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I have the following PDE: $$2u_t+9u_{xx}-2u_x=0$$ $$u(x,T)=e^x$$ and I get that $$dX(s)=-dS+3dW(s)$$ $$X(t)=x$$

But how do I get the expected value of $e^x$? I tried substituting $Y(s)=e^x$, but I get the wrong answer... The answer should be $u(x,t)=e^{(x+\frac{7}{2}(T-t))}$.

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  • $\begingroup$ What is $X, S, W$ ? $\endgroup$ – Cesareo Dec 18 '18 at 20:42
  • $\begingroup$ Sorry, X is a stochastic process, W is a Wiener process and S is time $\endgroup$ – Natalie_94 Dec 19 '18 at 10:34
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Hint:

Your SDE results in the following expression:

$X(T) = x - (T - t) + 3(W(T) - W(t))$

Now you just need to find $E[e^{X(T)}]$ Can you do that?

Hint2: MGF

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