Infinitesimal generator (stochastic processes)

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[1] ${\displaystyle (A,D(A))}$ by

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

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

Here ${\displaystyle C_{0}(E)}$ 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. However, the Feller generator is always closed and densely defined. If ${\displaystyle X}$ is ${\displaystyle \mathbb {R} ^{d}}$-valued and ${\displaystyle D(A)}$ contains the test functions (compactly supported smooth functions) then[1]

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

where ${\displaystyle c(x)\geq 0}$, and ${\displaystyle (l(x),Q(x),N(x,\cdot ))}$ is a Lévy triplet for fixed ${\displaystyle x\in \mathbb {R} ^{d}}$.

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 ${\displaystyle -\psi }$.

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

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