A monotone scheme for G-equations with application to the explicit convergence rate of robust central limit theorem

We propose a monotone approximation scheme for a class of fully nonlinear PDEs called G-equations. Such equations arise often in the characterization of G-distributed random variables in a sublinear expectation space. The proposed scheme is constructed recursively based on a piecewise constant approximation of the viscosity solution to the G-equation. We establish the convergence of the scheme and determine the convergence rate with an explicit error bound, using the comparison principles for both the scheme and the equation together with a mollification procedure. The first application is obtaining the convergence rate of Peng's robust central limit theorem with an explicit bound of Berry-Esseen type. The second application is an approximation scheme with its convergence rate for the Black-Scholes-Barenblatt equation

Topics in Stochastic Control with Applications to Finance.

This thesis is devoted to PDE characterization for stochastic control problems when the classical methodology of dynamic programming does not work. Under the framework of viscosity solutions, a dynamic programming principle (DPP) serves as the tool to associate a (nonlinear) PDE to a stochastic control problem. Unfortunately, a DPP is in general difficult to prove, and may fail to be true in some cases. In this thesis, we investigate three different scenarios where classical dynamic programming does not work. The first one is quantile hedging in the presence of arbitrage, the second one is robust growth-optimal trading, and the third one is a stochastic differential game of control and stopping. In each of the cases, we develop new methods to circumvent the lack of a classical DPP.PhDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/99933/1/jayhuang_1.pd