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Let $X_t$ be a solution of the stochastic differential equation $ dX_t= -\frac{c-1}{2 X_t}dt+ dB_t, \, \qquad X_0=x_0$ where $c$ is a real constant and $B_t$ is a Brownian motion. Can you give me an example of a non-constant real function $u$ such that $u(X_t)$ is a local martingale and tell me how I can compute the ruin probability $\mathbb{P}(T_a < T_b)$ for real $a, b$ with $x_0 \in [a, b]$ where $T_a$ and $T_b$ denote the first hitting times of $a$ and $b$. Can you also tell me how to compute the first exit time $\mathbb{E}[\mbox{min}(T_a, T_b)]$.

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    First hitting times by $X_t$ or by $u(X_t)$?2011-12-06

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