does somebody know a reference, where I can find the value of the expectation of the running infimum of a geometric Brownian motion, namely:
Given a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in[0,T]},P)$, let $$S_t=S_0\exp((\mu-\sigma^2/2)t+\sigma W_t),\quad S_0,\mu,\sigma\in\mathbb{R}$$ where $W$ is a $P-$Brownian Motion.
I now calculated the following expectation under $P$ $$E_P(\inf_{0\leq s\leq T} S_s).$$ To calculate the expectation under an equivalent martingale measure $Q$ I can e.g. rely on Musiela/Rutkowski. I would prefer not to write down this calculation but to refer to a corresponding book or article.