SOC-MartNet: A Martingale Neural Network for the Hamilton-Jacobi-Bellman Equation without Explicit inf H in Stochastic Optimal Controls
CoRR(2024)
摘要
In this work, we propose a martingale based neural network, SOC-MartNet, for
solving high-dimensional Hamilton-Jacobi-Bellman (HJB) equations where no
explicit expression is needed for the Hamiltonian inf_u ∈ U H(t,x,u,
z,p), and stochastic optimal control problems with controls on both drift and
volatility. We reformulate the HJB equations into a stochastic neural network
learning process, i.e., training a control network and a value network such
that the associated Hamiltonian process is minimized and the cost process
becomes a martingale.To enforce the martingale property for the cost process,
we employ an adversarial network and construct a loss function based on the
projection property of conditional expectations. Then, the control/value
networks and the adversarial network are trained adversarially, such that the
cost process is driven towards a martingale and the minimum principle is
satisfied for the control.Numerical results show that the proposed SOC-MartNet
is effective and efficient for solving HJB-type equations and SOCP with a
dimension up to 500 in a small number of training epochs.
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