Let
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$
- $M$ be a real-valued continuous local $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$
- $X$ be a real-valued $\mathcal F$-predictable process on $(\Omega,\mathcal A,\operatorname P)$ with $$A_t:=\int_0^tX_s^2\:{\rm d}[M]_s<\infty\;\;\;\text{almost surely for all }t\ge0$$
Now, let $$Y_t:=\int_0^tX_s\:{\rm d}M_s\;\;\;\text{for }t\ge0.$$
What can we say about the distribution of the $Y_t$?
For example, if $X$ is non-random and $M$ is the standard Brownian motion, then the $Y_t$ are normally distributed with mean $0$ and variance $A_t$.
Are similar conclusions possible in more general cases?