Abstract
We consider a general class of mean field control problems described by stochastic delayed differential equations of McKean-Vlasov type. Two numerical algorithms are provided based on deep learning techniques, one is to directly parameterize the optimal control using neural networks, the other is based on numerically solving the McKean-Vlasov forward anticipated backward stochastic differential equation (MV-FABSDE) system. In addition, we establish a necessary and sufficient stochastic maximum principle for this class of mean field control problems with delay based on the differential calculus on function of measures, as well as existence and uniqueness results for the associated MV-FABSDE system.
Highlights
Stochastic games were introduced to study the optimal behaviors of agents interacting with each other
We prove the existence and uniqueness of the system of McKean–Vlasov forward anticipated backward stochastic differential equations (MV-FABSDE) under some suitable conditions using the method of continuation, which can be found in Zhang [6], Peng and Wu [8], Bensoussan et al [9], and Carmona et al [2]
The state dynamics is described by a McKean–Vlasov stochastic delayed differential equation
Summary
Stochastic games were introduced to study the optimal behaviors of agents interacting with each other. We may emphasize that the way that we present here for numerically computing conditional expectation may have a wide range of applications, and it is simple to implement We present another algorithm solving the mean field control problem by directly parameterizing the optimal control. This is illustrated on a simple linear-quadratic toy model, with delay in the control.
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