# Generalizations of the Reflection principle for Brownian motion

The reflection principle for Brownian motion roughly states that a Brownian motion reflected a stopping time is also a Brownian motion. More precisely, if $$W$$ is a Brownian motion and $$T$$ a stopping time then $$\hat{W}_t = W_t$$ on $$t\leq T$$ and $$\hat{W}_t = 2W_T-W_t$$ if $$t>T$$ is also a Brownian motion.

The proof of the reflection principle as far as I am aware only uses symmetry of the normal distribution, continuity of paths and the strong Markov property of $$W$$. My inclination is that the reflection would surely hold for a larger class of processes (possibly continuous symmetric stable processes?) than the Brownian motion, but I have yet to find a generalization.

My question is then if there are known generalizations of the reflection principle, or if not is it that continuous symmetric Markov processes are precisely Brownian motions?

• For deterministic times $T$ the reflection principle should hold, for instance, for any symmetric Lévy process and for stochastic integrals of the form $\int_0^t f(s) \, dW_s$ where $f$ is deterministic and $W$ a Brownian motion. – saz Mar 8 at 16:05