Let $W_{1}$ and $W_{2}$ be independent standard Brownian motions and set $$ X_t=\int_0^tW_{1}\ dW_{2} \text{ and } Y_t=\int_0^tW_{2}\ dW_{1}. $$
The product rule shows that $$ W_{1,t}W_{2,t}=X_t + Y_t. $$ For a fixed $t$ I would like to understand the nature of this additive decomposition of the random variable $W_{1,t}W_{2,t}$.
Questions:
What can be said about the (joint) distribution of $(X_t, Y_t)$?
More specifically:
- What is the marginal distribution of $X_t$ and $Y_t$?
- Can $X_t$ and $Y_t$ be expressed in terms of $W_{1,t}$ and $W_{2,t}$ alone?
- What is the correlation between $X_t$ and $Y_t$?
- What are the characteristic functions of $X_t$ and $Y_t$?
Or anything else which may shed some light on the joint distribution.
Beyond "raw" results I would appreciate references to literature, where I can learn more about ways to tackle those and similar questions.