Mean‐field backward stochastic differential equation with non‐Lipschitz coefficient

2019 ◽  
Vol 22 (5) ◽  
pp. 1986-1994
Author(s):  
Guangchen Wang ◽  
Huanjun Zhang
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Backward Stochastic Differential Equation ◽  
Mean Field

scholarly journals Linear-Quadratic-Gaussian mean-field controls of social optima

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zhenghong Qiu ◽  
Jianhui Huang ◽  
Tinghan Xie
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Backward Stochastic Differential Equation ◽  
Mean Field ◽  
Mean Field Approximation ◽  
Linear Quadratic ◽  
Linear Quadratic Gaussian ◽  
Individual Agent ◽  
State Process ◽  
Weakly Coupled

<p style='text-indent:20px;'>This paper investigates a class of unified stochastic linear-quadratic-Gaussian (LQG) social optima problems involving a large number of weakly-coupled interactive agents under a generalized setting. For each individual agent, the control and state process enters both diffusion and drift terms in its linear dynamics, and the control weight might be <i>indefinite</i> in cost functional. This setup is innovative and has great theoretical and realistic significance as its applications in mathematical finance (e.g., portfolio selection in mean-variation model). Using some <i>fully-coupled</i> variational analysis under the person-by-person optimality principle, and the mean-field approximation method, the decentralized social control is derived by a class of new type consistency condition (CC) system for typical representative agent. Such CC system is some mean-field forward-backward stochastic differential equation (MF-FBSDE) combined with <i>embedding representation</i>. The well-posedness of such forward-backward stochastic differential equation (FBSDE) system is carefully examined. The related social asymptotic optimality is related to the convergence of the average of a series of weakly-coupled backward stochastic differential equation (BSDE). They are verified through some Lyapunov equations.</p>

scholarly journals Mean-field anticipated BSDEs driven by time-changed Lévy noises

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Youxin Liu ◽  
Yang Dai
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Stochastic Differential Equation ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equation ◽  
Mean Field ◽  
Iterative Sequence ◽  
Uniqueness Of The Solution ◽  
The Fixed Point Theorem

Abstract The objective of this work is to show a new kind of mean-field anticipated backward stochastic differential equation (in short MF-ABSDE) driven by time-changed Lévy noises. We give two methods to prove the existence and uniqueness of the solution of those equations by the fixed point theorem and the Picard iterative sequence. Finally, we obtain a comparison theorem for the solutions.

scholarly journals Lp (p ≥ 1) solutions of multidimensional BSDEs with monotone generators in general time intervals

2014 ◽  
Vol 15 (01) ◽  
pp. 1550002 ◽  
Author(s):  
Li-Shun Xiao ◽  
Sheng-Jun Fan ◽  
Na Xu
Keyword(s):  
Differential Equation ◽  
Weak Convergence ◽  
Stochastic Differential Equation ◽  
Existence And Uniqueness ◽  
Additional Assumption ◽  
Backward Stochastic Differential Equation ◽  
Time Interval ◽  
Time Intervals ◽  
Lipschitz Continuous ◽  
General Time

In this paper, we are interested in solving general time interval multidimensional backward stochastic differential equation in Lp (p ≥ 1). We first study the existence and uniqueness for Lp (p > 1) solutions by the method of convolution and weak convergence when the generator is monotonic in y and Lipschitz continuous in z both non-uniformly with respect to t. Then we obtain the existence and uniqueness for L1 solutions with an additional assumption that the generator has a sublinear growth in z non-uniformly with respect to t.

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