Solution of the Bellman Equation Associated with an Infinite Dimensional Stochastic Control Problem and Synthesis of Optimal Control

1983 ◽  
Vol 21 (4) ◽  
pp. 531-550 ◽  
Author(s):  
Viorel Barbu ◽  
Giuseppe Da Prato
Keyword(s):  
Optimal Control ◽  
Control Problem ◽  
Stochastic Control ◽  
Bellman Equation ◽  
Infinite Dimensional ◽  
Stochastic Control Problem

scholarly journals Bellman equation and viscosity solutions for mean-field stochastic control problem

2018 ◽  
Vol 24 (1) ◽  
pp. 437-461 ◽  
Author(s):  
Huyên Pham ◽  
Xiaoli Wei
Keyword(s):  
Dynamic Programming ◽  
Control Problem ◽  
Stochastic Control ◽  
Viscosity Solutions ◽  
Bellman Equation ◽  
Risk Model ◽  
Mean Field ◽  
Dynamic Programming Principle ◽  
Probability Measures ◽  
Stochastic Control Problem

We consider the stochastic optimal control problem of McKean−Vlasov stochastic differential equation where the coefficients may depend upon the joint law of the state and control. By using feedback controls, we reformulate the problem into a deterministic control problem with only the marginal distribution of the process as controlled state variable, and prove that dynamic programming principle holds in its general form. Then, by relying on the notion of differentiability with respect to probability measures recently introduced by [P.L. Lions, Cours au Collège de France: Théorie des jeux à champ moyens, audio conference 2006−2012], and a special Itô formula for flows of probability measures, we derive the (dynamic programming) Bellman equation for mean-field stochastic control problem, and prove a verification theorem in our McKean−Vlasov framework. We give explicit solutions to the Bellman equation for the linear quadratic mean-field control problem, with applications to the mean-variance portfolio selection and a systemic risk model. We also consider a notion of lifted viscosity solutions for the Bellman equation, and show the viscosity property and uniqueness of the value function to the McKean−Vlasov control problem. Finally, we consider the case of McKean−Vlasov control problem with open-loop controls and discuss the associated dynamic programming equation that we compare with the case of closed-loop controls.

A Lower Bounding Linear Programming approach to the Perimeter Patrol Stochastic Control Problem

2012 ◽  
Author(s):  
Krishnamoorthy Kalyanam ◽  
Swaroop Darbha ◽  
Myoungkuk Park ◽  
Meir Pachter ◽  
Phil Chandler ◽  
...  
Keyword(s):  
Linear Programming ◽  
Control Problem ◽  
Stochastic Control ◽  
Programming Approach ◽  
Linear Programming Approach ◽  
Lower Bounding ◽  
Stochastic Control Problem
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