Dynamic programming for optimal control of stochastic McKean-Vlasov dynamics
arXiv:1604.04057
Abstract
We study the optimal control of general stochastic McKean-Vlasov equation. Such problem is motivated originally from the asymptotic formulation of cooperative equilibrium for a large population of particles (players) in mean-field interaction under common noise. Our first main result is to state a dynamic programming principle for the value function in the Wasserstein space of probability measures, which is proved from a flow property of the conditional law of the controlled state process. Next, by relying on the notion of differentiability with respect to probability measures due to P.L. Lions [32], and It{ô}'s formula along a flow of conditional measures, we derive the dynamic programming Hamilton-Jacobi-Bellman equation, and prove the viscosity property together with a uniqueness result for the value function. Finally, we solve explicitly the linear-quadratic stochastic McKean-Vlasov control problem and give an application to an interbank systemic risk model with common noise.
33 pages, to appear in SIAM Journal on Control and Optimization
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- Linear quadratic optimal control of conditional McKean-Vlasov equation with random coefficients and applications *
- Linear Quadratic Mean Field Type Control and Mean Field Games with Common Noise, with Application to Production of an Exhaustible Resource