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

By Huyen Pham, Xiaoli Wei. 2015

180 citations recorded by the source corpus.

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 pro\-bability measures recently introduced by P.L. Lions in [32], and a special It{\^o} formula for flows of probability measures, we derive the (dynamic programming) Bellman equation for mean-field stochastic control problem, and prove a veri\-fication 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 visc-sity 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.

Subfields: Demography, Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/dccc585bc76740db55f4d4c313194fb3a08ee556

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Citations in the corpus

Master Bellman equation in the Wasserstein space: Uniqueness of viscosity solutions (2021, background): “Concerning second-order equations, papers [41, 42, 4, 17, 16] focus on the existence of viscosity solutions, proving that the value function solves in the viscosity sense the Master Bellman equation.”
Viscosity solutions to parabolic master equations and McKean–Vlasov SDEs with closed-loop controls (2018, methodology): “See, e.g. Pham & Wei [32].”
McKean–Vlasov optimal control: The dynamic programming principle (2019, methodology): “This was achieved in Pham and Wei [67] for the control problem V ◦ S in the setting with σ0 ≡ 0), and in Pham and Wei [66] for the control problem V ◦,B S (for Φt(x) := x(t), with E = R).”
Linear Quadratic Mean Field Type Control and Mean Field Games with Common Noise, with Application to Production of an Exhaustible Resource (2016, methodology): “We note that [56] reproduces and slightly extends the results of [60] using dynamic programming.”