A monotone scheme for high-dimensional fully nonlinear PDEs

DOI10.1214/14-AAP1030zbMath1321.65158arXiv1212.0466MaRDI QIDQ2346082

Jianfeng Zhang, Wenjie Guo, Jia Zhuo

Publication date: 29 May 2015

Published in: The Annals of Applied Probability (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1212.0466

zbMATH Keywords

numerical exampleparabolic partial differential equationsMonte Carlo methodsviscosity solutionsmonotone schemesemilinearleast square regressionfully nonlinear partial differential equations

Mathematics Subject Classification ID

Monte Carlo methods (65C05) Nonlinear parabolic equations (35K55) Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games (49L25) Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs (65M75)

Related Items

On the convergence of monotone schemes for path-dependent PDEs ⋮ Efficient spectral sparse grid approximations for solving multi-dimensional forward backward sdes ⋮ Numerical simulations for \(G\)-Brownian motion ⋮ Multilevel Picard approximations of high-dimensional semilinear partial differential equations with locally monotone coefficient functions ⋮ On the homotopy analysis method for backward/forward-backward stochastic differential equations ⋮ Numerical Solution of the Incompressible Navier-Stokes Equation by a Deep Branching Algorithm ⋮ Discrete‐time approximation for stochastic optimal control problems under the G‐expectation framework ⋮ Convergence of the Backward Deep BSDE Method with Applications to Optimal Stopping Problems ⋮ Numerical methods for backward stochastic differential equations: a survey ⋮ A fully nonlinear Feynman-Kac formula with derivatives of arbitrary orders ⋮ A deep branching solver for fully nonlinear partial differential equations ⋮ Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks ⋮ A modified MSA for stochastic control problems ⋮ High Order Numerical Schemes for Second-Order FBSDEs with Applications to Stochastic Optimal Control ⋮ A worst-case risk measure by G-VaR ⋮ An efficient numerical method for forward-backward stochastic differential equations driven by \(G\)-Brownian motion ⋮ Donsker-type theorem for BSDEs: rate of convergence ⋮ Weak approximation of second-order BSDEs ⋮ Explicit deferred correction methods for second-order forward backward stochastic differential equations ⋮ On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations ⋮ Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations ⋮ Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning

Cites Work

• Unnamed Item
• Unnamed Item
• A splitting method for fully nonlinear degenerate parabolic PDEs
• Wellposedness of second order backward SDEs
• A probabilistic numerical method for fully nonlinear parabolic PDEs
• Discrete-time probabilistic approximation of path-dependent stochastic control problems
• The steepest descent method for forward-backward SDEs
• Solving forward-backward stochastic differential equations explicitly -- a four step scheme
• An analysis of a least squares regression method for American option pricing
• A numerical scheme for BSDEs
• Number of paths versus number of basis functions in American option pricing
• A forward scheme for backward SDEs
• Time discretization and Markovian iteration for coupled FBSDEs
• Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations
• Controlled Markov processes and viscosity solutions
• A forward-backward stochastic algorithm for quasi-linear PDEs
• Numerical methods for forward-backward stochastic differential equations
• A regression-based Monte Carlo method to solve backward stochastic differential equations
• Monte-Carlo Valuation of American Options: Facts and New Algorithms to Improve Existing Methods
• Least-Squares Monte Carlo for Backward SDEs
• Discretization of forward–backward stochastic differential equations and related quasi-linear parabolic equations
• Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs
• Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations
• On Numerical Approximations of Forward-Backward Stochastic Differential Equations
• User’s guide to viscosity solutions of second order partial differential equations
• Consistency of Generalized Finite Difference Schemes for the Stochastic HJB Equation
• Numerical solution of quasilinear parabolic equations and backward stochastic differential equations
• Solving Backward Stochastic Differential Equations Using the Cubature Method: Application to Nonlinear Pricing
• Valuing American Options by Simulation: A Simple Least-Squares Approach
• A QUANTIZATION TREE METHOD FOR PRICING AND HEDGING MULTIDIMENSIONAL AMERICAN OPTIONS

This page was built for publication: A monotone scheme for high-dimensional fully nonlinear PDEs