Diffusion process

A diffusion process is a Markov process with continuous sample paths for which the Kolmogorov forward equation is the Fokker–Planck equation.{{cite web|title=9. Diffusion processes|url=http://math.nyu.edu/faculty/varadhan/stochastic.fall08/sec10.pdf|access-date=October 10, 2011}}

A diffusion process is defined by the following properties.

Let $a^\{ij\}(x,t)$ be uniformly continuous coefficients and $b^\{i\}(x,t)$ be bounded, Borel measurable drift terms. There is a unique family of probability measures $\backslash mathbb\{P\}^\{\backslash xi,\backslash tau\}\_\{a;b\}$ (for $\backslash tau\; \backslash ge\; 0$, $\backslash xi\; \backslash in\; \backslash mathbb\{R\}^d$) on the canonical space $\backslash Omega\; =\; C([0,\backslash infty),\; \backslash mathbb\{R\}^d)$, with its Borel $\backslash sigma$-algebra, such that:

1. (Initial Condition) The process starts at $\backslash xi$ at time $\backslash tau$: $\backslash mathbb\{P\}^\{\backslash xi,\backslash tau\}\_\{a;b\}[\backslash psi\; \backslash in\; \backslash Omega\; :\; \backslash psi(t)\; =\; \backslash xi\; \backslash text\{\; for\; \}\; 0\; \backslash le\; t\; \backslash le\; \backslash tau]\; =\; 1.$

2. (Local Martingale Property) For every $f\; \backslash in\; C^\{2,1\}(\backslash mathbb\{R\}^d\; \backslash times\; [\backslash tau,\backslash infty))$, the process

$M\_t^\{[f]\}\; =\; f(\backslash psi(t),t)\; -\; f(\backslash psi(\backslash tau),\backslash tau)\; -\; \backslash int\_\backslash tau^t\; \backslash bigl(L\_\{a;b\}\; +\; \backslash tfrac\{\backslash partial\}\{\backslash partial\; s\}\backslash bigr)\; f(\backslash psi(s),s)\backslash ,ds$

is a local martingale under $\backslash mathbb\{P\}^\{\backslash xi,\backslash tau\}\_\{a;b\}$ for $t\; \backslash ge\; \backslash tau$, with $M\_t^\{[f]\}\; =\; 0$ for $t\; \backslash le\; \backslash tau$.

This family $\backslash mathbb\{P\}^\{\backslash xi,\backslash tau\}\_\{a;b\}$ is called the $\backslash mathcal\{L\}\_\{a;b\}$-diffusion.

It is clear that if we have an $\backslash mathcal\{L\}\_\{a;b\}$-diffusion, i.e. $(X\_t)\_\{t\; \backslash ge\; 0\}$ on $(\backslash Omega,\; \backslash mathcal\{F\},\; \backslash mathcal\{F\}\_t,\; \backslash mathbb\{P\}^\{\backslash xi,\backslash tau\}\_\{a;b\})$, then $X\_t$ satisfies the SDE $dX\_t^i\; =\; \backslash frac\{1\}\{2\}\backslash ,\backslash sum\_\{k=1\}^d\; \backslash sigma^i\_k(X\_t)\backslash ,dB\_t^k\; +\; b^i(X\_t)\backslash ,dt$. In contrast, one can construct this diffusion from that SDE if $a^\{ij\}(x,t)\; =\; \backslash sum\_k\; \backslash sigma^k\_i(x,t)\backslash ,\backslash sigma^k\_j(x,t)$ and $\backslash sigma^\{ij\}(x,t)$, $b^i(x,t)$ are Lipschitz continuous.

To see this, let $X\_t$ solve the SDE starting at $X\_\backslash tau\; =\; \backslash xi$. For $f\; \backslash in\; C^\{2,1\}(\backslash mathbb\{R\}^d\; \backslash times\; [\backslash tau,\backslash infty))$, apply Itô's formula: $df(X\_t,t)\; =\; \backslash bigl(\backslash frac\{\backslash partial\; f\}\{\backslash partial\; t\}\; +\; \backslash sum\_\{i=1\}^d\; b^i\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\_i\}\; +\; v\; \backslash sum\_\{i,j=1\}^d\; a^\{ij\}\backslash ,\backslash frac\{\backslash partial^2\; f\}\{\backslash partial\; x\_i\; \backslash partial\; x\_j\}\backslash bigr)\backslash ,dt\; +\; \backslash sum\_\{i,k=1\}^d\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\_i\}\backslash ,\backslash sigma^i\_k\backslash ,dB\_t^k.$ Rearranging gives $f(X\_t,t)\; -\; f(X\_\backslash tau,\backslash tau)\; -\; \backslash int\_\backslash tau^t\; \backslash bigl(\backslash frac\{\backslash partial\; f\}\{\backslash partial\; s\}\; +\; L\_\{a;b\}f\backslash bigr)\backslash ,ds\; =\; \backslash int\_\backslash tau^t\; \backslash sum\_\{i,k=1\}^d\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\_i\}\backslash ,\backslash sigma^i\_k\backslash ,dB\_s^k,$ whose right‐hand side is a local martingale, matching the local‐martingale property in the diffusion definition. The law of $X\_t$ defines $\backslash mathbb\{P\}^\{\backslash xi,\backslash tau\}\_\{a;b\}$ on $\backslash Omega\; =\; C([0,\backslash infty),\; \backslash mathbb\{R\}^d)$ with the correct initial condition and local martingale property. Uniqueness follows from the Lipschitz continuity of $\backslash sigma\backslash !,\backslash !b$. In fact, $L\_\{a;b\}\; +\; \backslash tfrac\{\backslash partial\}\{\backslash partial\; s\}$ coincides with the infinitesimal generator $\backslash mathcal\{A\}$ of this process. If $X\_t$ solves the SDE, then for $f(\backslash mathbf\{x\},t)\; \backslash in\; C^2(\backslash mathbb\{R\}^d\; \backslash times\; \backslash mathbb\{R\}^+)$, the generator $\backslash mathcal\{A\}$ is $\backslash mathcal\{A\}f(\backslash mathbf\{x\},t)\; =\; \backslash sum\_\{i=1\}^d\; b\_i(\backslash mathbf\{x\},t)\backslash ,\backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\_i\}\; +\; v\backslash sum\_\{i,j=1\}^d\; a\_\{ij\}(\backslash mathbf\{x\},t)\backslash ,\backslash frac\{\backslash partial^2\; f\}\{\backslash partial\; x\_i\; \backslash partial\; x\_j\}\; +\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; t\}.$

• Stochastic differential equation
• Itô calculus
• Fokker–Planck equation
• Markov process
• Diffusion
• Itô diffusion
• Jump diffusion
• Sample-continuous process

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Category:Markov processes

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