I have the following non-linear PDE and I have no idea how to go about solving it using a finite difference scheme in Python. Can someone get me started and/or point me to an algorithm for doing this? It represents the price of a derivative in the Uncertain Volatility Model (where $\sigma \in [\sigma_{low}, \sigma_{high}]$).

$$\partial_t u(t,x) + H(x, D_x^2u(t,x)) = 0$$ $(t,x) \in [0,T) \times\mathbb{R}$


$$H(x,\Gamma) = \frac{1}{2}x^2\Sigma(\Gamma)^2\Gamma$$ $$\Sigma(\Gamma) = \sigma_{low}1_{\Gamma < 0} + \sigma_{high}1_{\Gamma \ge 0}$$


I don't know of any libraries for this. There is a pretty good literature on the problem you mention though. I suggest https://cs.uwaterloo.ca/~paforsyt/numuncert.pdf as a good paper to follow; they study numerical techniques, document pitfalls, and even prove something about convergence of their preferred approach.


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