On numerical approximations of high-dimensional nonlinear parabolic partial differential equations and of backward stochastic differential equations

• Applicant: Professor Dr. Martin Hutzenthaler
• Subject Area: Mathematics
• Term: from 2017 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 381158774

• Final Report Year: 2023

Final Report Abstract

High-dimensional forward-backward stochastic differential equations (FBSDEs) and high-dimensional secondorder parabolic partial differential equations (PDEs) are abundant in many important areas including financial engineering, economics, quantum mechanics, or statistical physics. We developed approximation methods which approximate solutions of FBSDEs with convergence rate nearly one-half under suitable assumptions. Our methods do not suffer from the curse of dimensionality and can thus also be applied if the forward process is very high-dimensional. These methods are the first and until now only implementable approximation methods for FBSDEs which have been mathematically demonstrated to overcome the curse for FBSDEs with general time horizon and general globally Lipschitz continuous terminal conditions and gradient-independent nonlinearities. Our approximation methods are based on multilevel Picard approximations – a nonlinear Monte Carlo method which we developed in the first project phase to approximate semilinear PDEs at fixed space-time points without suffering from the curse of dimensionality. In addition, our approximation methods are based on the multilevel approach of Stefan Heinrich. We show how to construct approximations of the full path of a sufficiently regular stochastic process given approximations at fixed time points without significantly reducing the order of convergence.

Publications

• Multilevel Picard approximations for highdimensional decoupled forward-backward stochastic differential equations
  Hutzenthaler, M. & Nguyen, T. A.
  
• Multilevel Picard approximations of high-dimensional semilinear partial differential equations with locally monotone coefficient functions. Applied Numerical Mathematics, 181, 151-175.
  Hutzenthaler, Martin & Nguyen, Tuan Anh
  
• Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations. Journal of Numerical Mathematics, 0(0).
  Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas & Anh, Nguyen Tuan
  
• Strong convergence rate of Euler-Maruyama approximations in temporal-spatial Hölder-norms. Journal of Computational and Applied Mathematics, 413, 114391.
  Hutzenthaler, Martin & Nguyen, Tuan Anh