diff --git a/spec.toml b/spec.toml index 496cdc9..372207e 100644 --- a/spec.toml +++ b/spec.toml @@ -1,6 +1,6 @@ database = "spec.db" document = "document.tex" -siteTitle = "Unnamed Website" +siteTitle = "Brownian Motion and Stochastic Calculus" [compiler] compileAll = false diff --git a/src/calculus/index.tex b/src/calculus/index.tex index acbd260..1d91e76 100644 --- a/src/calculus/index.tex +++ b/src/calculus/index.tex @@ -1,4 +1,4 @@ -\chapter{Integral Manifolds/Shenanigans} +\chapter{Integral Manifolds} \label{chap:integral} \input{./integrable.tex} diff --git a/src/index.tex b/src/index.tex index d151d7c..b2ad758 100644 --- a/src/index.tex +++ b/src/index.tex @@ -1,9 +1,14 @@ -\part{Diffusion Processes} +\part{Stochastic Processes} \label{part:diffusion} \input{./aws/index.tex} \input{./operator/index.tex} + + +\part{Stochastic Calculus} +\label{part:calculus} + \input{./diffusion/index.tex} -\input{./calculus/index.tex} \input{./sde/index.tex} -\input{./mal/index.tex} \ No newline at end of file +\input{./mal/index.tex} +\input{./calculus/index.tex} \ No newline at end of file diff --git a/src/mal/bridge.tex b/src/mal/bridge.tex new file mode 100644 index 0000000..f5a6e94 --- /dev/null +++ b/src/mal/bridge.tex @@ -0,0 +1,40 @@ +\section{The Brownian Bridge} +\label{section:brownian-bridge} + + +\begin{definition}[Brownian Bridge] +\label{definition:brownian-bridge} + Let $(\Omega, \cf, \bp)$ be a probability space, $a \in \real^d$, and $X: \Omega \to C([0, 1]; \real^d)$ be a Gaussian process, then $X$ is a \textbf{Brownian bridge} from $0$ to $a$ such that: + \begin{enumerate} + \item For each $t \in [0, 1]$, $\ev(X_t) = at$. + \item For each $s, t \in [0, 1]$, $\text{Cov}(X_s, X_t) = (s \wedge t - st)I$. + \end{enumerate} +\end{definition} + +\begin{theorem}[{{\cite[Theorem 40.3]{Rogers}}}] +\label{theorem:brownian-bridge} + Let $a \in \real^d$, then the following are equivalent definitions of the distribution of the Brownian bridge from $0$ to $a$. + \begin{enumerate} + \item There exists a Brownian motion $B$ such that for each $t \in [0, 1]$, + \[ + X_t = B_t + t(a - B_1) + \] + \item There exists a Brownian motion $B$ such that for each $t \in [0, 1]$, + \[ + X_t = at + (1 - t)B_{t/(1-t)} + \] + \item $X$ is a continuous process such that + \[ + Y_t = X_t - at + \int_0^1 \frac{X_s - as}{1 - s}ds + \] + + is a Brownian motion. + \item The distribution of $X$ is the $h$-transform of the classical Wiener measure on $C([0, 1]; \real^d)$, where + \[ + h(t, x) = \frac{1}{[2\pi(1 - t)]^{d/2}}\exp\braks{-\frac{\norm{x - a}_{\real^d}^2}{2(1 - t)}} + \] + \end{enumerate} + +\end{theorem} + + diff --git a/src/mal/brownian.tex b/src/mal/brownian.tex index dbba563..bd53f2d 100644 --- a/src/mal/brownian.tex +++ b/src/mal/brownian.tex @@ -1,5 +1,5 @@ -\section{$h$-Brownian Motions} -\label{section:h-brownian} +\section{Girsanov