From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Thu, 20 Oct 2022 18:25:31 +0200 Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex' --- source/know/concept/dynkins-formula/index.md | 81 ++++++++++++++-------------- 1 file changed, 41 insertions(+), 40 deletions(-) (limited to 'source/know/concept/dynkins-formula') diff --git a/source/know/concept/dynkins-formula/index.md b/source/know/concept/dynkins-formula/index.md index 53da86f..c0d20c5 100644 --- a/source/know/concept/dynkins-formula/index.md +++ b/source/know/concept/dynkins-formula/index.md @@ -8,12 +8,12 @@ categories: layout: "concept" --- -Given an [Itō diffusion](/know/concept/ito-calculus/) $X_t$ -with a time-independent drift $f$ and intensity $g$ -such that the diffusion uniquely exists on the $t$-axis. -We define the **infinitesimal generator** $\hat{A}$ -as an operator with the following action on a given function $h(x)$, -where $\mathbf{E}$ is a +Given an [Itō diffusion](/know/concept/ito-calculus/) $$X_t$$ +with a time-independent drift $$f$$ and intensity $$g$$ +such that the diffusion uniquely exists on the $$t$$-axis. +We define the **infinitesimal generator** $$\hat{A}$$ +as an operator with the following action on a given function $$h(x)$$, +where $$\mathbf{E}$$ is a [conditional expectation](/know/concept/conditional-expectation/): $$\begin{aligned} @@ -23,13 +23,13 @@ $$\begin{aligned} } \end{aligned}$$ -Which only makes sense for $h$ where this limit exists. -The assumption that $X_t$ does not have any explicit time-dependence -means that $X_0$ need not be the true initial condition; -it can also be the state $X_s$ at any $s$ infinitesimally smaller than $t$. +Which only makes sense for $$h$$ where this limit exists. +The assumption that $$X_t$$ does not have any explicit time-dependence +means that $$X_0$$ need not be the true initial condition; +it can also be the state $$X_s$$ at any $$s$$ infinitesimally smaller than $$t$$. -Conveniently, for a sufficiently well-behaved $h$, -the generator $\hat{A}$ is identical to the Kolmogorov operator $\hat{L}$ +Conveniently, for a sufficiently well-behaved $$h$$, +the generator $$\hat{A}$$ is identical to the Kolmogorov operator $$\hat{L}$$ found in the [backward Kolmogorov equation](/know/concept/kolmogorov-equations/): $$\begin{aligned} @@ -44,7 +44,7 @@ $$\begin{aligned} The general definition of resembles that of a classical derivative, -and indeed, the generator $\hat{A}$ can be thought of as a differential operator. +and indeed, the generator $$\hat{A}$$ can be thought of as a differential operator. In that case, we would like an analogue of the classical fundamental theorem of calculus to relate it to integration. Such an analogue is provided by **Dynkin's formula**: -for a stopping time $\tau$ with a finite expected value $\mathbf{E}[\tau|X_0] < \infty$, +for a stopping time $$\tau$$ with a finite expected value $$\mathbf{E}[\tau|X_0] < \infty$$, it states that: $$\begin{aligned} @@ -109,7 +110,7 @@ $$\begin{aligned} A common application of Dynkin's formula is predicting -when the stopping time $\tau$ occurs, and in what state $X_\tau$ this happens. +when the stopping time $$\tau$$ occurs, and in what state $$X_\tau$$ this happens. Consider an example: -for a region $\Omega$ of state space with $X_0 \in \Omega$, -we define the exit time $\tau \equiv \inf\{ t : X_t \notin \Omega \}$, -provided that $\mathbf{E}[\tau | X_0] < \infty$. +for a region $$\Omega$$ of state space with $$X_0 \in \Omega$$, +we define the exit time $$\tau \equiv \inf\{ t : X_t \notin \Omega \}$$, +provided that $$\mathbf{E}[\tau | X_0] < \infty$$. -To get information about when and where $X_t$ exits $\Omega$, -we define the *general reward* $\Gamma$ as follows, -consisting of a *running reward* $R$ for $X_t$ inside $\Omega$, -and a *terminal reward* $T$ on the boundary $\partial \Omega$ where we stop at $X_\tau$: +To get information about when and where $$X_t$$ exits $$\Omega$$, +we define the *general reward* $$\Gamma$$ as follows, +consisting of a *running reward* $$R$$ for $$X_t$$ inside $$\Omega$$, +and a *terminal reward* $$T$$ on the boundary $$\partial \Omega$$ where we stop at $$X_\tau$$: $$\begin{aligned} \Gamma = \int_0^\tau R(X_t) \dd{t} + \: T(X_\tau) \end{aligned}$$ -For example, for $R = 1$ and $T = 0$, this becomes $\Gamma = \tau$, -and if $R = 0$, then $T(X_\tau)$ can tell us the exit point. -Let us now define $h(X_0) = \mathbf{E}[\Gamma | X_0]$, +For example, for $$R = 1$$ and $$T = 0$$, this becomes $$\Gamma = \tau$$, +and if $$R = 0$$, then $$T(X_\tau)$$ can tell us the exit point. +Let us now define $$h(X_0) = \mathbf{E}[\Gamma | X_0]$$, and apply Dynkin's formula: $$\begin{aligned} @@ -168,11 +169,11 @@ $$\begin{aligned} &= \mathbf{E}\big[ T(X_\tau) | X_0 \big] + \mathbf{E}\bigg[ \int_0^\tau \hat{L}\{h(X_t)\} + R(X_t) \dd{t} \bigg| X_0 \bigg] \end{aligned}$$ -The two leftmost terms depend on the exit point $X_\tau$, -but not directly on $X_t$ for $t < \tau$, -while the rightmost depends on the whole trajectory $X_t$. +The two leftmost terms depend on the exit point $$X_\tau$$, +but not directly on $$X_t$$ for $$t < \tau$$, +while the rightmost depends on the whole trajectory $$X_t$$. Therefore, the above formula is fulfilled -if $h(x)$ satisfies the following equation and boundary conditions: +if $$h(x)$$ satisfies the following equation and boundary conditions: $$\begin{aligned} \boxed{ @@ -183,8 +184,8 @@ $$\begin{aligned} } \end{aligned}$$ -In other words, we have just turned a difficult question about a stochastic trajectory $X_t$ -into a classical differential boundary value problem for $h(x)$. +In other words, we have just turned a difficult question about a stochastic trajectory $$X_t$$ +into a classical differential boundary value problem for $$h(x)$$. -- cgit v1.2.3