Convergence analysis of monotone schemes for second-order non-linear parabolic PDEs and their applications in sublinear expectation

Second-order non-linear parabolic partial differential equations have been a central research area for decades due to their various applications to physics, engineering, finance and others. To solve such equations, numerical approximations are usually designed to converge to a weak version of solutions called viscosity solutions. This thesis proposes monotone approximation schemes for two specific types of non-linear parabolic equations arising in applied mathematics, and establishes the convergence and convergence rate to viscosity solutions. The proposed schemes involve only semi-discretization in time and the rates of convergence are represented in the form of some exponent of the time step scaled by a constant. We first propose a monotone scheme for a class of semi-linear parabolic equations that are convex and coercive in their gradients arising from utility indifference pricing in mathematical finance. The proposed scheme is based on a splitting method and its convergence rate is determined by combining Krylov’s shaking coefficients technique and the Barles-Jakobsen optimal switching approximation. An extension to variational inequalities is also studied using an obstacle switching system. We then build a piece-wise constant monotone scheme for a class of fully non-linear equations called G-equations arising from Knightian uncertainty in statistics, and determine its convergence rate with an explicit error bound. We present three applications under a sublinear expectation framework: a convergence rate for Peng’s robust central limit theorem with an explicit bound of Berry-Esseen type, a monoton scheme for the Black-Scholes-Barenblatt (BSB) equation that is a natural generalization of Cox-Ross-Rubinstein (CIR) binomial tree approximation to the case with model ambiguity, and an optimal switching approximation to G-normal distribution