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 Itō 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

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

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

$Af(x)=\sum _{i}b_{i}(x){\frac {\partial f}{\partial x_{i}}}(x)+{\frac {1}{2}}\sum _{i,j}{\big (}\sigma (x)\sigma (x)^{\top }{\big )}_{i,j}(x){\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}(x),$

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

$Af(x)=b(x)\cdot \nabla _{x}f(x)+{\frac {1}{2}}{\big (}\sigma (x)\sigma (x)^{\top }{\big )}:\nabla _{x}\nabla _{x}f(x).$

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

$Af(t,x)={\frac {\partial f}{\partial t}}(t,x)+{\frac {1}{2}}{\frac {\partial ^{2}f}{\partial x^{2}}}(t,x).$

• The Ornstein-Uhlenbeck process on R, which satisfies the stochastic differential equation dX_t = μX_t dt + σ dB_t, has generator

$Af(x)=\mu xf'(x)+{\frac {\sigma ^{2}}{2}}f''(x).$

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

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

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

$Af(x)=rxf'(x)+{\frac {1}{2}}\alpha ^{2}x^{2}f''(x).$

References

• Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth edition ed.). Berlin: Springer. ISBN 3-540-04758-1. {{cite book}}: |edition= has extra text (help) (See Section 7)