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 $f(T,S)=S^4$?

  • $\begingroup$ Your PDE seems to assume $r=0$. Is that the case? $\endgroup$ Nov 19 '20 at 17:24
  • $\begingroup$ @DaneelOlivaw No. r is not 0 $\endgroup$
    – Moh514
    Nov 19 '20 at 17:47

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