I am trying to calculate the following integral:
$$\int_{0}^{\infty}\int_{-\infty}^{0} e^{\sigma y}e^{c(x+y)-Tc^2/2}\frac{2}{\sqrt{2\pi T^{3}}}e^{-\frac{(x-y)^2}{2T}}dydx,$$ where $c =\frac{r-\sigma^{2}/2}{\sigma}$ and $T,\sigma,r>0$.
I have tried to integrate by parts over $y$ to get rid of the $(x-y)$ but it doesn't seem to lead anywhere. Any hints?
Hint:
By the change of variable $z:=x-y+a$, with well chosen $a$, you will end-up with a Gaussian in $z$ times an exponential in $y$. At the same time, the domain will be transformed to a dihedral.
By integration on $y$, you will obtain the exponential of a piecewise linear function of $z$, times the Gaussian factor. Then by suitable shifts, pure Gaussians.
In the end, the solution should be given by a combination of error functions.