Multi-dimensional reflected backward stochastic differential equations and the comparison theorem

2010 ◽  
Vol 30 (5) ◽  
pp. 1819-1836 ◽  
Author(s):  
Wu Zhen ◽  
Xiao Hua
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Comparison Theorem ◽  
Backward Stochastic Differential Equations

Comparison Theorem for any Solutions of Backward Stochastic Differential Equations and its Application

2012 ◽  
Vol 524-527 ◽  
pp. 3801-3804
Author(s):  
Shi Yu Li ◽  
Wu Jun Gao ◽  
Jin Hui Wang
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Comparison Theorem ◽  
Backward Stochastic Differential Equations ◽  
Stochastic Equations ◽  
Local Martingale ◽  
One Dimensional ◽  
Lipschitz Continuous ◽  
The One ◽  
Uniformly Lipschitz

ƒIn this paper, we study the one-dimensional backward stochastic equations driven by continuous local martingale. We establish a generalized the comparison theorem for any solutions where the coefficient is uniformly Lipschitz continuous in z and is equi-continuous in y.

scholarly journals A general comparison theorem for backward stochastic differential equations

2010 ◽  
Vol 42 (3) ◽  
pp. 878-898 ◽  
Author(s):  
Samuel N. Cohen ◽  
Robert J. Elliott ◽  
Charles E. M. Pearce
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Comparison Theorem ◽  
Backward Stochastic Differential Equations ◽  
Nonlinear Expectations

A useful result when dealing with backward stochastic differential equations is the comparison theorem of Peng (1992). When the equations are not based on Brownian motion, the comparison theorem no longer holds in general. In this paper we present a condition for a comparison theorem to hold for backward stochastic differential equations based on arbitrary martingales. This theorem applies to both vector and scalar situations. Applications to the theory of nonlinear expectations are also explored.

scholarly journals Backward stochastic differential equations with unbounded generators

2019 ◽  
Vol 19 (01) ◽  
pp. 1950008 ◽  
Author(s):  
Bujar Gashi ◽  
Jiajie Li
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Unique Solution ◽  
Comparison Theorem ◽  
Linear Growth ◽  
Sufficient Conditions ◽  
Backward Stochastic Differential Equations ◽  
Type Condition ◽  
Solution Pair ◽  
Picard Iterations

In this paper, we consider two classes of backward stochastic differential equations (BSDEs). First, under a Lipschitz-type condition on the generator of the equation, which can also be unbounded, we give sufficient conditions for the existence of a unique solution pair. The method of proof is that of Picard iterations and the resulting conditions are new. We also prove a comparison theorem. Second, under the linear growth and continuity assumptions on the possibly unbounded generator, we prove the existence of the solution pair. This class of equations is more general than the existing ones.

scholarly journals A general comparison theorem for backward stochastic differential equations

2010 ◽  
Vol 42 (03) ◽  
pp. 878-898 ◽  
Author(s):  
Samuel N. Cohen ◽  
Robert J. Elliott ◽  
Charles E. M. Pearce
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Comparison Theorem ◽  
Backward Stochastic Differential Equations ◽  
Nonlinear Expectations

A useful result when dealing with backward stochastic differential equations is the comparison theorem of Peng (1992). When the equations are not based on Brownian motion, the comparison theorem no longer holds in general. In this paper we present a condition for a comparison theorem to hold for backward stochastic differential equations based on arbitrary martingales. This theorem applies to both vector and scalar situations. Applications to the theory of nonlinear expectations are also explored.

Converse Comparison Theorems for Backward Stochastic Differential Equations

2013 ◽  
Vol 411-414 ◽  
pp. 1400-1403
Author(s):  
Xiao Qin Huang ◽  
Wei Hua Jiang ◽  
Xiao Jie Liu
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Comparison Theorem ◽  
Backward Stochastic Differential Equations ◽  
Comparison Theorems

In this note, we establish a converse comparison theorem for backward stochastic differential equations (BSDEs).

Some Properties of BSDEs Driven by a Simple Lévy Process with Continuous Coeffcient

2011 ◽  
Vol 50-51 ◽  
pp. 288-292
Author(s):  
Shi Qiu Zheng ◽  
Dian Chuan Jin ◽  
Shuai Zhang ◽  
Yan Mei Yang ◽  
Jin Peng Wang
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Comparison Theorem ◽  
Lévy Process ◽  
Linear Growth ◽  
Backward Stochastic Differential Equations ◽  
Levy Process

In this paper, we mainly study the properties of solutions of backward stochastic differential equations (BSDEs) driven by a simple Lévy process, whose coefficient coeffcient is continuous with linear growth. A comparison theorem for solutions of the equations are obtained, we also show the equation has either one or uncountably many solutions.

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