Does there exist a variance formula for stochastic integrals?
Suppose we have
$dX = \sigma (X) dW + \mu(X) dt$
Do we have a formula for $Var(X_t)$ or an intergral of $X$ against $B$
More speicfically, I am interested in this problem
$dY = \eta dB+(\dfrac{a}{Y} - bY) dt$
and I would like to work out the variance of $\int^t_0 Y_s dB_s$
here a are b are just constants, but I think this would be a bit specific.
In case of the general Itô Diffusion it may help to use Dynkin's formula, i.e. for $\varphi \in C^2(\mathbb{R})$ you know that $$ \mathbb{E}\varphi(X_t) = \mathbb{E}\varphi(X_0) + \int_0^t \mathbb{E}A\varphi(X_s) \,ds $$ where $A$ is the infinitesimal generator associated with $(X_t)_{t\ge0}$. Maybe this is helpful to you if you consider this for $\varphi(x) = x$ and $\varphi(x) = x^2$.