Converse Comparison Theorems for Backward Stochastic Differential Equations

A class of backward doubly stochastic differential equations (BDSDEs) are studied. We obtain a comparison theorem of these multidimensional BDSDEs. As its applications, we derive the existence of solutions for this multidimensional BDSDEs with continuous coefficients. We can also prove that this solution is the minimal solution of the BDSDE.

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

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.524-527.3801 ◽

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.

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A general comparison theorem for backward stochastic differential equations

Advances in Applied Probability ◽

10.1239/aap/1282924067 ◽

2010 ◽

Vol 42 (3) ◽

pp. 878-898 ◽

Cited By ~ 9

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.

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Anticipated Generalized Backward Doubly Stochastic Differential Equations

Symmetry ◽

10.3390/sym14010114 ◽

2022 ◽

Vol 14 (1) ◽

pp. 114

Author(s):

Tie Wang ◽

Jiaxin Yu

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Uniqueness Theorem ◽

Comparison Theorem ◽

Existence And Uniqueness ◽

Comparison Theorems ◽

Existence And Uniqueness Theorem ◽

New Class ◽

Doubly Stochastic ◽

Increasing Process

In this paper, we explore a new class of stochastic differential equations called anticipated generalized backward doubly stochastic differential equations (AGBDSDEs), which not only involve two symmetric integrals related to two independent Brownian motions and an integral driven by a continuous increasing process but also include generators depending on the anticipated terms of the solution (Y, Z). Firstly, we prove the existence and uniqueness theorem for AGBDSDEs. Further, two comparison theorems are obtained after finding a new comparison theorem for GBDSDEs.

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Backward Doubly Stochastic Differential Equations with Markov Chains and a Comparison Theorem

Symmetry ◽

10.3390/sym12121953 ◽

2020 ◽

Vol 12 (12) ◽

pp. 1953

Author(s):

Ning Ma ◽

Zhen Wu

Keyword(s):

Differential Equations ◽

Markov Chains ◽

Stochastic Differential Equations ◽

Comparison Theorem ◽

Lipschitz Condition ◽

Existence And Uniqueness ◽

Comparison Theorems ◽

Yosida Approximation ◽

Uniqueness Of Solutions ◽

Doubly Stochastic

In this paper we study the existence and uniqueness of solutions for one kind of backward doubly stochastic differential equations (BDSDEs) with Markov chains. By generalizing the Itô’s formula, we study such problem under the Lipschitz condition. Moreover, thanks to the Yosida approximation, we solve such problem under monotone condition. Finally, we give the comparison theorems for such equations under the above two conditions respectively.

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Comparison Theorems for the Multi-Dimensional Backward Doubly Stochastic Differential Equations

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.166-169.3210 ◽

2012 ◽

Vol 166-169 ◽

pp. 3210-3213 ◽

Cited By ~ 1

Author(s):

Bao Yan Han

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Comparison Theorem ◽

Comparison Theorems ◽

Doubly Stochastic

A class of backward doubly stochastic differential equations are studied. We obtain a comparison theorem of these multi-dimensional backward doubly stochastic differential equations.

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