3 votes
Accepted

Ito integral over an indicator function

As the comment says, the question makes little sense for a geometric Brownian motion. So let $P$ be just a general Ito process with $P_t = \int_0^t a_s ds + \int_0^t b_s dW_s$. You want to write the ...
user avatar
  • 23.1k
1 vote

prove $P\{\int_{0}^{\infty}f(W_s)ds=\infty\}=1$

We need to show that $$ \int_0^T f(B_t) dt\to\infty, T\to\infty. $$ I will use the occupation density formula (UPD see a simpler argument below): $$ \int_0^T f(B_t) dt = \int_{\mathbb{R}} f(x) L^x_T(B)...
user avatar
  • 23.1k
1 vote

Basic application of Ito's Lemma

In the first formula you are using the Ito formula for non-autonomous transformations $y=f(t,x)$ with the increments $$ dY_t = f_t\,dt+f_x\,dX+\frac12f_{xx}\,d\langle X\rangle_t. $$ In the example ...
user avatar

Only top scored, non community-wiki answers of a minimum length are eligible