About Stochastic and Not-So-Stochastic Integrals

Here we consider 3 kinds of integrals. Suppose W is a Brownian motion, f is a stochastic function and g is a deterministic function.

1. $\int_t^T W_s ds$
This entity turns out to be a normally distributed random variable, distributed as N(0, $T^3 /3$). [Missing: proof , can be found in Crack's HOTS pp. 154]

2. $\int_t^T g dW_s$
This is not only a martingale, but also a normally distributed random variable N(0,$\int_t^T g^2 ds$).

3. $\int_t^T f dW_s$
This is a martingale, but the distribution is not (necessarily) normal.

One implication of 2. versus 3. is this.