scholarly journals A BSDE Approach to Stochastic Differential Games with Regime Switching

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
J. Y. Li ◽  
M. N. Tang
Keyword(s):  
Dynamic Programming ◽  
Regime Switching ◽  
Time Horizon ◽  
Backward Stochastic Differential Equation ◽  
Stochastic Differential Games ◽  
Dynamic Programming Principle ◽  
Value Functions ◽  
Finite Time Horizon ◽  
Hamilton Jacobi Bellman ◽  
Zero Sum

In this paper, we study a two-player zero-sum stochastic differential game with regime switching in the framework of forward-backward stochastic differential equations on a finite time horizon. By means of backward stochastic differential equation methods, in particular that of the notion from stochastic backward semigroups, we prove a dynamic programming principle for both the upper and the lower value functions of the game. Based on the dynamic programming principle, the upper and the lower value functions are shown to be the unique viscosity solutions of the associated upper and lower Hamilton–Jacobi–Bellman–Isaacs equations.

scholarly journals The Dynamic Programming Method of Stochastic Differential Game for Functional Forward-Backward Stochastic System

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Shaolin Ji ◽  
Chuanfeng Sun ◽  
Qingmeng Wei
Keyword(s):  
Dynamic Programming ◽  
Differential Game ◽  
Backward Stochastic Differential Equation ◽  
Transformation Method ◽  
Dynamic Programming Method ◽  
Stochastic Differential Game ◽  
Value Functions ◽  
Girsanov Transformation ◽  
Path Dependent ◽  
Hamilton Jacobi Bellman

This paper is devoted to a stochastic differential game (SDG) of decoupled functional forward-backward stochastic differential equation (FBSDE). For our SDG, the associated upper and lower value functions of the SDG are defined through the solution of controlled functional backward stochastic differential equations (BSDEs). Applying the Girsanov transformation method introduced by Buckdahn and Li (2008), the upper and the lower value functions are shown to be deterministic. We also generalize the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations to the path-dependent ones. By establishing the dynamic programming principal (DPP), we derive that the upper and the lower value functions are the viscosity solutions of the corresponding upper and the lower path-dependent HJBI equations, respectively.

scholarly journals Probabilistic interpretation of a system of coupled Hamilton-Jacobi-Bellman-Isaacs equations

2020 ◽  
Author(s):  
Juan Li ◽  
Wenqiang Li ◽  
Qingmeng Wei
Keyword(s):  
Dynamic Programming ◽  
Value Function ◽  
Backward Stochastic Differential Equation ◽  
Stopping Times ◽  
Dynamic Programming Principle ◽  
Probabilistic Interpretation ◽  
Isaacs Equations ◽  
Hamilton Jacobi Bellman ◽  
Cost Functionals ◽  
Strong Dynamic

By introducing a stochastic differential game whose dynamics and multi-dimensional cost functionals form a multi-dimensional coupled forward-backward stochastic differential equation with jumps, we give a probabilistic interpretation to a system of coupled Hamilton-Jacobi-Bellman-Isaacs equations. For this, we generalize the definition of the lower value function  initially defined only for deterministic times $t$ and states $x$ to  stopping times $\tau$ and random variables $\eta\in L^2(\Omega,\mathcal {F}_\tau,P; \mathbb{R})$. The generalization plays a key role in the proof of a strong dynamic programming principle. This strong dynamic programming principle allows us to show that the lower value function is a viscosity solution of our system of multi-dimensional coupled Hamilton-Jacobi-Bellman-Isaacs equations. The uniqueness is obtained for a particular but important case.

scholarly journals Dynamic programming principle and associated Hamilton-Jacobi-Bellman equation for stochastic recursive control problem with non-Lipschitz aggregator

2018 ◽  
Vol 24 (1) ◽  
pp. 355-376 ◽  
Author(s):  
Jiangyan Pu ◽  
Qi Zhang
Keyword(s):  
Dynamic Programming ◽  
Control Problem ◽  
Bellman Equation ◽  
Backward Stochastic Differential Equation ◽  
Dynamic Programming Principle ◽  
Unknown Variable ◽  
Hamilton Jacobi Bellman Equation ◽  
Hamilton Jacobi Bellman ◽  
The Stability ◽  
The Value Function

