Infinitesimal generator (stochastic processes)

This article is about infinitesimal generator for general stochastic processes. For generators for continuous time Markov chains, see transition rate matrix.

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  Rn×m are the drift and diffusion fields respectively. For a point x  Rn, let Px denote the law of X given initial datum X0 = x, and let Ex denote expectation with respect to Px.

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 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

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

\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).
A f(x) = \theta(\mu - x) f'(x) + \frac{\sigma^{2}}{2} f''(x).
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 f(x) = r x f'(x) + \frac1{2} \alpha^{2} x^{2} f''(x).

See also

References

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