In the previous section, we have seen that if is the solution of a stochastic differential equation
then is a Markov process, that is for every ,
where . It is remarkable that this property still holds when is now any finite stopping time. This property is called the strong Markov property.
The key lemma is the following:
Lemma. Let be a standard Brownian motion and let be a finite stopping time. The process, is a standard Brownian motion independent from .
Proof. Let be a finite stopping time of the filtration . We first assume bounded. Let us consider the process Let , . Applying Doob’s stopping theorem to the martingale with the stopping times and , yields:
The increments of are therefore independent and stationary. The conclusion then easily follows. If is not bounded almost surely, then we can consider the stopping time and from the previous result the finite dimensional distributions do not depend on and are the same as a Brownian motion. We can then let to conclude
Theorem. For every , is a strong Markov process with semigroup : For every Borel function with polynomial growth, every , and every finite stopping time ,
Proof. The proof is identical to the proof of the usual Markov property with the additional ingredient given by the previous proposition
The strong Markov property for solutions of stochastic differential equations is useful to solve boundary value problems in partial differential equations theory. Let be a bounded closed set in . For , we denote . If is bounded Borel function such that , we define
The proof of the following theorem is let to the reader.
Theorem. Let be a bounded Borel function and assume that the function is . Then is the unique solution of the Dirirchlet boundary value problem
in , with the initial condition
and the boundary condition