In this work we study the stochastic recursive control problem, in which the aggregator (or generator) of the backward stochastic differential equation describing the running cost is continuous but not necessarily Lipschitz with respect to the first unknown variable and the control, and monotonic with respect to the first unknown variable. The dynamic programming principle and the connection between the value function and the viscosity solution of the associated Hamilton-Jacobi-Bellman equation are established in this setting by the generalized comparison theorem for backward stochastic differential equations and the stability of viscosity solutions. Finally we take the control problem of continuous-time Epstein−Zin utility with non-Lipschitz aggregator as an example to demonstrate the application of our study.

scholarly journals Dynamic Set Values for Nonzero-Sum Games with Multiple Equilibriums

2021 ◽  
Author(s):  
Zachary Feinstein ◽  
Birgit Rudloff ◽  
Jianfeng Zhang
Keyword(s):  
Dynamic Programming ◽  
Open Loop ◽  
Dynamic Programming Principle ◽  
Value Of The Game ◽  
State Dependent ◽  
Path Dependent ◽  
Duality Approach ◽  
Finite Time Horizon ◽  
Zero Sum ◽  
Set Value

Nonzero sum games typically have multiple Nash equilibriums (or no equilibrium), and unlike the zero-sum case, they may have different values at different equilibriums. Instead of focusing on the existence of individual equilibriums, we study the set of values over all equilibriums, which we call the set value of the game. The set value is unique by nature and always exists (with possible value [Formula: see text]). Similar to the standard value function in control literature, it enjoys many nice properties, such as regularity, stability, and more importantly, the dynamic programming principle. There are two main features in order to obtain the dynamic programming principle: (i) we must use closed-loop controls (instead of open-loop controls); and (ii) we must allow for path dependent controls, even if the problem is in a state-dependent (Markovian) setting. We shall consider both discrete and continuous time models with finite time horizon. For the latter, we will also provide a duality approach through certain standard PDE (or path-dependent PDE), which is quite efficient for numerically computing the set value of the game.

scholarly journals A Constrained Markovian Diffusion Model for Controlling the Pollution Accumulation

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1466
Author(s):  
Beatris Adriana Escobedo-Trujillo ◽  
José Daniel López-Barrientos ◽  
Javier Garrido-Meléndez
Keyword(s):  
Dynamic Programming ◽  
Dirichlet Problem ◽  
Stochastic Control ◽  
Finite Time ◽  
Time Horizon ◽  
Closed Loop ◽  
Programming Techniques ◽  
Pollution Accumulation ◽  
Finite Time Horizon ◽  
The Cost

This work presents a study of a finite-time horizon stochastic control problem with restrictions on both the reward and the cost functions. To this end, it uses standard dynamic programming techniques, and an extension of the classic Lagrange multipliers approach. The coefficients considered here are supposed to be unbounded, and the obtained strategies are of non-stationary closed-loop type. The driving thread of the paper is a sequence of examples on a pollution accumulation model, which is used for the purpose of showing three algorithms for the purpose of replicating the results. There, the reader can find a result on the interchangeability of limits in a Dirichlet problem.

scholarly journals Finite state N-agent and mean field control problems

2021 ◽  
Author(s):  
Alekos Cecchin
Keyword(s):  
Control Problem ◽  
Value Function ◽  
Mean Field ◽  
Control Problems ◽  
Value Functions ◽  
Limit Value ◽  
Field Control ◽  
Finite State ◽  
Finite Time Horizon ◽  
Hamilton Jacobi Bellman

We examine mean field control problems  on a finite state space, in continuous time and over a finite time horizon. We characterize the value function of the mean field control problem as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation in the simplex. In absence of any convexity assumption, we exploit this characterization to prove convergence, as $N$ grows, of the value functions of the centralized $N$-agent optimal control problem to the limit mean field control problem  value function, with a convergence rate of order $\frac{1}{\sqrt{N}}$. Then, assuming convexity, we show that the limit value function is smooth and establish propagation of chaos, i.e.  convergence of the $N$-agent optimal trajectories to the unique limiting optimal trajectory, with an explicit rate.

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