Transforms} +\label{section:girsanov} \begin{theorem}[Cameron-Martin-Girsanov, {{\cite[Theorem 38.5]{Rogers}}}] \label{theorem:cameron-martin} @@ -45,132 +45,3 @@ \end{enumerate} \end{theorem} -\begin{definition}[Mean Value Property] -\label{definition:mean-value-property} - Let $\bp$ be the distribution of a strong Markov process on $D([0, T]; \real^d)$. For each $0 \le s \le t \le T$ and $x \in \real^d$, let $P_{s, t}(x, \cdot)$ be its transition distribution, and $h: (0, \infty) \times \real^d \to [0, \infty)$ be a measurable function, then $h$ satisfies the \textbf{mean-value property} with respect to $\bp$ if for any $0 \le s \le t \le T$ and $x \in \real^d$, - \[ - h(s, x) = \int_{\real^d}h(t, y)P_{s, t}(x, dy) - \] -\end{definition} - - -\begin{lemma} -\label{lemma:h-density-martingale} - Let $\bp$ be the distribution of a strong Markov process on $D([0, T]; \real^d)$ with transition functions $P_{s, t}(x, \cdot)$, $\bracs{X_t|t \in [0, T]}$ be the canonical process, $\bracs{\cf_t|t \in [0, T]}$ be its natural filtration, $h: (0, \infty) \times \real^d \to [0, \infty)$ be a function satisfying the mean-value property with respect to $\bp$, and $A_0 = \bracs{h(0, \cdot) > 0}$, then the process - \[ - \one_{\bracs{X_0 \in A_0}} \frac{h(t, X_t)}{h(0, X_0)} - \] - - is a $\bracs{\mathcal{F}_t}$-martingale with respect to $\bp$. -\end{lemma} -\begin{proof} - Let $t \in [0, T]$ and $\mu$ be the initial distribution of $\bp$, then - \[ - \ev_\bp\braks{\one_{\bracs{X_0 \in A_0}} \frac{h(t, X_t)}{h(0, X_0)}} = - \int_{A_0} \frac{1}{h(0, x)}\int_{\real^d} h(t, y)P_{0, t}(x, dy) \mu(dx) = \mu(A_0) - \] - - so the given process is integrable. Now, since $h$ satisfies the mean-value property, - \begin{align*} - \ev_\bp\braks{\one_{\bracs{X_0 \in A_0}} \frac{h(T, X_T)}{h(0, X_0)} \bigg | \cf_t} &= \frac{\one_{\bracs{X_0 \in A_0}}}{h(0, X_0)} \cdot \ev_\bp\braks{h(t, X_t)| \cf_t} \\ - &= \one_{\bracs{X_0 \in A_0}} \frac{h(t, X_t)}{h(0, X_0)} - \end{align*} - - -\end{proof} - - -\begin{proposition} -\label{proposition:space-time-regular-measure} - Let $\bp$ be the distribution of a strong Markov process on $D([0, T]; \real^d)$ with transition functions $P_{s, t}(x, \cdot)$, $\mu$ be its initial distribution, $\bracs{X_t|t \in [0, T]}$ be the canonical process, $\bracs{\cf_t|t \in [0, T]}$ be its natural filtration, $h: (0, \infty) \times \real^d \to [0, \infty)$ be a function satisfying the mean-value property with respect to $\bp$, then there exists a probaiblity measure $\bp^h$ on $D([0, \infty); \real^d)$ such that: - \begin{enumerate} - \item For each $t \ge 0$, - \[ - \frac{d \bp^h}{d \bp} \bigg |_{\cf_{t^+}} = \frac{\one_{\bracs{X_0 \in A_0}}}{\mu(A_0)}\frac{h(t, X_t)}{h(0, X_0)} - \] - - where $A_0 = \bracs{h(0, \cdot) > 0}$. - - \item Under $\bp^h$, $X_t$ is a Markov process with