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Consider the following PDE on $[0,T]\times \mathbb{R}$: $$ \begin{cases} \dfrac{\partial F}{\partial t}+\mu(t,x) \dfrac{\partial F}{\partial x}+ \frac12 \sigma^2(t,x)\dfrac{\partial^2 F}{\partial x^2} = 0 \\ F(T,x)=g(x) \end{cases} $$ with $\mu(t,x)=-\frac{1}{1-t}, \ \sigma(t,x)=1, \ g(x)=x^2$.

Question: What is the solution of the problem?

I know that if $X=\{X_t: t \geq 0\}$ satisfies $$dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dW_t$$ then (by Feynman-Kac's theorem) $$F(t,x)=\mathbb{E}^Q[g(X_T)|X_t=x]$$

I know that $$X_T=X_t-\int_{t}^T\frac{1}{1-s}ds+\int_{t}^TdW_s$$ but then I don't know how to deal with $(X_T)^2$. How can I get the solution?

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  • $\begingroup$ Try to expand $X_T^2$ and see what you can have. $\endgroup$ – Gordon May 19 at 16:55
  • $\begingroup$ That's what I've tried but I get stuck... I know that $$\int_{t}^TdW_s=W_T-W_t$$ but I don't know for example how to get rid of $$\int_{t}^T \frac{1}{1-s}ds$$ $\endgroup$ – Babado May 19 at 18:00

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