An efficient discretization to simulate the solution of linear–quadratic stochastic boundary control problem

Key Points

The presented discretization reduces complexities in the solution of the stochastic heat equation control problem.
Conditional expectations are no longer needed, leading to faster computations in boundary control scenarios.
Backward SPDEs are usually expensive to solve, but the reformulation offers the same convergence rates at a lower cost.
The method supports efficient simulation and optimality analysis across relevant control contexts.

Abstract

Abstract We present a fast, implementable discretization for the Dirichlet boundary control problem associated with the stochastic heat equation and show its space–time convergence with rates. After space–time discretization the discrete optimality conditions involve the discretization of a backward SPDE, whose numerical solution is well known to be costly since it requires the computation of conditional expectations. In this work we give a reformulation of the discrete optimality conditions, which avoids the need to simulate conditional expectations and therefore significantly reduces complexities if compared with regression-based simulation while keeping the same convergence rate.

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