2

I am confronted with the following expectation

$$E_t \left[\int_t^Tg(S_s)dS_s \right]$$

Where $S_t$ is a stochastic process. How would we go about computing this quantity? If we can't do so in the general case, are there particular forms of $g$ or $S_t$ that allow us to say something intelligent about the expectation?

For example if $S_t$ has stochastic differential $dS_t=r S_t dt+\sigma S_t dW_t$ and $g$ has certain nice properties, can we make progress?

Gestall
  • 21
  • 1
    Well if S is martingale and $g$ as nice properties like is bounded or L2 usually the stochastic integral is at least a local martingale and usually a martingale so in this last situation the expectation should be null. Once again it depends on $S$ and $g$. – TheBridge Sep 08 '21 at 12:35
  • Interesting, thanks! What's an example? Better yet, anywhere I can look for a proof in the case that S is martingale? – Gestall Sep 09 '21 at 01:27
  • Well is S is defined by an SDE if there is no drift and a solution exists it's a local maritngale more on criteria that assure you that a process is a martingale can be found here https://math.stackexchange.com/questions/38908/criteria-for-being-a-true-martingale/38947#38947 – TheBridge Sep 09 '21 at 11:12

0 Answers0