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Infinitesimal generator (stochastic processes)

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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 → 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

  • 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
\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 dXt = θ (μ − Xt) dt + σ dBt, 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 dXt = rXt dt + αXt dBt, has generator
A f(x) = r x f'(x) + \frac1{2} \alpha^{2} x^{2} f''(x).

See also

References

  • (See Section 7.3)
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