Comparison Theorems for the Multidimensional BDSDEs and Applications

Author(s):

Bo Zhu ◽

Baoyan Han

Keyword(s):

Differential Equations ◽

Comparison Theorem ◽

Existence Of Solutions ◽

Minimal Solution ◽

Comparison Theorems ◽

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.

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 ◽

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.

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 ◽

Comparison Theorem ◽

Lipschitz Condition ◽

Existence And Uniqueness ◽

Comparison Theorems ◽

Yosida Approximation ◽

Uniqueness Of Solutions ◽

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.

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 ◽

Author(s):

Bao Yan Han

Keyword(s):

Differential Equations ◽

Comparison Theorem ◽

Comparison Theorems ◽

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

Comparison theorems for anticipated backward doubly stochastic differential equations with non-Lipschitz coefficients

Random Operators and Stochastic Equations ◽

10.1515/rose-2020-2026 ◽

2020 ◽

Vol 28 (1) ◽

pp. 19-26

Author(s):

Sadibou Aidara

Keyword(s):

Differential Equations ◽

Comparison Theorems ◽

AbstractIn this work, we prove some comparison theorems of anticipated backward doubly stochastic differential equations with non-Lipschitz coefficients.

Comparison Theorems of Backward Doubly Stochastic Differential Equations and Applications

Stochastic Analysis and Applications ◽

10.1081/sap-200044444 ◽

2005 ◽

Vol 23 (1) ◽

pp. 97-110 ◽

Cited By ~ 39

Author(s):

Yufeng Shi ◽

Yanling Gu ◽

Kai Liu

Keyword(s):

Differential Equations ◽

Comparison Theorems ◽

A comparison theorem and uniqueness theorem of backward doubly stochastic differential equations

Acta Mathematicae Applicatae Sinica English Series ◽

10.1007/s10255-011-0057-y ◽

2011 ◽

Vol 27 (2) ◽

pp. 223-232 ◽

Cited By ~ 3

Author(s):

Qian Lin ◽

Zhen Wu

Keyword(s):

Differential Equations ◽

Uniqueness Theorem ◽

Comparison Theorem ◽

Existence of solution for Mean-field Reflected Discontinuous Backward Doubly Stochastic Differential Equation

Applied Mathematics and Nonlinear Sciences ◽

10.2478/amns.2020.2.00038 ◽

2020 ◽

Vol 5 (2) ◽

pp. 205-216

Author(s):

Mostapha Abdelouahab Saouli

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Stochastic Differential Equation ◽

Comparison Theorem ◽

Mean Field ◽

Existence Of Solution ◽

Maximal Solution ◽

Existence Of A Solution ◽

AbstractIn this paper we prove the existence of a solution for mean-field reflected backward doubly stochastic differential equations (MF-RBDSDEs) with one continuous barrier and discontinuous generator (left-continuous). By a comparison theorem establish here for MF-RBDSDEs, we provide a minimal or a maximal solution to MF-RBDSDEs.

A NOTE ON HOMEOMORPHISM FOR BACKWARD DOUBLY SDEs AND APPLICATIONS

Stochastics and Dynamics ◽

10.1142/s021949371000308x ◽

2010 ◽

Vol 10 (04) ◽

pp. 549-560 ◽

Author(s):

A. AMAN ◽

M. N'ZI ◽

J. M. OWO

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Comparison Theorem ◽

Stochastic Partial Differential Equations ◽

Real Parameter ◽

Partial Differential ◽

Doubly Stochastic ◽

Terminal Values

In this note, we study the class of backward doubly stochastic differential equations (BDSDEs). In our framework, the terminal values depend on a real parameter. Under suitable assumptions and by the help of strict comparison theorem, we show homeomorphism property for the solution. This result is used to study homeomorphism property for quasi-linear stochastic partial differential equations.

BDSDE with Poisson jumps under stochastic Lipschitz and linear growth conditions

Stochastics and Dynamics ◽

10.1142/s0219493718500399 ◽

2018 ◽

Vol 18 (05) ◽

pp. 1850039 ◽

Author(s):

Ahmadou Bamba Sow ◽

Yaya Sagna

Keyword(s):

Differential Equations ◽

Comparison Theorem ◽

Existence And Uniqueness ◽

Linear Growth ◽

Growth Conditions ◽

Minimal Solution ◽

Poisson Jumps ◽

Stochastic Growth ◽

Doubly Stochastic ◽

Lipschitz Conditions

In this paper, we deal with a backward doubly stochastic differential equations with jumps. Under stochastic Lipschitz conditions on the coefficients, we prove the existence and uniqueness of solution and provide a comparison theorem. Using this comparison theorem, we show the existence of a minimal solution when the drift satisfy a stochastic growth condition.