Let (X_{t}) be as in Problem 19.5 and set (widehat{tau}_{b}:=inf left{t geqslant 0: X_{t}=b ight}). a) Use

Let \(X_{t}\) be as in Problem 19.5 and set \(\widehat{\tau}_{b}:=\inf \left\{t \geqslant 0: X_{t}=b\right\}\).

a) Use Problem 19.5 to find the probability density of \(\widehat{\tau}_{b}\) if \(\alpha, b>0\).

b) Find \(\mathbb{P}\left(\widehat{\tau}_{b})\) for \(\alpha

Data From Problem 19.5

Transcribed Image Text:

E[(Mt Ms) | Fs]=E[(M) - (M)s | Fs] for all s < t < T.