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Abstract
We address a class of McKean-Vlasov (MKV) control problems with common noise, called polynomial conditional MKV, and extending the known class of linear quadratic stochastic MKV control problems. We show how this polynomial class can be reduced by suitable Markov embedding to finite-dimensional stochastic control problems, and provide a discussion and comparison of three probabilistic numerical methods for solving the reduced control problem: quantization, regression by control randomization, and regress-later methods. Our numerical results are illustrated on various examples from portfolio selection and liquidation under drift uncertainty, and a model of interbank systemic risk with partial observation.
Highlights
The optimal control of McKean-Vlasov dynamics is a rather new topic in the area of stochastic control and applied probability, which has been knowing a surge of interest with the emergence of the mean-field game theory
This is a MKV control problem under partial observation, but notice that it does not belong to the class of linear quadratic (LQ) MKV problems due to the control α which appears in a multiplicative form with the state
We have investigated how to use probabilistic numerical methods for some classes of mean field control problem via Markovian embedding
Summary
The optimal control of McKean-Vlasov ( called mean-field) dynamics is a rather new topic in the area of stochastic control and applied probability, which has been knowing a surge of interest with the emergence of the mean-field game theory. We make the standard Lipschitz condition on the coefficients bpx, μ, aq, σpx, μ, aq, σ0px, μ, aq with respect to px, μq in Rn P2 pRnq, uniformly in a P A, where P2 pRnq is the set of all probability measures on pRn, BpRnqq with a finite second-order moment, endowed with the 2-Wasserstein metric W2 This ensures the well-posedness of the controlled MKV stochastic differential equation (SDE) (1.1). We consider the value function for the conditional MKV control problem (1.2), defined on r0, T s P2 pRnq by vpt, μq sup Jpt, μ, αq sup. The HJB equation (1.6) is a fully nonlinear partial differential equation (PDE) in the infinite-dimensional Wasserstein space This PDE does not have an explicit solution except in the notable important class of linear-quadratic MKV control problem. Our purpose is to investigate a class of MKV control problems which can be reduced to finite-dimensional problems in view of numerical resolution
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