# 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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