I have two questions:


Let's say I had an estimate of $\mu$ in an sde:

$dZ = \mu Z dt + \sigma Z dW$

If I ran monte carlo I could price say an option like $Call_t = e^{-rT}E_t[(Z-k)+]$

Feynman-Kac says that given a PDE of a certain form (see https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula)

then $Call_t = e^{-rT}E_t[(Z-k)+]$ and $dZ = \mu(X, t) dt + \sigma(X,t)dW$

The question is could I use that PDE to solve the above sde? Does feynman kac allow us to say if we have an SDE and an expectation it implies a PDE?


Is there a way besides feynman-kac to write the PDE for the above SDE and expectation? For example, in the black-scholes paper they use delta hedging to write out a PDE

$d\Pi = dCall - \frac{dCall}{ds}S$ show that it has no risk so it must drift a $r\Pi dt$ then set the drifts equal to eachother to get the black scholes formula.

But there doesn't seem to be an equivalent argument for the physical measure.

Anyways I may be confusing myself, any help would be appreciated.


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