for an exercise I need to calculate the following problem:
Define
$F(Y,t)=\mathbb{E}[e^{-\int_0^T r(Y(s))ds}\mid Y(t)]$
where $Y(t)=(Y_1(t), Y_2(t), ..., Y_N(t))$ follows the diffusion process: $dY(t)=\mu(Y(t))dt + \sigma(Y(t))dW(t)$
and
$r(t)=\delta_0+\delta_y^\top Y(t)$
where $\delta_0$ is a constant and $\delta_y$ is a real valued N dimensional vector.
My question is now how I can calculate:
$\mathcal{D}F(Y,t)=F_t(Y,t) + F_Y(Y,t) \mu(Y) + \frac{1}{2} tr[F_{YY}(Y,t)\sigma(Y)\sigma(Y)^\top]$
I know that the answer should be $r(Y(t))F(Y,t)$, but I do not know how to calculate the differentials.
Thank you in advance for your help. Best wishes