Henry Schellhorn's Abstract

Claremont Mathematics Weekend

January 30 - 31, 2016

Speaker: Henry Schellhorn
Title: Semi-group Solution of Second Order Fully Nonlinear Parabolic Partial Differential Equations
Abstract:

The Hamilton-Jacobi-Bellman equation (HJB) of stochastic control is an example of a fully nonlinear parabolic second-order PDE. As far as we know, a semi-group theory has not yet been proposed for this type of equation. One application of the semi-group theory is to give conditions for  local existence of the PDE and, in some very particular cases, for global existence. The advantage of the semi-group method is that the existence proof is constructive, and thus leads automatically to Picard-type algorithms for the numerical computation of solutions. For the HJB equation, this method is similar to the policy iteration method, which was developed only in the discrete-time setting.

We obtain a semi-group solution by combining two new results from stochastic analysis. First, we use a result by Cheridito, Soner, Touzi, and Victoir (2006) which characterizes the solution of that PDE as a second-order backward stochastic differential (2BSDE). Since there is a martingale solution to the 2BSDE, the solution of the PDE is directly related to the martingale solving the corresponding BSDE. Secondly,  we reuse one of our results  ( Jin, Peng, and Schellhorn 2015) , where we gave a new representation of a Brownian martingale as the exponential of a time-dependent generator, applied to the terminal value.