An unbiased Ito type stochastic representation for transport PDEs: A Toy Example

We propose a stochastic representation for a simple class of transport PDEs based on Ito representations. We detail an algorithm using an estimator stemming for the representation that, unlike regularization by noise estimators, is unbiased. We rely on recent developments on branching diffusions, regime switching processes and their representations of PDEs. There is a loose relation between our technique and regularization by noise, but contrary to the latter, we add a perturbation and immediately its correction. The method is only possible through a judicious choice of the diffusion coefficient $\sigma$. A key feature is that our approach does not rely on the smallness of $\sigma$, in fact, our $\sigma$ is strictly bounded from below which is in stark contrast with standard perturbation techniques. This is critical for extending this method to non-toy PDEs which have nonlinear terms in the first derivative where the usual perturbation technique breaks down. The examples presented show the algorithm outperforming alternative approaches. Moreover, the examples point toward a potential algorithm for the fully nonlinear case where the method of characteristics breaks down.