Backward stochastic differential equations driven byG-Brownian motion

Abstract

We design a class of numerical schemes for backward stochastic differential equation driven by G-Brownian motion (G-BSDE), which is related to a fully nonlinear PDE. Based on Peng's central limit theorem, we employ the CLT method to approximate G-distributed. Rigorous stability and convergence analysis are also carried out. It is shown that the θ-scheme admits a half order convergence rate in the general case. In particular, for the case of θ 1 ∈ [0, 1] and θ 2 = 0, the scheme can reach first-order in the deterministic case. Several numerical tests are given to support our theoretical results.

Quadratic backward stochastic differential equations driven by G-Brownian motion: Discrete solutions and approximation

“…In particular, the uniqueness of K is impressive in the light of G-martingale estimates found in [39]. The results in [11] breaks new ground in the G-expectation theory. In the accompanying paper [12], Hu et al discussed fundamental properties of the above GBSDE: the comparison theorem, the fully nonlinear Feynman-Kac formula and the related Girsanov transformation.…”

“…We now compare the result of [11] with the profound works [37,38] by Soner et al, in which the so-called second order backward stochastic differential equations (2BSDEs) are deeply studied. This type of equation is highly related to the GBSDE and it is defined on the Wiener space as follows:…”

“…[30] and [10]). However, the challenging problem of wellposedness for backward GSDEs (GBSDEs) remained open until a complete theorem has been proved by Hu et al [11].…”

“…Then, the partition of unity theorem was employed in [11] to proceed a type of Galerkin approximation to solutions of GBSDEs with general parameters. Besides, the uniqueness was deduced in [11] based on a priori estimates. In particular, the uniqueness of K is impressive in the light of G-martingale estimates found in [39].…”

“…The main objective of this paper is to provide the existence and uniqueness result for scalar-valued quadratic GBSDEs adapted to the setting of [11]. Without loss of generality, we consider only the following type of GBSDE:…”