Let $B_{\mu}(t)$ be a one-dimensional Brownian motion with drift $\mu \geq 0.$ For $a > 0,$ let $$T_a = \inf\{t > 0: B_{\mu}(t) = a\}$$ denote the first hitting time of $B.$ The Laplace transform of $T_a$ well known, $$\mathbb{E}(e^{-\lambda T_a}) = \exp(\mu a - |a|\sqrt{\mu^2+2\lambda}),$$ which can be found using optional stopping theorem and Girsanov's transform (see Karatzas and Shreve). Is the Laplace transform for $T_a^2$ also known in closed form? We can find this by finding a martingale for BM of the form $$\exp(f - \lambda t^2) =: u(x,t)$$ and applying the optional stopping theorem (with Girsanov), where $f(t,x,\lambda)$ would be some function making $u(B_0(t), t)$ a martingale. Does anyone have a reference or know a closed form solution for the Laplace transform of $T_a^2?$


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