Empirical Regression Method for Backward Doubly Stochastic Differential Equations

Abstract

In this paper we design a numerical scheme for approximating Backward Doubly Stochastic Differential Equations (BDSDEs for short) which represent solution to Stochastic Partial Differential Equations (SPDEs). We first use a time-discretization and then, we decompose the value function on a functions basis. The functions are deterministic and depend only on time-space variables, while decomposition coefficients depend on the external Brownian motion B. The coefficients are evaluated through a empirical regression scheme, which is performed conditionally to B. We establish non asymptotic error estimates, conditionally to B, and deduce how to tune parameters to obtain a convergence conditionally and unconditionally to B. We provide numerical experiments as well.

Probabilistic interpretation for solutions of fully nonlinear stochastic PDEs

“…This introduces additional non trivial difficulties. The same type of problems were already pointed out in [3], where the authors analyze regression schemes for approximating BDSDEs as well as their convergence, and obtain non-asymptotic error estimates, conditionally to the external noise (that is W in our context). Similarly to the classical 2BSDEs, the solution of a 2BDSDE has to be represented as a supremum of solutions to standard BDSDEs.…”

“…Combining these two approaches should in principle allow to obtain efficient numerical schemes for computing solutions to 2BDSDEs, and therefore for fully non-linear SPDEs. As far as we know, there are no literature on the subject, except the cases of semilinear and quasilinear SPDEs (see [30], [32], [33], [4], [3]), and so our results could prove to be a non-negligible progress.…”

“…It is our conviction that they open a new path for possible numerical simulations of solution to fully-nonlinear SPDEs. Indeed, numerical schemes for classical BDSDEs are by now well-known (see for instance [3]), and numerical procedures for solving second-order BSDEs have also been successfully implemented in the recent years, see for instance Possamaï and Tan [57] or Ren and Tan [60]. Combining these two approaches should in principle allow to obtain efficient numerical schemes for computing solutions to 2BDSDEs, and therefore for fully non-linear SPDEs.…”

A Nonintrusive Stratified Resampler for Regression Monte Carlo: Application to Solving Nonlinear Equations

“…It takes the form of a discrete Gronwall lemma, which easily allows to derive the following upper bound (see [2,Appendix A.3]):…”

Numerical computation for backward doubly SDEs with random terminal time

“…Several generalizations to investigate more general nonlinear SPDEs have been developed following different approaches of the notion of weak solutions: the technique of stochastic flow (Bally and Matoussi [8], Matoussi et al [38]); the approach based on Dirichlet forms and their associated Markov processes (Denis and Stoica [21], Bally, Pardoux and Stoica [9], Denis, Matoussi and Stoica [19,20]); stochastic viscosity solution for SPDEs (Buckdahn and Ma [16,15], Lions and Souganidis [35,36,34]). Above approaches have allowed the study of numerical schemes for the Sobolev solution of semilinear SPDEs via Monte-Carlo methods (time discretization and regression schemes [6,5,4]).…”