# Expected value of max of a Stochastic process

Ciao all,

I'm working on some stochastic processes and I'm stuck on this problem. Let $S_t$ be the stochastic process defined by: $$dS_t = \sigma S_t dW_t$$ with initial data $S_0 \in \mathbb{R}^+$.

I'm trying to compute the expected value: $$\mathbb{E}\left[ \max_{t \in [0, T]}S_t \right]$$

but it's not clear how to fight the problem. For example I cannot use Ito's Lemma since the function is not $C^1$ w.r.t. $S_t$.

I've started from a simpler case:

\begin{align} \mathbb{E} [\max(S_t, S_T)] & = \mathbb{E}[S_t]\mathbb{P}(S_t > S_T) + \mathbb{E}[S_T]\mathbb{P}(S_T > S_t)\\ & = S_0 \left(\mathbb{P}(S_T > S_t) + \mathbb{P}(S_t > S_T) \right) \end{align} At this point the two terms on the rhs can be computed directly. My idea was to extend this computation for all $t$ but I'm sure I have to take in account in somehow that the expected value at a future time is conditionated by all the past times.

Can you help be, even with some ideas? Thank you, ciao!

AM