Using the reflection principle, it's easily shown that the infimum $\displaystyle \inf_{u \in [0, T]} B_u$ of a non-drifting Brownian motion $B_t$ has the same distribution as $-| B_T |$. The Cameron–Martin theorem tells us that a drifting Brownian motion $X_t = B_t + c t$ is a non-drifting one under a certain change of measure. Combining these two results, it seems to me that the p.d.f. of $\displaystyle \inf_{u \in [0, T]} X_u$ is
$$\frac{2 \exp \left(-\frac{(x – c T)^2}{2 T} \right)}{\sqrt{2 \pi T}} + 2 c \exp \left( 2 c x \right) \Phi \left( \frac{x + c T}{\sqrt{T}} \right)$$
but I'm unsure of the correctness of my derivation, let alone the correctness of the result.
I suspect it's wrong, but only because I can't get the correct result in a later calculation using this density function. Is the density function above wrong, and if so, what is the correct distribution?
Edit: Tried the derivation again more carefully and got a different density, which seems more reasonable. Still have lingering doubts, however.
Best Answer
Here is the answer :
You can check it in Chesney, Jeanblanc, and Yor's book Mathematical Models for Financial Markets on page 147
For $a\leq 0$ : $$ P(\inf_{s\in[0,t]} X_s> a)=\Phi\left(\frac{-a+c.t}{\sqrt{t}}\right)-e^{2c.a}.\Phi\left(\frac{a+c.t}{\sqrt{t}}\right). $$
Regards