An example of Feynman-Kac

I’ve been learning about Feynman-Kac recently and I understand the underlying ideas. I am stuck however in actually computing explicit solutions for specific problems.

For example, assume that $S_t$ is the price of an asset with SDE $dS_t = rS_tdt+
sigma S_tdW_t$
, where $r$ and $sigma$ are positive numbers, and $W_t$ is a standard Brownian motion under some measure. Consider the function $f(t, S_t)$, dependent on time $t$ and on the price $S_t$. How to solve the following boundary problem where the domain is $[0,T]times mathbb{R}$:
$$ f_t +dfrac{1}{2}sigma^2 S^2 f_{SS}=0$$
with terminal condition $f(T,S)=S^4$?

Quantitative Finance Asked by Moh514 on February 6, 2021

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