U \times V; L(E; F))$ ($k \ge 2$). + + For any $\xi, \eta \in C^k(U \times V; E)$, denote + \[ + \Xi(x, y) = (\xi(x, y), f(x, y) \cdot \xi(x, y)) \quad H(x, y) = (\eta(x, y), f(x, y) \cdot \eta(x, y)) + \] + + If for every $\xi, \eta \in C^k(U \times V; E)$ and $(x, y) \in U \times V$, + \[ + [Df(x, y) \cdot \Xi(x, y)] \cdot \eta(x, y) = [Df(x, y) \cdot H(x, y)] \cdot \xi(x, y) + \] + + 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 + \begin{enumerate} + \item For every $y \in V_0$, $\alpha(x_0, y) = y$. + \item For every $(x, y) \in U_0 \times V_0$, + \[ + (\partial_x \alpha)(x, y) = f(x, \alpha(x, y)) + \] + \end{enumerate} +\end{theorem} +\begin{proof} + Using translation, assume without loss of generality that $x_0 = 0$ and $y_0 = 0$. Let $B \in \cn_E(0)$ and + \[ + g: (-\eps, \eps) \times B \times V \to F \quad (t, z, y) \mapsto f(tz, y) \cdot z + \] + + then by \autoref{lemma:split-ode}, there exists $\delta > 0$, $B_0 \in \cn_E(0)$, and $\beta: (-\delta, \delta) \times B_0 \times V_0$ such that + \begin{enumerate} + \item[(a)] For each $(z, y) \in B_0 \times V_0$, $\beta(0, z, y) = y$. + \item[(b)] For each $(t, z, y) \in (-\delta, \delta) \times B_0 \times V_0$, + \[ + \frac{d}{dt}\beta(t, z, y) = f(tz, \beta(t, z, y)) \cdot z + \] + + \item[(c)] 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 $h \in E$, + \begin{align*} + (\partial_t\partial_z\beta)(t, z)\cdot h &= t(\partial_xf)(tz, \beta(t, z))\cdot h \cdot z + + f(tz, \beta(t, z))\cdot h \\ + &+ (\partial_yf)(tz, \beta(t, z)) \circ (\partial_z \beta)(t, z)\cdot h \cdot z + \end{align*} + \end{enumerate} + + Following (c), let $k(t, z) = (\partial_z\beta)(t, z)\cdot h - tf(tz, \beta(t, z))\cdot h$, then + \begin{align*} + \frac{d}{dt}k(t) &= (\partial_t\partial_z\beta)(t, z)\cdot h - f(tz, \beta)\cdot h \\ + &- t [(\partial_xf)(tz, \beta)]\cdot z \cdot h - t[(\partial_y f)(tz, \beta)]f(tz, \beta) \cdot z\cdot h + \end{align*} + + where by assumption, + \begin{align*} + &[(\partial_xf)(tz, \beta)]\cdot z \cdot h - [(\partial_y f)(tz, \beta)]f(tz, \beta) \cdot z \cdot h \\ + &= Df(tz, \beta) \cdot (z, f(tz, \beta) \cdot z) \cdot h \\ + &= Df(tz, \beta) \cdot (h, f(tz, \beta) \cdot h) \cdot z \\ + &= [(\partial_xf)(tz, \beta)]\cdot h \cdot z - [(\partial_y f)(tz, \beta)]f(tz, \beta) \cdot h\cdot z + \end{align*} + + Therefore by (c), + \begin{align*} + \frac{d}{dt}k(t) &= (\partial_yf)(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 \\ + &= (\partial_y f)(tz, \beta)[\partial_z \beta(t, z) - tf(tz, \beta)] \cdot h \cdot z \\ + &= (\partial_y f)(tz, \beta) \cdot k(t) \cdot z + \end{align*} + + 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 $z \in B_0$, + \[ + \partial_z \beta(t, z) = tf(tz, \beta(t, z)) + \] + + 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 + \[ + \partial_x\alpha(x, y) = \partial_z\beta(1, x, y) = f(z, \beta(1, x, y)) = f(x, \alpha(x, y)) + \] +\end{proof} + + + diff --git a/src/diffusion/martingale.tex b/src/diffusion/martingale.tex index 3ddda8f..9a81676 100644 --- a/src/diffusion/martingale.tex +++ b/src/diffusion/martingale.tex @@ -30,6 +30,67 @@ is a progressively measurable process. \end{lemma} + +\begin{proposition} +\label{proposition:martingale-quadratic} + Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, + \[ + L_t(\omega)u = \frac{1}{2}\dpn{A_t(\omega), u}{\real^{d \times d}} + \dpn{B_t(\omega), u}{\real^d} + \] + + 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 + \[ + Y_t = f(t, X_t) - \int_0^t [(\partial_r + L_r)f](r, X_r)dr + \] + and + \[ + W_t = f(t, X_t)^2 - \int_0^t [(\partial_r + L_r)(f^2)](r, X_r)dr + \] + + are martingales, then + \[ + [X]_t = \frac{1}{2}\int_0^t \dpn{Df, A(r)(Df)}{\real^d}(r, X_r)dr + \] + +\end{proposition} +\begin{proof} + Firstly, since $f$ admits bounded derivatives, the integral term is of locally bounded variation. Therefore if $Z_t = f(t, X_t)$, then $[X]_t = \angles{Z}_t$. + + Let $\bracs{t_j}_0^n \subset [0, t]$ be a partition, then + \[ + Z_t^2 - Z_0^2 = \sum_{j = 1}^{n}Z_{t_j}^2 - Z_{t_{j-1}}^2 = \sum_{j = 1}^n 2Z_{t_{j-1}}(Z_{t_j} - Z_{t_{j-1}}) + \sum_{j = 1}^n (Z_{t_j} - Z_{t_{j-1}})^2 + \] + + As the above holds for every partition, + \[ + Z_t^2 - Z_0^2 = 2\int_0^t Z_rZ(dr) + \angles{Z}_t + \] + + Since $Z_t = Y_t + \int_0^t(\partial_r + L_r)fdr$, + \begin{align*} + \int_0^t Z_rZ(dr) &= \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 &= 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*} + + On the other hand, + \[ + Z_t^2 - Z_0^2 = W_t - W_0 + \int_0^t [(\partial_r + L_r)(f^2)](r, X_r)dr + \] + + Therefore + \begin{align*} + \angles{Z}_t &= W_t - W_0 + \int_0^t [(\partial_r + L_r)(f^2)](r, X_r)dr \\ + &- 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 \\ + &= W_t - W_0 - 2 \int_0^t f(r, X_r) Y(dr) \\ + &+ \int_0^t [L_r(f^2)(r, X_r) - 2f(r, X_r)(L_rf)(r, X_r)]dr \\ + &= 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*} + + 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 \cite[Lemma 5.13]{Baudoin}. + +\end{proof} + + \begin{theorem}[Integration by Parts] \label{theorem:martingale-ibp} 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: @@ -81,7 +142,7 @@ \begin{align*} Y_t^{x, g} &= \exp\braks{\dpn{x, X_t - X_0 - \int_0^t b(r)dr}{\real^d} + g(t, X_t)} \\ &\cdot \exp\braks{-\frac{1}{2}\int_0^t \dpn{x + Dg, A(r)(x + Dg)}{\real^d}(r, X_r)dr} \\ - &\cdot \exp\braks{\int_0^t [(\partial_r + L_r)g](r, X_r)}dr + &\cdot \exp\braks{-\int_0^t [(\partial_r + L_r)g](r, X_r)}dr \end{align*} is a $\bracs{\mathcal{F}_t}$-martingale. @@ -179,15 +240,7 @@ (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} \] - 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 $L^1$. - - - - - - - - + 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 $L^1$. \end{proof} diff --git a/src/index.tex b/src/index.tex index c4c202c..f80ff8d 100644 --- a/src/index.tex +++ b/src/index.tex @@ -1,6 +1,6 @@ \part{Diffusion Processes} \label{part:diffusion} - - -\input{./diffusion/index.tex} \ No newline at end of file +\input{./diffusion/index.tex} +\input{./calculus/index.tex} +\input{./sde/index.tex} \ No newline at end of file diff --git a/src/sde/exact.tex b/src/sde/exact.tex new file mode 100644 index 0000000..d9c1b69 --- /dev/null +++ b/src/sde/exact.tex @@ -0,0 +1,103 @@ +\section{Itô Existence and Uniqueness} +\label{section:exact} + +\begin{definition}[Lipschitz Coefficient] +\label{definition:lipschitz-coefficient} + Let $\sigma: [0, \infty) \times C([0, \infty); \real^d) \to \real^n$, then $\sigma$ is \textbf{Lipschitz} if there exists $C \ge 0$ such that for any $\theta, \eta \in C([0, \infty); \real^d)$, + \[ + \norm{\sigma(t, \theta) - \sigma(t, \eta)}_{\real^n} \le C\norm{\theta - \eta}_{u, [0, t]} + \] +\end{definition} + +\begin{lemma} +\label{lemma:exact-uniform-bound} + 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 + \[ + X_t = \xi + \int_0^t \sigma_s dB_s + \int_0^t b_s ds + \] + + then for any $T > 0$ and $p \ge 2$, there exists $C_{T, p, n} \ge 0$ such that for all $0 \le t \le T$, + \[ + \ev\braks{\norm{X}_{u, [0, t]}^p} \le C_{T, p, n}\braks{\ev(\norm{\xi}_{\real^n}^p) + + \ev\paren{\int_0^t \norm{\sigma_s}_{\real^n}^p + \norm{b_s}_{\real^n}^p ds }} + \] +\end{lemma} +\begin{proof}[Proof, \cite[Theorem 11.5]{Rogers}. ] + Assume without loss of generality that $\xi = 0$, then + \[ + \norm{X}_{u, [0, t]}^p \le C_p\paren{\sup_{0 \le s \le t}\int_0^s \sigma_rdB_r}^p + \paren{\int_0^t \norm{b_s}_{\real^n}ds}^p + \] + + where by Jensen's inequality, + \[ + \paren{\int_0^t \norm{b_s}ds}^p \le C_{t, p} \int_0^t \norm{b_s}_{\real^n}^p ds + \] + + and by the BDG inequality and Jensen's inequality, + \[ + \ev\braks{\paren{\sup_{0 \le s \le t}\int_0^s \sigma_rdB_r}^p} \le C_p \ev\braks{\paren{\int_0^t \norm{\sigma_s}_{\real^d}^2 ds}^{p/2}} \le C_p \int_0^t \norm{\sigma_s}_{\real^d}^p ds + \] +\end{proof} + +\begin{lemma} +\label{lemma:lipschitz-picard} + 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 \hyperref[Lipschitz condition]{definition:lipschitz-coefficient} with constant $K$. + + 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 + \[ + P(X, \xi)_t = \xi + \int_0^t \sigma(s, X)dB_s + \int_0^t b(s, X)ds + \] + + 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 $0 \le t \le T$, + \begin{align*} + &\ev\braks{\norm{P(X, \xi) - P(Y, \eta)}_{u, [0, t]}^p} \\ + &\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*} + +\end{lemma} + +\begin{theorem}[Itô] +\label{theorem:ito-existence-uniqueness} + 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 \hyperref[Lipschitz condition]{definition:lipschitz-coefficient}. If for each $T \ge 0$, + \[ + \sup_{0 \le s \le T}\norm{\sigma(s, 0)}_{L(\real^d; \real^n)} + \norm{b(s, 0)}_{\real^n} < \infty + \] + + then the SDE + \[ + X_t = \xi + \int_0^t \sigma(s, X)dB_s + \int_0^t b(s, X)ds + \] + + is exact and pathwise unique. +\end{theorem} +\begin{proof}[Proof of existence, \cite[Theorem 11.2]{Rogers}. ] + 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 $\mathcal{G}_t = \sigma(\mathcal{G}_t^\circ \cup \mathcal{N})$. + + Let $\xi \in L^\infty(\Omega, \cf_0; \real^n)$ and $X^{(0)} = \xi$. For each $m \in \natz$, define inductively + \[ + X^{(m+1)} = P(X^{(m)}, \xi)_t = \xi + \int_0^t \sigma(s, X^{(m)})dB_s + \int_0^t b(s, X^{(m)})ds + \] + + then $X^{(m)}$ has continuous sample paths, and for each $T > 0$, $\ev[{\normn{X^{(m)}}_{u, [0, s]}^2}] < \infty$. By \autoref{lemma:lipschitz-picard}, + \[ + \ev\braks{\norm{X^{(m+2)} - X^{(m+1)}}_{u, [0, T]}^2} \le C \int_0^t \ev\braks{\norm{X^{(m+1)} - X^{(m)}}_{u, [0, t]}^2} dt + \] + + Therefore + \[ + \ev\braks{\norm{X^{(m+1)} - X^{(m)}}_{u, [0, T]}^2} \le C_0 C \frac{T^n}{n!} + \] + + and $X^{(m)}$ converges uniformly in $L^2$ to a limiting process $X$, which satisfies the given equation. +\end{proof} +\begin{proof}[Proof of uniqueness. ] + Let $X$ and $Y$ be solutions of the SDE with the same setup, then by \autoref{lemma:lipschitz-picard}, + \[ + \ev\braks{\norm{X - Y}_{u, [0, T]}^2} \le C \int_0^t \ev\braks{\norm{X - Y}_{u, [0, t]}^2} dt + \] + + which implies that $X|_{[0, T]} = Y|_{[0, T]}$ almost surely. +\end{proof} + + + diff --git a/src/sde/index.tex b/src/sde/index.tex new file mode 100644 index 0000000..ca391a4 --- /dev/null +++ b/src/sde/index.tex @@ -0,0 +1,6 @@ +\chapter{Stochastic Differential Equations} +\label{chap:sde} + +\input{./setup} +\input{./exact} + diff --git a/src/sde/setup.tex b/src/sde/setup.tex new file mode 100644 index 0000000..06f7267 --- /dev/null +++ b/src/sde/setup.tex @@ -0,0 +1,90 @@ +\section{Definitions} +\label{section:sde-definitions} + +\begin{definition}[Previsible $\sigma$-Algebra] +\label{definition:previsible-sigma-algebra} + Let $(\Omega, \cf)$ be a measurable space and $\bracs{\cf_t|t \ge 0}$ be a filtration on $\Omega$, then the \textbf{previsible $\sigma$-algebra} $\mathscr{J}_\omega$ on $(0, \infty) \times \Omega$ associated with $\bracs{\mathcal{F}_t}$ is the $\sigma$-algebra generated by + \[ + \bracs{(s, t] \times A| 0 \le s < t < \infty, A \in \cf_s} + \] + + 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. +\end{definition} + +\begin{definition}[Previsible Path Functional] +\label{definition:previsible-path-functional} + 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 + \[ + \mathscr{X}_t = \sigma(\bracs{\pi_s|s \le t}) + \] + + and $\mathscr{J}_{C([0, \infty); \real^d)}$ be the previsible $\sigma$-algebra on $(0, \infty) \times C([0, \infty); \real^d)$, then a \textbf{previsible path functional} is a $\mathscr{J}$-measurable mapping on $(0, \infty) \times C([0, \infty); \real^d)$. +\end{definition} + +\begin{lemma} +\label{lemma:adapted-composition} + 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 + \begin{enumerate} + \item $X$ is $(\mathscr{J}_\Omega, \mathscr{J}_{C([0, \infty); \real^d)})$-measurable. + \item For any previsible path functional $\alpha: (0, \infty) \times C([0, \infty); \real^d)$, $\alpha(t, X)$ is $\bracs{\mathcal{F}_t}$-previsible. + \end{enumerate} +\end{lemma} + +\begin{definition}[Diffusion Type SDE] +\label{definition:diffusion-sde} + Let $\sigma: \real^d \to L(\real^d; \real^n)$ and $b: \real^n \to \real^n$ be measurable functions, then a \textbf{SDE of diffusion type} is the equation + \[ + X_t = \xi + \int_0^t \sigma(X_s) dB_s + \int_0^t b(X_s)ds + \] + + under the constraint + \begin{equation} + \int_0^t \norm{\sigma(X_s)}_{\real^n}^2 + \norm{b(X_s)}_{\real^n} ds < \infty \label{equation:diffusion-constraint} + \end{equation} + + for all $t > 0$, where + \begin{itemize} + \item $B$ is a standard Brownian motion on a filtered probability space $(\Omega, \bracs{\cf_t}, \bp)$. + \item $\xi$ is a $\cf_0$-measurable random variable. + \end{itemize} +\end{definition} + +\begin{definition}[Pathwise Uniqueness] +\label{definition:pathwise-uniqueness} + Let $\sigma: \real^d \to L(\real^d; \real^n)$ and $b: \real^n \to \real^n$ be measurable functions, then the diffusion SDE + \begin{equation} + X_t = \xi + \int_0^t \sigma(X_s) dB_s + \int_0^t b(X_s)ds \label{equation:diffusion-sde} + \end{equation} + + has \textbf{pathwise uniqueness} if given + \begin{itemize} + \item A filtered probability space $(\Omega, \bracs{\cf_t}, \bp)$, + \item $\bracs{\mathcal{F}_t}$-adapted standard Brownian motion $B$, + \item Continuous $\bracs{\mathcal{F}_t}$-semimartingales $X, Y: \Omega \to C([0, \infty); \real^d)$ satisfying \autoref{equation:diffusion-sde} and \autoref{equation:diffusion-constraint}, + \end{itemize} + + then $X = Y$ almost surely. +\end{definition} + +\begin{definition}[Pathwise Exact] +\label{definition:pathwise-exact} + Let $\sigma: \real^d \to L(\real^d; \real^n)$ and $b: \real^n \to \real^n$ be measurable functions, then the diffusion SDE + \begin{equation} + X_t = \xi + \int_0^t \sigma(X_s) dB_s + \int_0^t b(X_s)ds \label{equation:diffusion-sde} + \end{equation} + + is \textbf{pathwise exact} if given + \begin{itemize} + \item A filtered probability space $(\Omega, \bracs{\cf_t}, \bp)$, + \item $\bracs{\mathcal{F}_t}$-adapted standard Brownian motion $B$, + \item $\bracs{\mathcal{F}_t}$-semimartingales $X, Y: \Omega \to C([0, \infty); \real^d)$ satisfying \autoref{equation:diffusion-sde} and \autoref{equation:diffusion-constraint}, + \end{itemize} + + then for every $t \ge 0$, $X_t = Y_t$ almost surely. +\end{definition} + + + + + +