Large Deviations for Two-Time-Scale Systems Driven by Nonhomogeneous Markov Chains and Associated Optimal Control Problems

2011 ◽  
Vol 49 (4) ◽  
pp. 1737-1765 ◽  
Author(s):  
Qi He ◽  
George Yin ◽  
Qing Zhang
Keyword(s):  
Optimal Control ◽  
Markov Chains ◽  
Large Deviations ◽  
Time Scale ◽  
Optimal Control Problems ◽  
Control Problems ◽  
Nonhomogeneous Markov Chains

Hamilton-Jacobi Equations with Semilinear Costs and State Constraints, with Applications to Large Deviations in Games

2021 ◽  
Author(s):  
William H. Sandholm ◽  
Hung V. Tran ◽  
Srinivas Arigapudi
Keyword(s):  
Game Theory ◽  
Optimal Control ◽  
Large Deviations ◽  
Evolutionary Game Theory ◽  
State Constraints ◽  
Optimal Control Problems ◽  
Evolutionary Game ◽  
Control Problems ◽  
Optimal Solutions ◽  
Jacobi Equations

We characterize solutions of a class of time-homogeneous optimal control problems with semilinear running costs and state constraints as maximal viscosity subsolutions to Hamilton-Jacobi equations and show that optimal solutions to these problems can be constructed explicitly. We present applications to large deviations problems arising in evolutionary game theory.

scholarly journals Maximum Principle for Optimal Control Problems of Forward-Backward Regime-Switching Systems Involving Impulse Controls

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Shujun Wang ◽  
Zhen Wu
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Markov Chains ◽  
Regime Switching ◽  
Optimal Control Problems ◽  
Sufficient Conditions ◽  
Control Problems ◽  
Optimal Controls ◽  
Switching Systems ◽  
Impulse Controls

This paper is concerned with optimal control problems of forward-backward Markovian regime-switching systems involving impulse controls. Here the Markov chains are continuous-time and finite-state. We derive the stochastic maximum principle for this kind of systems. Besides the Markov chains, the most distinguishing features of our problem are that the control variables consist of regular and impulsive controls, and that the domain of regular control is not necessarily convex. We obtain the necessary and sufficient conditions for optimal controls. Thereafter, we apply the theoretical results to a financial problem and get the optimal consumption strategies.

On closed-loop equilibrium strategies for mean-field stochastic linear quadratic problems

2020 ◽  
Vol 26 ◽  
pp. 41
Author(s):  
Tianxiao Wang
Keyword(s):  
Optimal Control ◽  
Closed Loop ◽  
Optimal Control Problems ◽  
Mean Field ◽  
Open Loop ◽  
Time Inconsistency ◽  
Control Problems ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Equilibrium Strategies

This article is concerned with linear quadratic optimal control problems of mean-field stochastic differential equations (MF-SDE) with deterministic coefficients. To treat the time inconsistency of the optimal control problems, linear closed-loop equilibrium strategies are introduced and characterized by variational approach. Our developed methodology drops the delicate convergence procedures in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. When the MF-SDE reduces to SDE, our Riccati system coincides with the analogue in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. However, these two systems are in general different from each other due to the conditional mean-field terms in the MF-SDE. Eventually, the comparisons with pre-committed optimal strategies, open-loop equilibrium strategies are given in details.

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