Let
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$
- $X$ be a $\mathcal F$-progressive process on $(\Omega,\mathcal A,\operatorname P)$ with $$\sigma_t^2:=\operatorname E\left[\int_0^tX_s^2\:{\rm d}_s\right]<\infty\;\;\;\text{for all }t\ge0\tag1$$
- $W$ be a $\mathcal F$-Brownian motion on $(\Omega,\mathcal A,\operatorname P)$
- $Y:=X\cdot W$ denote the Itō integral process of $X$ with respect to $W$
Note that $$\operatorname E\left[Y_t\right]=0\tag2$$ and $$\operatorname{Var}\left[Y_t\right]=\sigma_t^2\tag3$$ for all $t\ge0$.
How can we show that $Y_t\sim\mathcal N(0,\sigma_t^2)$ for all $t\ge0$, if $X$ is non-random?
Is it possible to obtain the desired claim by showing that the characteristic function $$\varphi_t(s):=\operatorname E\left[e^{{\rm i}sY_t}\right]\;\;\;\text{for }s\in\mathbb R$$ of $Y_t$ is equal to the characteristic function $$\psi_t(s):=e^{\frac{-s^2\sigma_t^2}2}\;\;\;\text{for }s\in\mathbb R$$ of $\mathcal N(0,\sigma_t^2)$ for all $t\ge0$?
Question 2: What can we tell about the distribution of the $Y_t$ for general (random) $X$? And what can we tell under the weaker assumption $$\int_0^tX_s^2\:{\rm d}s<\infty\;\;\;\text{almost surely for all }t\ge0\tag4?$$