Ito Isometry formula with Brownian motion

Question

Is it possible to use the Ito Isometry formula to calculate,

$$E\left [ \left ( \int_{0}^{t}g(s,W_s)dW_s \right )^2 \right ]$$

for something like $g(t,W_t)=(1+t^2+W_t)$

Thanks for any help on this

Answer 1

For the Ito-isometry to be applicable for $\mathbb E[\int_0^t (\varphi_s dW_s)^2]$ the function $\varphi_s$ has to be progressively measurable with $\mathbb E \int_0^t \varphi_s^2 ds <\infty$ for any $t>0$. Note that $\varphi_s$ can be a random function, i.e. $\varphi_s = 1+s^2+W_s$.

Answer 2

In your particular case

$$g= 1+t^2+W_t$$ if you asume this brownian motion is the same as the one in the integrator, you have that $g$ is adapted to any admissible filtration.

Furthermore

$$\mathbb E \int_0^T (1+t^2+W_t)^2dt\leq 3\mathbb E\int_0^T1+t^4+W_t^2 dt=3\left( T+T^5/5+\int_0^T\mathbb E(W_t^2)dt\right)<\infty.$$

Hence you can use Ito's isometry.