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 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 ''L''² 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 $(X_t)_$ with Feller semigroup $T=(T_t)_$ and state space $E$ we define the generator $(A,D(A))$ by

:$D(A)=\left\$,
:$A f=\lim_ \frac$, for any $f\in D(A)$.
Here $C_(E)$ denotes the Banach space of continuous functions on $E$ vanishing at infinity, equipped with the supremum norm, and $T_t f(x)= \mathbb^x f(X_t)=\mathbb(f(X_t), X_0=x)$. 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 $\mathbb^d$-valued and $D(A)$ contains the test functions (compactly supported smooth functions) then
:$A f(x)=- c(x) f(x) + l (x) \cdot \nabla f(x) + \frac \text Q(x) \nabla f(x) + \int_ \left( f(x+y)-f(x)-\nabla f(x) \cdot y \chi(, y, ) \right) N(x,dy)$,
where $c(x) \geq 0$, and $(l(x), Q(x),N(x,\cdot))$ is a Lévy triplet for fixed $x \in \mathbb^d$.

Lévy processes

The generator of Lévy semigroup is of the form

$A f(x)= l \cdot \nabla f(x) + \frac \text Q \nabla f(x) + \int_ \left( f(x+y)-f(x)-\nabla f(x) \cdot y \chi(, y, ) \right) \nu(dy)$

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

$\int_ \min(, y, ^2,1) \nu(dy) < \infty$ and $0 \leq 1-\chi(s) \leq \kappa \min(s,1)$for some $\kappa >0$ with $s \chi(s)$ is bounded. If we define

$\psi(\xi)=\psi(0)-i l \cdot \xi + \frac \xi \cdot Q \xi + \int_ (1-e^+i\xi \cdot y \chi(, y, )) \nu(dy )$

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

$A f (x)= - \int e^ \psi (\xi) \hat(\xi) d \xi$

where $\hat$ 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 $L$ be a Lévy process with symbol $\psi$ (see above). Let $\Phi$ be locally Lipschitz and bounded. The solution of the SDE $d X_t = \Phi(X_) d L_t$ exists for each deterministic initial condition $x \in \mathbb^d$ 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 $d X_t = l(X_t) dt+ \sigma(X_t) dW_t$ 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

$q(x,\xi)=- i l(x)\cdot \xi + \frac \xi Q(x)\xi.$

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 $\mathbb^$, which satisfies the stochastic differential equation $dX_ = dB_$, has generator $\Delta$, where $\Delta$ denotes the Laplace operator.
* The two-dimensional process $Y$ satisfying:

::$\mathrm Y_ =$

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

::$\mathcalf(t, x) = \frac (t, x) + \frac1 \frac (t, x)$

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

::$\mathcal f(x) = \theta(\mu - x) f'(x) + \frac f''(x)$

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

::$\mathcal f(t, x) = \frac (t, x) + \theta(\mu - x) \frac (t, x) + \frac \frac (t, x)$

* A geometric Brownian motion on $\mathbb$, which satisfies the stochastic differential equation $dX_ = rX_dt + \alpha X_dB_$, has generator:

::$\mathcal f(x) = r x f'(x) + \frac1 \alpha^ x^ f''(x)$

See also

* Dynkin's formula

References

* (See Chapter 9)
* (See Section 7.3)

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Stochastic differential equations