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I want to derive density of minimum of drifted Brownian motion using density of B_t, i know how to drive using joint density but i am facing problem when i use non joint density.

Suppose $X_t$ = at + $B_t$ is a drifted Brownian motion where $B_t$ is standard Brownian motion. I am able to derive density for minimum of $B_t$ and it comes out to be $\frac{1}{\sqrt 2\pi T}\int_{-\infty}^{a} Z_t f_m(x) dx$ .

where f_m(x) = $2e^{-\frac{x^2}{2t}}/{\sqrt{2\pi t}}$ and $Z_t$=$e^{\alpha y - \alpha^2 t/2}$

P($\hat m \le a)$ = $\int_{-\infty}^{m} $$e^{\alpha y - \alpha^2 t/2}$ $2e^{-\frac{x^2}{2t}}/{\sqrt{2\pi t}}$ $dx$ .

How to proceed from above steps onward?

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Let $$ X_{t}^{\xi}=\xi t+W_{t} $$ be a drifting Brownian motion. Let $a$ be a real number and $$ \tau_{a\xi}=\inf\left\{ t\geq0\colon X_{t}^{\xi}=a\right\} $$ be the first time the drifting Brownian motion reaches level $a$. The density of $\tau_{a\xi}$ is [1] $$ f_{a\xi}(t)=\frac{\left|a\right|}{\sqrt{2\pi t^{3}}}\exp\left(-\frac{\left(a-\xi t\right)^{2}}{2t}\right). $$ Then, some tediuos calculations yield $$ \mathbb{P}(\tau_{a\xi}\leq t)=\int_{0}^{t}f_{a\xi}(t)dt=\frac{1}{2}\left\{ 1+\text{sgn}(a)\text{erf}\left(\frac{\xi t-a}{\sqrt{2t}}\right)+e^{2a\xi}\left[1-\text{sgn}(a)\text{erf}\left(\frac{\xi t+a}{\sqrt{2t}}\right)\right]\right\} . $$ Now, since $X_{0}^{\xi}=0$, $$ \mathbb{P}\left(\min_{0\leq s\leq t}X_{t}^{\xi}\leq a\right)=\begin{cases} 1 & \text{if }a\geq0\\ \mathbb{P}(\tau_{a\xi}\leq t) & \text{if }a<0. \end{cases} $$


  1. Shreve, Steven E., Stochastic calculus for finance. II: Continuous-time models., Springer Finance. New York, NY: Springer (ISBN 0-387-40101-6/hbk). xix, 550 p. (2004). ZBL1068.91041.
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