Lecture 29. The strong Markov property for solutions of stochastic differential equations

In the previous section, we have seen that if (X_t^x)_{t \ge 0} is the solution of a stochastic differential equation
X_t^{x} =x +\int_0^t b(X_s^{x}) ds + \int_0^t \sigma(X_s^{x}) dB_s,
then (X_t^x)_{t \ge 0} is a Markov process, that is for every t,T \ge 0,
\mathbb{E}(f(X_{t+T}^x) \mid \mathcal{F}_T)=(P_{t}f )(X_T^x),
where P_tf(x)=\mathbb{E}( f(X_t^x)). It is remarkable that this property still holds when T is now any finite stopping time. This property is called the strong Markov property.
The key lemma is the following:

Lemma. Let (B_t)_{t\ge 0} be a standard Brownian motion and let T be a finite stopping time. The process, (B_{T+t}-B_T)_{t\ge 0} is a standard Brownian motion independent from \mathcal{F}_T.

Proof. Let T be a finite stopping time of the filtration (\mathcal{F}_t)_{t \ge 0}. We first assume T bounded. Let us consider the process \tilde{B}_t=B_{T+t}-B_T, \quad t \ge 0. Let \lambda \in \mathbb{R}, 0\le s \le t. Applying Doob’s stopping theorem to the martingale \left( e^{i\lambda B_t +\frac{\lambda^2}{2} t}\right)_{t \ge 0}, with the stopping times t+T and s+T , yields:
\mathbb{E} \left( e^{i\lambda B_{T+t} +\frac{\lambda^2}{2} (T+t)}\mid \mathcal{F}_{T +s} \right)=e^{i\lambda B_{T+s} +\frac{\lambda^2}{2} (T+s)}.
Therefore
\mathbb{E} \left( e^{i\lambda (B_{T+t} -B_{T+s})}\mid \mathcal{F}_{T+s} \right)=e^{-\frac{\lambda^2}{2} (t-s) }.
The increments of (\tilde{B}_t)_{t \ge 0} are therefore independent and stationary. The conclusion then easily follows. If T is not bounded almost surely, then we can consider the stopping time T \wedge N and from the previous result the finite dimensional distributions (B_{ t_1 +T\wedge N}-B_{T \wedge N}, \cdots , B_{ t_n +T\wedge N}-B_{T \wedge N}) do not depend on N and are the same as a Brownian motion. We can then let N \to  + \infty to conclude \square

Theorem. For every x \in \mathbb{R}^n, (X_t^{x})_{t\ge 0, x \in \mathbb{R}^d} is a strong Markov process with semigroup (P_t)_{t \ge 0}: For every Borel function f:\mathbb{R}^n \to \mathbb{R} with polynomial growth, every t \ge 0, and every finite stopping time T,
\mathbb{E}(f(X_{t+T}^x) \mid \mathcal{F}_T)=(P_{t}f )(X_T^x).

Proof. The proof is identical to the proof of the usual Markov property with the additional ingredient given by the previous proposition \square

The strong Markov property for solutions of stochastic differential equations is useful to solve boundary value problems in partial differential equations theory. Let K be a bounded closed set in \mathbb{R}^n. For x \in \Omega, we denote T_x= \inf \{ t \ge 0, X_t \in \partial K \}. If f is bounded Borel function such that f_{\partial K}=0, we define
P^K_t f(x)=\mathbb{E}\left( f(X^x_t) \mathbf{1}_{t \le T_x } \right).
The proof of the following theorem is let to the reader.

Theorem. Let f:K \to \mathbb{R} be a bounded Borel function and assume that the function u(t,x)=(P^K_tf)(x) is C^{1,2}. Then u is the unique solution of the Dirirchlet boundary value problem
\frac{\partial u}{\partial t} (t,x)=Lu(t,x)
in [0,+\infty) \times \mathbb{R}^n , with the initial condition
u(0,x)=f(x),
and the boundary condition
u(t,x)=0, \quad x \in \partial K.

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