Infinitesimal generator (stochastic processes)

In mathematics — specifically, in stochastic analysis — the infinitesimal generator of a stochastic process is a partial differential operator that encodes a great deal of information about the process. The generator is used in evolution equations such as the Kolmogorov backward equation (which describes the evolution of statistics of the process); its L² Hermitian adjoint is used in evolution equations such as the Fokker–Planck equation (which describes the evolution of the probability density functions of the process).

Definition

Let X : [0, +∞) × Ω → Rⁿ defined on a probability space (Ω, Σ, P) be an Itô diffusion satisfying a stochastic differential equation of the form

$\mathrm{d} X_{t} = b(X_{t}) \, \mathrm{d} t + \sigma (X_{t}) \, \mathrm{d} B_{t},$

where B is an m-dimensional Brownian motion and b : Rⁿ → Rⁿ and σ : Rⁿ → R^n×m are the drift and diffusion fields respectively. For a point x ∈ Rⁿ, let P^x denote the law of X given initial datum X₀ = x, and let E^x denote expectation with respect to P^x.

The infinitesimal generator of X is the operator A, which is defined to act on suitable functions f : Rⁿ → R by

$A f (x) = \lim_{t \downarrow 0} \frac{\mathbf{E}^{x} [f(X_{t})] - f(x)}{t}.$

The set of all functions f for which this limit exists at a point x is denoted D_A(x), while D_A 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 D_A and that

$A f (x) = \sum_{i} b_{i} (x) \frac{\partial f}{\partial x_{i}} (x) + \frac1{2} \sum_{i, j} \big( \sigma (x) \sigma (x)^{\top} \big)_{i, j} \frac{\partial^{2} f}{\partial x_{i} \, \partial x_{j}} (x),$

or, in terms of the gradient and scalar and Frobenius inner products,

$A f (x) = b(x) \cdot \nabla_{x} f(x) + \frac1{2} \big( \sigma(x) \sigma(x)^{\top} \big) : \nabla_{x} \nabla_{x} f(x).$

Generators of some common processes

• Standard Brownian motion on Rⁿ, which satisfies the stochastic differential equation dX_t = dB_t, has generator ½Δ, where Δ denotes the Laplace operator.
• The two-dimensional process Y satisfying

$\mathrm{d} Y_{t} = { \mathrm{d} t \choose \mathrm{d} B_{t} } ,$

where B is a one-dimensional Brownian motion, can be thought of as the graph of that Brownian motion, and has generator

$A f(t, x) = \frac{\partial f}{\partial t} (t, x) + \frac1{2} \frac{\partial^{2} f}{\partial x^{2}} (t, x).$

• The Ornstein–Uhlenbeck process on R, which satisfies the stochastic differential equation dX_t = θ (μ − X_t) dt + σ dB_t, has generator

$A f(x) = \theta(\mu - x) f'(x) + \frac{\sigma^{2}}{2} f''(x).$

• Similarly, the graph of the Ornstein–Uhlenbeck process has generator

$A f(t, x) = \frac{\partial f}{\partial t} (t, x) + \theta(\mu - x) \frac{\partial f}{\partial x} (t, x) + \frac{\sigma^{2}}{2} \frac{\partial^{2} f}{\partial x^{2}} (t, x).$

• A geometric Brownian motion on R, which satisfies the stochastic differential equation dX_t = rX_t dt + αX_t dB_t, has generator

$A f(x) = r x f'(x) + \frac1{2} \alpha^{2} x^{2} f''(x).$

See also

• Dynkin's formula

References

• Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. doi:10.1007/978-3-642-14394-6. ISBN 3-540-04758-1. (See Section 7.3)