diff --git a/preamble.sty b/preamble.sty index a9281ac..9f291b9 100644 --- a/preamble.sty +++ b/preamble.sty @@ -167,6 +167,7 @@ \newcommand{\cd}{\mathcal{D}} \newcommand{\calr}{\mathcal{R}} \newcommand{\scp}{\mathscr{P}} +\newcommand{\wien}{\mathcal{W}} % Jokes \newcommand{\lol}{\boxed{\text{LOL.}}} diff --git a/spec.db b/spec.db index 06e8d46..290f0cb 100644 Binary files a/spec.db and b/spec.db differ diff --git a/spec.toml b/spec.toml index 09470aa..496cdc9 100644 --- a/spec.toml +++ b/spec.toml @@ -16,6 +16,8 @@ primaryColour = "blue" neutralColour = "grey" searchLimit = 16 maxSearchPages = 48 -recentChanges = 10 -tableOfContentsDepth = 2 +recentChanges = 0 +tableOfContentsDepth = 1 +hoverPreview = false +copyLabelButton = false advertiseSpec = true diff --git a/src/sde/exact.tex b/src/sde/exact.tex index d9c1b69..431a7e8 100644 --- a/src/sde/exact.tex +++ b/src/sde/exact.tex @@ -100,4 +100,50 @@ \end{proof} +\begin{lemma}[Gronwall] +\label{lemma:gronwall} + Let $f \in C([0, T]; \real)$, and $c, K > 0$ such that + \[ + f(t) \le c + K\int_0^t f(s)ds + \] + + for all $t \in [0, T]$, then + \[ + \frac{d}{dt}\braks{e^{-Kt}\int_0^t f(s)ds} \le ce^{-Kt} + \] + + and $f(t) \le ce^{Kt}$. +\end{lemma} + + +\begin{theorem}[Blagoveshchenskii-Blagoveshchensk] +\label{theorem:strong-solution-properties} + 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 + \[ + X_t = \xi + \int_0^t \sigma(s, X)dB_s + \int_0^t b(s, X)ds + \] + + admits a strong solution + \[ + F: \real^n \times C([0, \infty); \real^d) \to C([0, \infty); \real^n) + \] + + such that: + \begin{enumerate} + \item For each $\theta \in C([0, \infty); \real^d)$, $F(\cdot, \theta): \real^d \to C([0, \infty); \real^n)$ is continuous. + \item For each solution $X^y$ initial condition $y \in \real^n$ and $s \ge 0$, + \[ + X_{s + t}^y = F(X_s^y, \tau_{-s}B)_t \quad \forall t \ge 0 + \] + + almost surely, where $(\tau_{-s}B)_r = B_{r+s}$. + \item For each $y \in \real^n$, $\bracs{X_t^y|t \ge 0}$ is a Markov process. + \end{enumerate} + +\end{theorem} diff --git a/src/sde/index.tex b/src/sde/index.tex index ca391a4..856d66f 100644 --- a/src/sde/index.tex +++ b/src/sde/index.tex @@ -3,4 +3,6 @@ \input{./setup} \input{./exact} +\input{./weak} +\input{./martingale} diff --git a/src/sde/martingale.tex b/src/sde/martingale.tex new file mode 100644 index 0000000..7144afd --- /dev/null +++ b/src/sde/martingale.tex @@ -0,0 +1,81 @@ +\section{The Martingale Formulation} +\label{section:weak-martingale} + + +\begin{definition}[Martingale Problem] +\label{definition:martingale-problem} + Let $a: [0, \infty) \times C([0, \infty); \real^n) \to L(\real^n; \real^n)$ and $b: [0, \infty) \times C([0, \infty); \real^n) \to \real^n$ be previsible path functionals. For each $u \in C_c^\infty(\real^n)$, let + \[ + Lu = \frac{1}{2}\dpn{A, D^2f}{\real^{n \times n}} + \dpn{b, Df}{\real^n} + \] + + then for any $y \in \real^n$, filtered probability space $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$, and $X: \Omega \to C([0, \infty); \real^n)$, $X$ is a \textbf{solution to the martingale problem for $(a, b)$ starting at $y$} if: + \begin{enumerate} + \item $X_0 = y$ almost surely. + \item For each $f \in C_c^\infty(\real^n)$, the process + \[ + C_t^f = f(X_t) - f(X_0) - \int_0^t Lf(s, X_s)ds + \] + + is a $\bracs{\mathcal{F}_t}$-martinagle. + \end{enumerate} + + If the distribution of $X$ on $C([0, \infty); \real^n)$ is the unique distribution satisfying the above, then the solution for the martingale problem is unique. If for each $y \in \real^n$, such a solution exists, then the martingale problem is well posed. +\end{definition} + +\begin{theorem}[Equivalence of Formulations] +\label{theorem:equivalence-of-formulations} + Let $(\Omega, \bracsn{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, $B$ be a $\bracs{\mathcal{F}_t}$-Brownian motion, $\sigma: [0, \infty) \times C([0, \infty); \real^n) \to L(\real^n; \real^n)$ and $b: [0, \infty) \times C([0, \infty); \real^n) \to \real^n$ be bounded measurable functions such that $\sigma_t$ is invertible for all $t \ge 0$. + + Let $y \in \real^n$ and $X: \Omega \to C([0, \infty); \real^n)$ be a solution to the martingale problem for $(\sigma^*\sigma, b)$ starting at $y$, then there exists a weak solution of the SDE + \[ + Y_t = Y_0 + \int_0^t \sigma(s, Y) dB_s + \int_0^t b(s, Y)ds + \] + + starting at $y$ whose distribution is the same as $X$. +\end{theorem} +\begin{proof}[Proof, {{\cite[Theorem 20.1]{Rogers}}}. ] + By truncation, for each $1 \le i \le n$, + \[ + M_t = X_t - \int_0^t b(s, X)ds + \] + + and + \[ + M_tM_t^T - \int_0^t a(s, X)ds + \] + + are local martingales. In which case, by Lévy's characterisation of Brownian motion, + \[ + \td B_t = \int_0^t \sigma(s, X)^{-1}dM_s + \] + + is a standard Brownian motion, and $X$ and $\td B$ satisfy the SDE. +\end{proof} + +\begin{theorem} +\label{theorem:martingale-markov} + Let $a: \real^n \to L(\real^n; \real^n)$ and $b: \real^n \to \real^n$ be bounded measurable functions such that the martingale problem for $(a, b)$ is well-posed, that is, let + \[ + Lu = \frac{1}{2}\dpn{A, D^2f}{\real^{n \times n}} + \dpn{b, Df}{\real^n} + \] + + then for each $y \in \real^n$, there exists a unique probability measure $\bp^y$ on $C([0, \infty); \real^n)$ such that: + \begin{enumerate} + \item $\bp^y(x_0 = y) = 1$. + \item For each $f \in C_c^\infty$, + \[ + C_t^f = f(\pi_t) - f(x_0) - \int_0^t Lf(x_s)ds + \] + + is a $\bp^y$-martingale. + \end{enumerate} + + then + \begin{enumerate} + \item $\bracs{x_t|t \ge 0}$ is a time-homogeneous strong Markov process with respect to $\bp^y$. + \item If $a = \sigma \sigma^*$ and $(\sigma, b)$ satisfy the \hyperref[Lipschitz condition]{definition:lipschitz-coefficient}, then the generator of the above process is $L$. + \end{enumerate} +\end{theorem} + + diff --git a/src/sde/setup.tex b/src/sde/setup.tex index 06f7267..92df5f4 100644 --- a/src/sde/setup.tex +++ b/src/sde/setup.tex @@ -21,6 +21,12 @@ 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{definition}[Augmentation] +\label{definition:augmentation} + Let $(\Omega, \bracs{\cf_t}, \bp)$ be a filtered probability space and $\mathcal{N}$ be the collection of all $\bp$-null sets in $\cf = \sigma(\bracs{\cf_t|t \ge 0})$. For each $t \ge 0$, let $\ol{\cf}_t = \sigma(\cf_t \cup \mathcal{N})$, then the filtration $\bracsn{\ol{\cf}_t|t \ge 0}$ is the \textbf{$\bp$-augmentation} of $\bracs{\mathcal{F}_t}$. +\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 @@ -51,9 +57,9 @@ \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 + 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, then the 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} + X_t = \xi + \int_0^t \sigma(s, X) dB_s + \int_0^t b(s, X) ds\label{equation:diffusion-sde} \end{equation} has \textbf{pathwise uniqueness} if given @@ -68,10 +74,10 @@ \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} + 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, then the SDE + \[ + X_t = \xi + \int_0^t \sigma(s, X) dB_s + \int_0^t b(s, X) ds + \] is \textbf{pathwise exact} if given \begin{itemize} @@ -86,5 +92,37 @@ +\begin{definition}[Strong Solution] +\label{definition:sde-strong-solution} + 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, then a mapping + \[ + F: \real^n \times C([0, \infty); \real^d) \to C([0, \infty); \real^n) + \] + + is a \textbf{strong solution} to the SDE + \begin{equation} + \label{eq:sde-strong-solution} + X_t = \xi + \int_0^t \sigma(s, X) dB_s + \int_0^t b(s, X) ds + \end{equation} + + if + \begin{enumerate} + \item For each $t \ge 0$, let + \[ + \mathcal{F}_t = \sigma(\bracs{\pi_s: C([0, \infty); \real^d) \to \real^d|0 \le s \le t}) \quad \mathcal{G}_t = \sigma(\bracs{\pi_s: C([0, \infty); \real^n) \to \real^n|0 \le s \le t}) + \] + + Let $\wien^d$ and $\wien^n$ be the classical Wiener measure on $C([0, \infty); \real^d)$ and $C([0, \infty); \real^n)$, respectively, and $\bracsn{\ol{\mathcal{G}}_t|t \ge 0}$ be the $\wien^n$-augmentation of $\bracs{\mathcal{G}_t|t \ge 0}$, then + \[ + F^{-1}(\mathcal{F}_t) \subset \mathcal{B}(\real^n) \times \overline{\mathcal{G}_t} + \] + + for all $t \ge 0$. + + \item For any filtered probability space $(\Omega, \bracs{\mathcal{H}_t}, \bp)$, random variable $\xi: \Omega \to \real^n$, and $\bracs{\mathcal{H}_t}$-Brownian motion $B$, the process $X = F(\xi, B)$ solves \autoref{eq:sde-strong-solution}. + \end{enumerate} + +\end{definition} + diff --git a/src/sde/weak.tex b/src/sde/weak.tex new file mode 100644 index 0000000..7aeb7f2 --- /dev/null +++ b/src/sde/weak.tex @@ -0,0 +1,54 @@ +\section{Weak Solutions} +\label{section:weak-solution} + + +\begin{definition}[Weak Solution] +\label{definition:sde-weak-solution} + Let $\mu$ be a Borel probability measure on $\real^n$, $\sigma: [0, \infty) \times C([0, \infty); \real^d) \to L(\real^d; \real^n)$ and $b: [0, \infty) \times C([0, \infty); \real^d) \to \real^n$ be previsible path functionals, then the SDE + \[ + X_t = X_0 + \int_0^t \sigma(s, X) dB_s + \int_0^t b(s, X) ds + \] + + has a \textbf{weak solution with initial distribution} $\mu$ if there exists a filtered probability space $(\Omega, \bracs{\cf_t}, \bp)$, a $\bracs{\mathcal{F}_t}$-Brownian motion $B$, and a $\bracs{\mathcal{F}_t}$-semimartingale $X: \Omega \to C([0, \infty); \real^d)$ such that: + \begin{enumerate} + \item $X_0$ has distribution $\mu$. + \item For each $t > 0$, + \[ + \int_0^t \norm{\sigma(s, X)}_{\real^n}^2 + \norm{\sigma(s, X)}_{\real^n}^2 ds < \infty + \] + + almost surely. + \item For each $t \ge 0$, + \[ + X_t = X_0 + \int_0^t \sigma(s, X) dB_s + \int_0^t b(s, X) ds + \] + \end{enumerate} +\end{definition} + +\begin{definition}[Uniqueness in Distribution] +\label{definition:uniqueness-in-distribution} + 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, then the SDE + \[ + X_t = X_0 + \int_0^t \sigma(s, X) dB_s + \int_0^t b(s, X) ds + \] + + has \textbf{uniqueness in law} if for any solutions $X$ and $X'$ such that the distributions of $X_0$ and $X_0'$ are the same, the distributions of $X$ and $X'$ are the same. +\end{definition} + +\begin{theorem}[Yamada and Watanabe, {{\cite[Theorem 17.1]{Rogers}}}] +\label{theorem:yamada-watanabe} + 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, and + \[ + X_t = X_0 + \int_0^t \sigma(s, X) dB_s + \int_0^t b(s, X) ds + \] + + then the above SDE is exact if and only if the following conditions hold: + \begin{enumerate} + \item The SDE has a weak solution. + \item The SDE has the pathwise-uniqueness property. + \end{enumerate} + + If the above holds, then the SDE also has uniqueness in law. +\end{theorem} + +