Infinitesimal generator (stochastic processes)

This article is about infinitesimal generator for general stochastic processes. For generators for the special case of finite-state continuous time Markov chains, see transition rate matrix.

In mathematics — specifically, in stochastic analysis — the infinitesimal generator of a Feller process (i.e. a continuous-time Markov process satisfying certain regularity conditions) is a Fourier multiplier 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

General case

For a Feller process $(X_{t})_{t\geq 0}$ with Feller semigroup $T=(T_{t})_{t\geq 0}$ and state space $E$ we define the generator $(A,D(A))$ by

$D(A)=\left\{f\in C_{0}(E):\lim _{t\downarrow 0}{\frac {T_{t}f-f}{t}}{\text{ exists as uniform limit}}\right\}$,

$Af=\lim _{t\downarrow 0}{\frac {T_{t}f-f}{t}}$, for any $f\in D(A)$.

Here $C_{0}(E)$ denotes the Banach space of continuous functions on $E$ vanishing at infinity, equipped with the supremum norm, and $T_{t}f(x)=\mathbb {E} ^{x}f(X_{t})=\mathbb {E} (f(X_{t})|X_{0}=x)$. In general, it is not easy to describe the domain of the Feller generator but it is always closed and densely defined. If $X$ is $\mathbb {R} ^{d}$-valued and $D(A)$ contains the test functions (compactly supported smooth functions) then

$Af(x)=-c(x)f(x)+l(x)\cdot \nabla f(x)+{\frac {1}{2}}{\text{div}}Q(x)\nabla f(x)+\int _{\mathbb {R} ^{d}\setminus {\{0\}}}\left(f(x+y)-f(x)-\nabla f(x)\cdot y\chi (|y|)\right)N(x,dy)$, where $c(x)\geq 0$, and $(l(x),Q(x),N(x,\cdot ))$ is a Lévy triplet for fixed $x\in \mathbb {R} ^{d}$.

Lévy processes

The generator of Lévy semigroup is of the form

$Af(x)=l\cdot \nabla f(x)+{\frac {1}{2}}{\text{div}}Q\nabla f(x)+\int _{\mathbb {R} ^{d}\setminus {\{0\}}}\left(f(x+y)-f(x)-\nabla f(x)\cdot y\chi (|y|)\right)\nu (dy)$

where $l\in \mathbb {R} ^{d},Q\in \mathbb {R} ^{d\times d}$ is positive semidefinite and $\nu$ is a Lévy measure satisfying

$\int _{\mathbb {R} ^{d}\setminus \{0\}}\min(|y|^{2},1)\nu (dy)<\infty$ and $0\leq 1-\chi (s)\leq \kappa \min(s,1)$for some $\kappa >0$ with $s\chi (s)$ is bounded. If we define

$\psi (\xi )=\psi (0)-il\cdot \xi +{\frac {1}{2}}\xi \cdot Q\xi +\int _{\mathbb {R} ^{d}\setminus {\{0\}}}(1-e^{iy\cdot \xi }+i\xi \cdot y\chi (|y|))\nu (dy)$

for $\psi (0)\geq 0$ then the generator can be written as

$Af(x)=-\int e^{ix\cdot \xi }\psi (\xi ){\hat {f}}(\xi )d\xi$

where ${\hat {f}}$ denotes the Fourier transform. So the generator of a Lévy process (or semigroup) is a Fourier multiplier operator with symbol $-\psi$.

Stochastic differential equations driven by Lévy processes

Let ${\textstyle L}$ be a Lévy process with symbol $\psi$ (see above). Let $\Phi$ be locally Lipschitz and bounded. The solution of the SDE $dX_{t}=\Phi (X_{t-})dL_{t}$ exists for each deterministic initial condition $x\in \mathbb {R} ^{d}$ and yields a Feller process with symbol $q(x,\xi )=\psi (\Phi ^{\top }(x)\xi ).$

Note that in general, the solution of an SDE driven by a Feller process which is not Lévy might fail to be Feller or even Markovian.

As a simple example consider ${\textstyle dX_{t}=l(X_{t})dt+\sigma (X_{t})dW_{t}}$ with a Brownian motion driving noise. If we assume $l,\sigma$ are Lipschitz and of linear growth, then for each deterministic initial condition there exists a unique solution, which is Feller with symbol

$q(x,\xi )=-il(x)\cdot \xi +{\frac {1}{2}}\xi Q(x)\xi .$

Generators of some common processes

• For finite-state continuous time Markov chains the generator may be expressed as a transition rate matrix
• Standard Brownian motion on $\mathbb {R} ^{n}$, which satisfies the stochastic differential equation $dX_{t}=dB_{t}$, has generator ${\textstyle {1 \over {2}}\Delta }$, where $\Delta$ 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:

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

• The Ornstein–Uhlenbeck process on $\mathbb {R}$, which satisfies the stochastic differential equation ${\textstyle dX_{t}=\theta (\mu -X_{t})dt+\sigma dB_{t}}$, has generator:

${\mathcal {A}}f(x)=\theta (\mu -x)f'(x)+{\frac {\sigma ^{2}}{2}}f''(x)$

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

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

• A geometric Brownian motion on $\mathbb {R}$, which satisfies the stochastic differential equation ${\textstyle dX_{t}=rX_{t}dt+\alpha X_{t}dB_{t}}$, has generator:

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

See also

• Dynkin's formula

References

• Calin, Ovidiu (2015). An Informal Introduction to Stochastic Calculus with Applications. Singapore: World Scientific Publishing. p. 315. ISBN 978-981-4678-93-3. (See Chapter 9)
• Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications. Universitext (Sixth ed.). Berlin: Springer. doi:10.1007/978-3-642-14394-6. ISBN 3-540-04758-1. (See Section 7.3)