G-Expectation for Conformable Backward Stochastic Differential Equations

Jin-Rong Wang,Donal O’Regan,Mei Luo,Michal Fečkan

doi:10.3390/axioms11020075

Abstract

In this paper, we study the applications of conformable backward stochastic differential equations driven by Brownian motion and compensated random measure in nonlinear expectation. From the comparison theorem, we introduce the concept of g-expectation and give related properties of g-expectation. In addition, we find that the properties of conformable backward stochastic differential equations can be deduced from the properties of the generator g. Finally, we extend the nonlinear Doob–Meyer decomposition theorem to more general cases.

Highlights

The initial research motivation of nonlinear expectations came from risk measurement and option pricing in financial applications
In 1997, Peng [2] introduced a new nonlinear expectation, namely the g-expectation, based on the backward stochastic differential equation driven by Brownian motion
Royer [4] studied the backward stochastic differential equation driven by Brownian motion and Poisson random measure, and introduced the corresponding g-expectation and a large number of studies show that this g-expectation can be applied to financial problems

Summary

Introduction

The initial research motivation of nonlinear expectations came from risk measurement and option pricing in financial applications. We study g-expectation for conformable backward stochastic differential equations. (see [2] (Definition 3.1)) A functional E : L2(Ω, FT, P) → R is called a nonlinear expectation if it satisfies the following properties: (i) Strict monotonicity: if X1 ≥ X2 a.s., E [X1] ≥ E [X2], and if X1 ≥ X2 a.s., E [X1] = E [X2] ⇔ X1 = X2 a.s.

Results

Conclusion