The NNStopping algorithm
Problems Supported:
The general idea 💡
Similar to DeepSplitting and DeepBSDE, NNStopping evaluates the PDE as a Stochastic Differential Equation. Consider an Obstacle PDE of the form:
\[ max\lbrace\partial_t u(t,x) + \mu(t, x) \nabla_x u(t,x) + \frac{1}{2} \sigma^2(t, x) \Delta_x u(t,x) , g(t,x) - u(t,x)\rbrace\]
Such PDEs are commonly used as representations for the dynamics of stock prices that can be exercised before maturity, such as American Options.
Using the Feynman-Kac formula, the underlying SDE will be:
\[dX_{t}=\mu(X,t)dt + \sigma(X,t)\ dW_{t}^{Q}\]
The payoff of the option would then be:
\[sup\lbrace\mathbb{E}[g(X_\tau, \tau)]\rbrace\]
Where τ is the stopping (exercising) time. The goal is to retrieve both the optimal exercising strategy (τ) and the payoff.
We approximate each stopping decision with a neural network architecture, inorder to maximise the expected payoff.