# Solving a PDE from the Feymann-Kac formula

Let $$u:[0,T]\times\mathbb{R} \mapsto \mathbb{R}_+$$, $$T>0$$, $$r>0$$, $$\mu>0$$, $$\sigma>0$$ and $$A>0$$.

What is the solution to the following PDE: $$$$u_t(t,x)+\mu x u_x(t,x)+\frac{\sigma^2x^2}{2}u_{xx}(t,x)-ru(t,x)+\frac{1}{A+x}=0$$$$ with the terminal condition $$u(T,x)=0$$.

By Feymann-Kac theorem, $$u$$ has the probabilistic representation $$u(t,x)=\mathbb{E}\left[\int_t^Te^{-r(u-t)}\frac{1}{A+S_u}du|S_t=x\right]$$ where $$S$$ follows the SDE $$dS_t=\mu S_tdt+\sigma S_tdW_t.$$ However, to find $$u$$ by first solving $$\mathbb{E}\left[\frac{1}{A+S_u}\right]$$ does not lead to an analytical solution because the random variable $$\frac{1}{A+S_u}$$ follows a logit normal distribution which is known for not having analytical expression for the mean.

• There is a $r$ in the first equation. Is that a typo? Should it be $t$? May 18 at 11:23
• Sorry, I forgot to declare $r>0$, now it is edited. May 19 at 9:03
• Function $1/(A+x)$ is not integrable. Is it proven that a solution exists? May 19 at 16:22
• Since $S_t>0$ a.s. for all $t\in[0,T]$, then $0<\mathbb{E}[\frac{1}{A+S_u}]\leq \frac{1}{A}$, hence, it has finite moment. A solution $u$ should exist because the probabilistic representation is well-defined. May 20 at 4:17
• Based on a past question you asked, don't you rather mean $\frac{1}{A+\exp(\pm x)}$? As it stands, note that $\mathbb{E}\big(\frac{1}{A+S_u}\big)$ does not exist, since the integral does not converge May 20 at 5:50