Wednesday, April 14, 2010

Monte Carlo for PDEs on steroids

Fresh from his PhD defense - itself a memorable event that took place at Fields last week, with a star-filled examining committee consisting of two professors transplanted from France, one from Iran and two from Canada - Arash Fahim used the visitors seminar this week to describe the contents of his thesis to the rest of us.

Together with Nizar Touzi and others, he developed a probabilistic scheme to numerically approximate the solution of a certain type of fully nonlinear PDE. This generalizes the well-known use of Monte Carlo to solve linear parabolic PDEs through the Feymann-Kac formula, as well as the semi-linear case, which corresponds to approximating the solution of a BSDE. In the fully nonlinear case one is lead to consider the so-called 2BSDEs, as well as clever ways to approximate derivatives inside expectations, leading to what Nizar likes to call a Monte Carlo/Finite Differences scheme.

Arash showed several convincing numerical examples, but it is clear that this is a vast area with plenty of room for a lot more work.

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