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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 @@ $$\begin{aligned} <label for="proof-kolmogorov">Proof</label> <div class="hidden" markdown="1"> <label for="proof-kolmogorov">Proof.</label> -We define a new process $Y_t \equiv h(X_t)$, and then apply Itō's lemma, leading to: +We define a new process $$Y_t \equiv h(X_t)$$, and then apply Itō's lemma, leading to: $$\begin{aligned} \dd{Y_t} @@ -53,7 +53,7 @@ $$\begin{aligned} &= \hat{L}\{h(X_t)\} \dd{t} + \pdv{h}{x} g(X_t) \dd{B_t} \end{aligned}$$ -Where we have recognized the definition of $\hat{L}$. +Where we have recognized the definition of $$\hat{L}$$. Integrating the above equation yields: $$\begin{aligned} @@ -63,14 +63,14 @@ $$\begin{aligned} As always, the latter [Itō integral](/know/concept/ito-integral/) is a [martingale](/know/concept/martingale/), so it vanishes -when we take the expectation conditioned on the "initial" state $X_0$, leaving: +when we take the expectation conditioned on the "initial" state $$X_0$$, leaving: $$\begin{aligned} \mathbf{E}[Y_t | X_0] = Y_0 + \mathbf{E}\bigg[ \int_0^t \hat{L}\{h(X_s)\} \dd{s} \bigg| X_0 \bigg] \end{aligned}$$ -For suffiently small $t$, the integral can be replaced by its first-order approximation: +For suffiently small $$t$$, the integral can be replaced by its first-order approximation: $$\begin{aligned} \mathbf{E}[Y_t | X_0] @@ -78,22 +78,23 @@ $$\begin{aligned} \end{aligned}$$ Rearranging this gives the following, -to be understood in the limit $t \to 0^+$: +to be understood in the limit $$t \to 0^+$$: $$\begin{aligned} \hat{L}\{h(X_0)\} \approx \frac{1}{t} \mathbf{E}[Y_t - Y_0| X_0] \end{aligned}$$ + </div> </div> The general definition of resembles that of a classical derivative, -and indeed, the generator $\hat{A}$ can be thought of as a differential operator. +and indeed, the generator $$\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 @@ $$\begin{aligned} <div class="hidden" markdown="1"> <label for="proof-dynkin">Proof.</label> The proof is similar to the one above. -Define $Y_t = h(X_t)$ and use Itō’s lemma: +Define $$Y_t = h(X_t)$$ and use Itō’s lemma: $$\begin{aligned} \dd{Y_t} @@ -118,7 +119,7 @@ $$\begin{aligned} &= \hat{L} \{h(X_t)\} \dd{t} + \pdv{h}{x} g(X_t) \dd{B_t} \end{aligned}$$ -And then integrate this from $t = 0$ to the provided stopping time $t = \tau$: +And then integrate this from $$t = 0$$ to the provided stopping time $$t = \tau$$: $$\begin{aligned} Y_\tau @@ -127,7 +128,7 @@ $$\begin{aligned} All [Itō integrals](/know/concept/ito-integral/) are [martingales](/know/concept/martingale/), -so the latter integral's conditional expectation is zero for the "initial" condition $X_0$. +so the latter integral's conditional expectation is zero for the "initial" condition $$X_0$$. The rest of the above equality is also a martingale: $$\begin{aligned} @@ -135,30 +136,30 @@ $$\begin{aligned} = \mathbf{E}\bigg[ Y_\tau - Y_0 - \int_0^\tau \hat{L}\{h(X_t)\} \dd{t} \bigg| X_0 \bigg] \end{aligned}$$ -Isolating this equation for $\mathbf{E}[Y_\tau | X_0]$ then gives Dynkin's formula. +Isolating this equation for $$\mathbf{E}[Y_\tau | X_0]$$ then gives Dynkin's formula. </div> </div> A common application of Dynkin's formula is predicting -when the stopping time $\tau$ occurs, and in what state $X_\tau$ this happens. +when the stopping time $$\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)$$. |