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!



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.