Itô diffusion

In addition to being (sample) continuous, an Itô diffusion X satisfies the stronger requirement to be a Feller-continuous process.

The precise mathematical formulation of this statement requires some additional notation: Let Σ∗ denote the natural filtration of (Ω, Σ) generated by the Brownian motion B: for t ≥ 0, It is easy to show that X is adapted to Σ∗ (i.e. each Xt is Σt-measurable), so the natural filtration F∗ = F∗X of (Ω, Σ) generated by X has Ft ⊆ Σt for each t ≥ 0.

Formally, the infinitesimal generator of an Itô diffusion X is the operator A, which is defined to act on suitable functions f : Rn → R by The set of all functions f for which this limit exists at a point x is denoted DA(x), while DA denotes the set of all f for which the limit exists for all x ∈ Rn.

One can show that any compactly-supported C2 (twice differentiable with continuous second derivative) function f lies in DA and that or, in terms of the gradient and scalar and Frobenius inner products, The generator A for standard n-dimensional Brownian motion B, which satisfies the stochastic differential equation dXt = dBt, is given by i.e., A = Δ/2, where Δ denotes the Laplace operator.

If Ex[τU] = +∞ for all open sets U containing x, define The characteristic operator and infinitesimal generator are very closely related, and even agree for a large class of functions.

Conversely, a given second-order linear partial differential equation of the form Λf = 0 may be hard to solve directly, but if Λ = A∗ for some Itô diffusion X, and an invariant measure for X is easy to compute, then that measure's density provides a solution to the partial differential equation.

An invariant measure is comparatively easy to compute when the process X is a stochastic gradient flow of the form where β > 0 plays the role of an inverse temperature and Ψ : Rn → R is a scalar potential satisfying suitable smoothness and growth conditions.

The expression for the variance may be interpreted as follows: large values of κ mean that the potential well Ψ has "very steep sides", so Xt is unlikely to move far from the minimum of Ψ at m; similarly, large values of β mean that the system is quite "cold" with little noise, so, again, Xt is unlikely to move far away from m. In general, an Itô diffusion X is not a martingale.

The proof is quite simple: it follows from the usual expression of the action of the generator on smooth enough functions f and Itô's lemma (the stochastic chain rule) that Since Itô integrals are martingales with respect to the natural filtration Σ∗ of (Ω, Σ) by B, for t > s, Hence, as required, since Ms is Fs-measurable.

For example, canonical Brownian motion on the real line starting at 0 exits the interval (−R, +R) at a random time τR with expected value Dynkin's formula provides information about the behaviour of X at a fairly general stopping time.

For more information on the distribution of X at a hitting time, one can study the harmonic measure of the process.

In many situations, it is sufficient to know when an Itô diffusion X will first leave a measurable set H ⊆ Rn.

The harmonic measure satisfies an interesting mean value property: if f : Rn → R is any bounded, Borel-measurable function and φ is given by then, for all Borel sets G ⊂⊂ H and all x ∈ G, The mean value property is very useful in the solution of partial differential equations using stochastic processes.

Let A be a partial differential operator on a domain D ⊆ Rn and let X be an Itô diffusion with A as its generator.