Positive invariant set

29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church. 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

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

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

$Af(x)=\sum _i{b}_i(x)\frac{\partial f}{\partial x_i}(x)+\frac{1}{2}\sum _{i,j}(\sigma (x)\sigma (x{)}^{\mathrm{\top }}{)}_{i,j}\frac{{\partial }^{2}f}{\partial x_i\partial x_j}(x),$

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

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

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

$\mathrm{d}{Y}_t=(\genfrac{}{}{0ex}{}{\mathrm{d}t}{\mathrm{d}{B}_t}),$

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 = θ (μ − X_t) dt + σ dB_t, has generator

$Af(x)=\theta (\mu -x)f^{\prime }(x)+\frac{{\sigma }^2}{2}{f}^{″}(x).$

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

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

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

$Af(x)=rx{f}^{\prime }(x)+\frac{1}{2}{\alpha }^2{x}^2{f}^{″}(x).$

See also

• Dynkin's formula

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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