Forward‐backward doubly stochastic differential equations with random jumps and related games

2020 ◽  
Author(s):  
Qingfeng Zhu ◽  
Yufeng Shi ◽  
Bin Teng
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Doubly Stochastic ◽  
Random Jumps

scholarly journals A Type of Time-Symmetric Stochastic System and Related Games

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 118
Author(s):  
Qingfeng Zhu ◽  
Yufeng Shi ◽  
Jiaqiang Wen ◽  
Hui Zhang
Keyword(s):  
Nash Equilibrium ◽  
Time Delay ◽  
Differential Equations ◽  
Equilibrium Point ◽  
Stochastic Differential Equations ◽  
Stochastic System ◽  
Stochastic Systems ◽  
Necessary Condition ◽  
Nash Equilibrium Point ◽  
Doubly Stochastic

This paper is concerned with a type of time-symmetric stochastic system, namely the so-called forward–backward doubly stochastic differential equations (FBDSDEs), in which the forward equations are delayed doubly stochastic differential equations (SDEs) and the backward equations are anticipated backward doubly SDEs. Under some monotonicity assumptions, the existence and uniqueness of measurable solutions to FBDSDEs are obtained. The future development of many processes depends on both their current state and historical state, and these processes can usually be represented by stochastic differential systems with time delay. Therefore, a class of nonzero sum differential game for doubly stochastic systems with time delay is studied in this paper. A necessary condition for the open-loop Nash equilibrium point of the Pontriagin-type maximum principle are established, and a sufficient condition for the Nash equilibrium point is obtained. Furthermore, the above results are applied to the study of nonzero sum differential games for linear quadratic backward doubly stochastic systems with delay. Based on the solution of FBDSDEs, an explicit expression of Nash equilibrium points for such game problems is established.

scholarly journals Comparison Theorems for the Multidimensional BDSDEs and Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Bo Zhu ◽  
Baoyan Han
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Comparison Theorem ◽  
Existence Of Solutions ◽  
Minimal Solution ◽  
Comparison Theorems ◽  
Doubly Stochastic

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.

L2-regularity result for solutions of backward doubly stochastic differential equations

2019 ◽  
Vol 20 (02) ◽  
pp. 2050012
Author(s):  
Achref Bachouch ◽  
Anis Matoussi
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Rate Of Convergence ◽  
Numerical Scheme ◽  
Regularity Result ◽  
Time Discretization ◽  
Lipschitz Continuous ◽  
Doubly Stochastic

We prove an [Formula: see text]-regularity result for the solutions of Forward Backward doubly stochastic differential equations (F-BDSDEs) under globally Lipschitz continuous assumptions on the coefficients. As an application of our result, we derive the rate of convergence in time for the (Euler time discretization-based) numerical scheme for F-BDSDEs proposed by Bachouch et al. (2016) under only globally Lipschitz continuous assumptions.

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

2020 ◽  
Vol 28 (1) ◽  
pp. 19-26
Author(s):  
Sadibou Aidara
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Comparison Theorems ◽  
Doubly Stochastic

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

scholarly journals An optimal control of a risk-sensitive problem for backward doubly stochastic differential equations with applications

2020 ◽  
Vol 28 (1) ◽  
pp. 1-18
Author(s):  
Dahbia Hafayed ◽  
Adel Chala
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Maximum Principle ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Control Problem ◽  
Stochastic Differential Equations ◽  
Doubly Stochastic ◽  
Risk Sensitive ◽  
Performance Functional

AbstractIn this paper, we are concerned with an optimal control problem where the system is driven by a backward doubly stochastic differential equation with risk-sensitive performance functional. We generalized the result of Chala [A. Chala, Pontryagin’s risk-sensitive stochastic maximum principle for backward stochastic differential equations with application, Bull. Braz. Math. Soc. (N. S.) 48 2017, 3, 399–411] to a backward doubly stochastic differential equation by using the same contribution of Djehiche, Tembine and Tempone in [B. Djehiche, H. Tembine and R. Tempone, A stochastic maximum principle for risk-sensitive mean-field type control, IEEE Trans. Automat. Control 60 2015, 10, 2640–2649]. We use the risk-neutral model for which an optimal solution exists as a preliminary step. This is an extension of an initial control system in this type of problem, where an admissible controls set is convex. We establish necessary as well as sufficient optimality conditions for the risk-sensitive performance functional control problem. We illustrate the paper by giving two different examples for a linear quadratic system, and a numerical application as second example.

Strong uniqueness for an SPDE via backward doubly stochastic differential equations

2013 ◽  
Vol 83 (10) ◽  
pp. 2186-2190 ◽  
Author(s):  
Alejandro Gomez ◽  
Kijung Lee ◽  
Carl Mueller ◽  
Ang Wei ◽  
Jie Xiong
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Strong Uniqueness ◽  
Doubly Stochastic
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