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Infinitesimal generator (stochastic processes)

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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).

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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

d X t = b ( X t ) d t + σ ( X t ) 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 ↓ 0 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 ) = ∑ i b i ( x ) ∂ f ∂ x i ( x ) + 1 2 ∑ i , j ( σ ( x ) σ ( x ) ⊤ ) i , j ∂ 2 f ∂ x i ∂ x j ( x ) ,

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

A f ( x ) = b ( x ) ⋅ ∇ x f ( x ) + 1 2 ( σ ( x ) σ ( x ) ⊤ ) : ∇ x ∇ x f ( x ) .

Generators of some common processes

  • Standard Brownian motion on Rn, which satisfies the stochastic differential equation dXt = dBt, has generator ½Δ, where Δ denotes the Laplace operator.
  • The two-dimensional process Y satisfying
    • where B is a one-dimensional Brownian motion, can be thought of as the graph of that Brownian motion, and has generator
  • The Ornstein–Uhlenbeck process on R, which satisfies the stochastic differential equation dXt = θ (μ − Xt) dt + σ dBt, has generator
  • Similarly, the graph of the Ornstein–Uhlenbeck process has generator
  • A geometric Brownian motion on R, which satisfies the stochastic differential equation dXt = rXt dt + αXt dBt, has generator

    • References

    Infinitesimal generator (stochastic processes) Wikipedia
    (Text) CC BY-SA