Can an Itō integral be $\infty$?

Question

In other words, can $\int_0^t f(s)dW(s)$ = $\infty$?

Thanks!

Answer 1

It is not well defined. When you put something like $\int_0^t f(s) dW(s)$, it has to be finite and usually $f(s)$ belongs to the space $$ \Theta:= \{ f: f \mbox{ is progressive measurable or adapted, } \int_0^T f^2_t dt <+\infty\}. $$ However, $$ E\left(\int_0^t f(s) dW(s) \right) $$ can be infinity, when the stochastic integral is only a local martingale, not a true martingale.