IJCRR - Vol 4 Issue 12, June
Date of Publication: 22-Jun-2012
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BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS UNDER G-BROWNIAN MOTION WITH DISCONTINUOUS DRIFT COEFFICIENTS
Author: Faiz Faizullah, Rahman Ullah
Category: General Sciences
Abstract:The main objective of this paper is to introduce the upper and lower solutions method for backward stochastic differential equations under G-Brownian motion (G-BSDEs). The existence of solutions for backward stochastic differential equations under G-Brownian motion having a discontinuous drift coefficient is shown with the method of upper and lower solutions. As an example, a scalar stochastic differential equation under G-Brownian motion having the unit step function as a drift coefficient is considered.
Keywords: Upper and lower solutions, backward stochastic differential equations, G-Brownian motion, discontinuous drift coefficient, existence.
To measure super hedging and risk in finance under volatility uncertainty, the G-Brownian motion and the related stochastic calculus were introduced by Peng [16, 17]. He introduced the backward stochastic differential equations under G-Brownian motion (G-BSDEs) and developed the existence and uniqueness of solutions for GBSDEs with Lipschitz continuous coefficients, see  chapter IV page 83 or the appendix of this paper. Later, X. Bai and Y. Lin extended the existence and uniqueness theory of the GBSDEs to the integral Lipschitz coefficients . Also see  for the stability theorem of GBSDEs. Now here in contrast to the above, we introduce the method of upper and lower solutions and establish the existence theory for G-BSDEs with discontinuous drift coefficients, such as in the following scalar G-BSDE
CONCLUSION Upper and lower solutions method is a very useful technique for the existence theory of boundary value problems (BVP). This method is widely used in ordinary and partial differential equations [3, 6, 12]. But a very limited literature is available on the method of upper and lower solutions for stochastic differential equations [8, 9]. The mentioned method for stochastic differential equations under G-Brownian motion (G-SDEs) was established by Faizullah and Piao in . Furthermore, this is still an open problem to develop the method of upper and lower solutions for classical backward stochastic differential equations.
The research of F. Faizullah is supported by the China Scholarships Council (CSC) and partially by National University of Sciences and Technology (NUST) Pakistan. We are very grateful to Prof. Daxiong Piao for his motivations and some useful suggestions for this work. We also thank to the anonymous reviewers for their valuable suggestions. Authors acknowledge the immense help received from the scholars whose articles are cited and included in references of this manuscript. The authors are also grateful to authors / editors / publishers of all those articles, journals and books from where the literature for this article has been reviewed and discussed.
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