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Let $Y$ be an exponential random variable with rate parameter $\lambda$. Let $T_{a}$ be the first hitting time of a Brownian Motion. I want to find $ P(\min(T_{a}, T_{-a}) < Y) $

In order to compute this, I want to find a martingale, then apply the stopping theorem. Can you suggest me a martingale to begin with?

edit: I used the exponential martingale and I got stuck in this equation: $ E[e^{cB_{min(T_{a}, T_{-a})} - \frac{c^{2}}{2}min(T_{a}, T_{-a})}] = P(T_{a}T_{-a})e^{-ca}E[e^{-\frac{c^{2}}{2}T_{-a}}] $

Then $ P(T_{a}>T_{-a}) = P(T_{a}

Then what to do?

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    @MichaelHardy : I corrected it2012-02-03

1 Answers 1

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Here is an idea, we have P[T_a\wedge T_{-a} where T'_a is the first hitting time of $|B_t|$, then we have if my calculation is right :

P[T_a' (where $p$ is the density of the law of $T_a'$)

$=\frac{1}{\mu}\int_{\mathbb{R}^+}[\int_{x}^\infty e^{-\mu.y}dy]p(x)dx$ =\int_{\mathbb{R}^+}e^{-\mu.x}p(x)dx=L_{T'_a}(\mu)

The third line gives the result as the Laplace Transform of T'_a at $\mu$, this is I think probably available in (for example) Karatzas and Shreve "Brownian Motion and Stochastic Calculus".

Best regards