Under a probability space $(\Omega, (\mathcal F_t), \mathbb P)$ consider two processes $X$ and $Y$ following the SDEs
$$ \frac{dX_t}{X_t} = \mu_t \ dt + \sigma(t,T) \ dW^1_t $$ $$ \frac{dY_t}{Y_t} = (\mu_t-\alpha_t )\ dt + \beta_t \ dW^2_t $$
with initial conditions $X_0(T)$ and $Y$. The standard Brownian motions $W^1$ and $W^2$ are such that $\langle W^1, W^2\rangle_t =\rho \ dt$.
Now consider and equivalent probability measure $\mathbb P^{\tau}$ such that $$ \left. \frac{d\mathbb P^{\tau}}{d\mathbb P}\right|_{\mathcal F_t} = \frac{X_t(\tau)}{e^{\mu_t}X_0(\tau)} = \mathcal E_t \left( \int_0 ^. \sigma(s,\tau) \ dW^1_s\right) $$
and the $\mathbb P^{\tau}$-Brownian motions $$ W^{1,\tau}_. := W^1_. + \int_0 ^. \sigma(s,\tau) \ ds $$
$$ W^{2,\tau}_. := W^2_. + \rho \int_0 ^. \sigma(s,\tau) \ ds $$
Is there a smart way to calculate the following expectation?
$$ \mathbb E^{\mathbb P^{\tau}}_t\left[ X_{\tau}(T) \ Y_{\tau}\right] $$