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Hard · Stochastic Calculus · Quant Trader interview question · Brownian Motion, Reflection Principle, Probability, Stochastic Processes
Let $W_t$ be a standard Brownian motion. The reflection principle provides a way to relate probabilities involving the maximum of $W_s$ for $0 \le s \le t$ to probabilities involving $W_t$ itself. Suppose you are given that $b \le a$. Express $P(\max_{s \le t} W_s \ge a, W_t \le b)$ in terms of a probability involving only $W_t$.