Numerical analysis of strongly nonlinear PDEs

Numerical Methods for Nonlinear PDEs in Finance

Many problems in finance can be posed in terms of an optimal stochastic control. Some well-known examples include transaction cost/uncertain volatility models [17, 2, 25], passport options [1, 26], unequal borrowing/lending costs in option pricing [9], risk control in reinsurance [23], optimal withdrawals in variable annuities[13], optimal execution of trades [20, 19], and asset allocation [28,...

Numerical Analysis of Hyperbolic PDEs

Numerical Solution of Nonlinear PDEs by Using Two-Level Iterative Techniques and Radial Basis Functions

‎Radial basis function method has been used to handle linear and‎ ‎nonlinear equations‎. ‎The purpose of this paper is to introduce the method of RBF to‎ ‎an existing method in solving nonlinear two-level iterative‎ ‎techniques and also the method is implemented to four numerical‎ ‎examples‎. ‎The results reveal that the technique is very effective‎ ‎and simple. Th...

Gap in Nonlinear Equivalence for Numerical Methods for PDEs

For a large class of nonlinear evolution PDEs, and more generally, of nonlinear semigroups, as well as their approximating numerical methods, two rather natural stability type convergence conditions are given, one being necessary, while the other is sufficient. The gap between these two stability conditions is analyzed, thus leading to a general nonlinear equivalence between stability and conve...

A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs

We consider the probabilistic numerical scheme for fully nonlinear PDEs suggested in [10], and show that it can be introduced naturally as a combination of Monte Carlo and finite differences scheme without appealing to the theory of backward stochastic differential equations. Our first main result provides the convergence of the discrete-time approximation and derives a bound on the discretizat...