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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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For the first part, you can use Ito's formula to find $d(u(X_t))$ to obtain a stochastic differential equation in terms of $dt$ and $dB_t$, which will be a local martingale when the $dt$ term is $0$. This yields a differential equation which $u$ will have to solve. Things are simplified somewhat by the assumption (given your notation) that $u$ depend on the value of $X_t$ and not on $t$ itself. For the remainder of the question, you will use the function $u$, and the fact that you have a local martingale. –  Aaron Dec 5 '11 at 23:29
    
First hitting times by $X_t$ or by $u(X_t)$? –  Ilya Dec 6 '11 at 7:32
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