transition function - \[ - P_{s, t}^h(x, dy) = \begin{cases} - \frac{h(t, y)}{h(s, x)}P_{s, t}(x, dy) &h(s, x) > 0 \\ - 0 & h(s, x) = 0 - \end{cases} - \] - \item If $\bp$ is the classical Wiener measure, then the process - \[ - \tilde X_t = X_t - \int_0^t \frac{\partial_x h(s, X_s)}{h(s, X_s)}ds - \] - - is a $\bp^h$-Brownian motion. - \end{enumerate} - - The distribution $\bp^h$ is the \textbf{$h$-transform} of $\mathbf{P}$. - -\end{proposition} - - -\begin{theorem}[Reuter, {{\cite[Theorem 39.66]{Rogers}}}] -\label{theorem:reuter} - Let $(\Omega, \cf, \bp)$ be a filtered probability space, $X: \Omega \to C([0, \infty); \real^d)$ be a standard $\bracs{\mathcal{F}_t}$-Brownian motion with drift $\mu$ starting at $0$. If $\tau = \inf\bracs{t \ge 0: |X_t| = 1}$, then $X_\tau$ and $\tau$ are independent. -\end{theorem} -\begin{proof} - If $\mu = 0$, then $X_\tau$ and $\tau$ are independent by rotational invariance. - - Otherwise, let $\wien$ be the classical Wiener measure on $C([0, \infty); \real^d)$, $\mathcal{V}$ be the distribution of $X$, and $Y$ be the canonical process on $C([0, \infty); \real^d)$. For each $t \ge 0$, let $\mathcal{G}_t = \sigma(\bracs{Y_s|0 \le s \le t})$, then by the Cameron-Martin formula, - \[ - h(t, Y_t) = \frac{d \mathcal{V}}{d \wien} \bigg |_{\mathcal{G}_t} = \exp\braks{\angles{\mu, Y_t}_{\real^d} - \frac{1}{2}\norm{\mu}_{\real^d}^2 t} - \] - - Since $h(t \wedge \tau, X_{t \wedge \tau})$ is a uniformly integrable martingale, - \[ - \frac{d \mathcal{V}}{d\wien} \bigg |_{\mathcal{G}_{T^+}} = h(\tau, Y_\tau) - \] - - In which case, for any measurable functions $f: \mathbb{S}^{d} \to [0, 1]$ and $g: [0, \infty) \to [0, 1]$, - \begin{align*} - \ev^{\mathcal{V}}[f(Y_\tau)g(\tau)] &= \ev^{\wien}\braks{f(Y_\tau)g(\tau)\exp(\angles{\mu, Y_\tau}_{\real^d} - \frac{1}{2}\norm{\mu}_{\real^d}^2 \tau)} \\ - &= \ev^\wien[f(Y_\tau)e^{\angles{\mu, Y_\tau}_{\real^d}}]\ev^\wien[g(\tau)e^{- \frac{1}{2}\norm{\mu}_{\real^d}^2 \tau}] \\ - &= \ev^{\mathcal{V}}(f(Y_\tau))\ev^{\mathcal{V}}(g(\tau)) - \end{align*} -\end{proof} - - -\begin{definition}[Brownian Bridge] -\label{definition:brownian-bridge} - Let $(\Omega, \cf, \bp)$ be a probability space, $a \in \real^d$, and $X: \Omega \to C([0, 1]; \real^d)$ be a Gaussian process, then $X$ is a \textbf{Brownian bridge} from $0$ to $a$ such that: - \begin{enumerate} - \item For each $t \in [0, 1]$, $\ev(X_t) = at$. - \item For each $s, t \in [0, 1]$, $\text{Cov}(X_s, X_t) = (s \wedge t - st)I$. - \end{enumerate} -\end{definition} - -\begin{theorem}[{{\cite[Theorem 40.3]{Rogers}}}] -\label{theorem:brownian-bridge} - Let $a \in \real^d$, then the following are equivalent definitions of the distribution of the Brownian bridge from $0$ to $a$. - \begin{enumerate} - \item There exists a Brownian motion $B$ such that for each $t \in [0, 1]$, - \[ - X_t = B_t + t(a - B_1) - \] - \item There exists a Brownian motion $B$ such that for each $t \in [0, 1]$, - \[ - X_t = at + (1 - t)B_{t/(1-t)} - \] - \item $X$ is a continuous process such that - \[ - Y_t = X_t - at + \int_0^1 \frac{X_s - as}{1 - s}ds - \] - - is a Brownian motion. - \item The distribution of $X$ is the $h$-transform of the classical Wiener measure on $C([0, 1]; \real^d)$, where - \[ - h(t, x) = \frac{1}{[2\pi(1 - t)]^{d/2}}\exp\braks{-\frac{\norm{x - a}_{\real^d}^2}{2(1 - t)}} - \] - \end{enumerate} - -\end{theorem} - - - diff --git a/src/mal/h.tex b/src/mal/h.tex new file mode 100644 index 0000000..d871ac4 --- /dev/null +++ b/src/mal/h.tex @@ -0,0 +1,93 @@ +\section{$h$-Transforms} +\label{section:h-transform} + + +\begin{definition}[Mean Value Property] +\label{definition:mean-value-property} + Let $\bp$ be the distribution of a strong Markov process on $D([0, T]; \real^d)$. For each $0 \le s \le t \le T$ and $x \in \real^d$, let $P_{s, t}(x, \cdot)$ be its transition distribution, and $h: (0, \infty) \times \real^d \to [0, \infty)$ be a measurable function, then $h$ satisfies the \textbf{mean-value property} with respect to $\bp$ if for any $0 \le s \le t \le T$ and $x \in \real^d$, + \[ + h(s, x) = \int_{\real^d}h(t, y)P_{s, t}(x, dy) + \] +\end{definition} + + +\begin{lemma} +\label{lemma:h-density-martingale} + Let $\bp$ be the distribution of a strong Markov process on $D([0, T]; \real^d)$ with transition functions $P_{s, t}(x, \cdot)$, $\bracs{X_t|t \in [0, T]}$ be the canonical process, $\bracs{\cf_t|t \in [0, T]}$ be its natural filtration, $h: (0, \infty) \times \real^d \to [0, \infty)$ be a function satisfying the mean-value property with respect to $\bp$, and $A_0 = \bracs{h(0, \cdot) > 0}$, then the process + \[ + \one_{\bracs{X_0 \in A_0}} \frac{h(t, X_t)}{h(0, X_0)} + \] + + is a $\bracs{\mathcal{F}_t}$-martingale with respect to $\bp$. +\end{lemma} +\begin{proof} + Let $t \in [0, T]$ and $\mu$ be the initial distribution of $\bp$, then + \[ + \ev_\bp\braks{\one_{\bracs{X_0 \in A_0}} \frac{h(t, X_t)}{h(0, X_0)}} = + \int_{A_0} \frac{1}{h(0, x)}\int_{\real^d} h(t, y)P_{0, t}(x, dy) \mu(dx) = \mu(A_0) + \] + + so the given process is integrable. Now, since $h$ satisfies the mean-value property, + \begin{align*} + \ev_\bp\braks{\one_{\bracs{X_0 \in A_0}} \frac{h(T, X_T)}{h(0, X_0)} \bigg | \cf_t} &= \frac{\one_{\bracs{X_0 \in A_0}}}{h(0, X_0)} \cdot \ev_\bp\braks{h(t, X_t)| \cf_t} \\ + &= \one_{\bracs{X_0 \in A_0}} \frac{h(t, X_t)}{h(0, X_0)} + \end{align*} + + +\end{proof} + +\begin{proposition} +\label{proposition:space-time-regular-measure} + Let $\bp$ be the distribution of a strong Markov process on $D([0, T]; \real^d)$ with transition functions $P_{s, t}(x, \cdot)$, $\mu$ be its initial distribution, $\bracs{X_t|t \in [0, T]}$ be the canonical process, $\bracs{\cf_t|t \in [0, T]}$ be its natural filtration, $h: (0, \infty) \times \real^d \to [0, \infty)$ be a function satisfying the mean-value property with respect to $\bp$, then there exists a probaiblity measure $\bp^h$ on $D([0, \infty); \real^d)$ such that: + \begin{enumerate} + \item For each $t \ge 0$, + \[ + \frac{d \bp^h}{d \bp} \bigg |_{\cf_{t^+}} = \frac{\one_{\bracs{X_0 \in A_0}}}{\mu(A_0)}\frac{h(t, X_t)}{h(0, X_0)} + \] + + where $A_0 = \bracs{h(0, \cdot) > 0}$. + + \item Under $\bp^h$, $X_t$ is a Markov process with transition function + \[ + P_{s, t}^h(x, dy) = \begin{cases} + \frac{h(t, y)}{h(s, x)}P_{s, t}(x, dy) &h(s, x) > 0 \\ + 0 & h(s, x) = 0 + \end{cases} + \] + \item If $\bp$ is the classical Wiener measure, then the process + \[ + \tilde X_t = X_t - \int_0^t \frac{\partial_x h(s, X_s)}{h(s, X_s)}ds + \] + + is a $\bp^h$-Brownian motion. + \end{enumerate} + + The distribution $\bp^h$ is the \textbf{$h$-transform} of $\mathbf{P}$. + +\end{proposition} + + +\begin{theorem}[Reuter, {{\cite[Theorem 39.66]{Rogers}}}] +\label{theorem:reuter} + Let $(\Omega, \cf, \bp)$ be a filtered probability space, $X: \Omega \to C([0, \infty); \real^d)$ be a standard $\bracs{\mathcal{F}_t}$-Brownian motion with drift $\mu$ starting at $0$. If $\tau = \inf\bracs{t \ge 0: |X_t| = 1}$, then $X_\tau$ and $\tau$ are independent. +\end{theorem} +\begin{proof} + If $\mu = 0$, then $X_\tau$ and $\tau$ are independent by rotational invariance. + + Otherwise, let $\wien$ be the classical Wiener measure on $C([0, \infty); \real^d)$, $\mathcal{V}$ be the distribution of $X$, and $Y$ be the canonical process on $C([0, \infty); \real^d)$. For each $t \ge 0$, let $\mathcal{G}_t = \sigma(\bracs{Y_s|0 \le s \le t})$, then by the Cameron-Martin formula, + \[ + h(t, Y_t) = \frac{d \mathcal{V}}{d \wien} \bigg |_{\mathcal{G}_t} = \exp\braks{\angles{\mu, Y_t}_{\real^d} - \frac{1}{2}\norm{\mu}_{\real^d}^2 t} + \] + + Since $h(t \wedge \tau, X_{t \wedge \tau})$ is a uniformly integrable martingale, + \[ + \frac{d \mathcal{V}}{d\wien} \bigg |_{\mathcal{G}_{T^+}} = h(\tau, Y_\tau) + \] + + In which case, for any measurable functions $f: \mathbb{S}^{d} \to [0, 1]$ and $g: [0, \infty) \to [0, 1]$, + \begin{align*} + \ev^{\mathcal{V}}[f(Y_\tau)g(\tau)] &= \ev^{\wien}\braks{f(Y_\tau)g(\tau)\exp(\angles{\mu, Y_\tau}_{\real^d} - \frac{1}{2}\norm{\mu}_{\real^d}^2 \tau)} \\ + &= \ev^\wien[f(Y_\tau)e^{\angles{\mu, Y_\tau}_{\real^d}}]\ev^\wien[g(\tau)e^{- \frac{1}{2}\norm{\mu}_{\real^d}^2 \tau}] \\ + &= \ev^{\mathcal{V}}(f(Y_\tau))\ev^{\mathcal{V}}(g(\tau)) + \end{align*} +\end{proof} diff --git a/src/mal/index.tex b/src/mal/index.tex index a62c2cd..bb0789b 100644 --- a/src/mal/index.tex +++ b/src/mal/index.tex @@ -1,4 +1,6 @@ -\chapter{Unknown Chapter} +\chapter{$h$-Transforms} \label{chap:mal} \input{./brownian.tex} +\input{./h.tex} +\input{./bridge.tex}