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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?

Lorenzo B.
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Patrick
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  • You should tidy up the integrand: get rid of the useless constants and rearrange as the exponential of a polynomial. –  Mar 13 '18 at 20:33
  • Expand out $(x-y)^2$, this should make the visually complicated integral more simple. Also, since $y$ is negative, it may help to transform $y\to-y$ and integrate from $0\to\infty$ instead of $-\infty\to0$. – MasterYoda Mar 13 '18 at 20:34

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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.