Infinitesimal generator (stochastic processes)

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^[1] 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 ${\displaystyle (X_{t})_{t\geq 0}}$ with Feller semigroup ${\displaystyle T=(T_{t})_{t\geq 0}}$ and state space ${\displaystyle E}$ we define the generator^[2] ${\displaystyle (A,D(A))}$ by

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

${\displaystyle Af=\lim _{t\downarrow 0}{\frac {T_{t}f-f}{t}},\quad \forall f\in D(A).}$ Where ${\displaystyle C_{\infty }(E)}$ denotes the Banach space of continuous functions on ${\displaystyle E}$ vanishing at infinity, equipped with the supremum norm and ${\displaystyle T_{t}f(x)=\mathbb {E} ^{x}f(X_{t})=\mathbb {E} (f(X_{t})|X_{0}=x)}$. In general, it is not easy to describe the domain of the Feller generator but it is always closed and densely defined. If ${\displaystyle X}$ is a ${\displaystyle \mathbb {R} ^{d}}$ valued and ${\displaystyle D(A)}$ contains the test functions (compactly supported smooth functions) then^[3]

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

Lévy processes[]

The generator of Lévy semigroup is of the form

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

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

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

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

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

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

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

Stochastic differential equations driven by Lévy processes[]

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

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 ${\displaystyle l,\sigma }$ are Lipschitz and of linear growth, then for each deterministic initial condition there exists a unique solution, which is Feller with symbol

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

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 ${\displaystyle \mathbb {R} ^{n}}$, which satisfies the stochastic differential equation ${\displaystyle dX_{t}=dB_{t}}$, has generator ${\textstyle {1 \over {2}}\Delta }$, where ${\displaystyle \Delta }$ denotes the Laplace operator.
• The two-dimensional process ${\displaystyle Y}$ satisfying:

${\displaystyle \mathrm {d} Y_{t}={\mathrm {d} t \choose \mathrm {d} B_{t}}}$

where ${\displaystyle B}$ is a one-dimensional Brownian motion, can be thought of as the graph of that Brownian motion, and has generator:

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

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

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

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

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

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

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

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 (Sixth ed.). Berlin: Springer. doi:10.1007/978-3-642-14394-6. ISBN 3-540-04758-1. (See Section 7.3)