We consider a new discretization in space (parameter $h>0$) and time (parameter $\tau>0$) of a stochastic optimal control problem, where a quadratic functional is minimized subject to a linear stochastic heat equation with linear noise. Its construction is based on the perturbation of a generalized difference Riccati equation to approximate the related feedback law. We prove a convergence rate of almost ${\mathcal O}(h^{2}+\tau )$ for its solution, and conclude from it a rate of almost ${\mathcal O}(h^{2}+\tau )$ resp. ${\mathcal O}(h^{2}+\tau ^{1/2})$ for computable approximations of the optimal state and control with additive resp. multiplicative noise.

中文翻译：

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SPDE 的基于 Riccati 的 SLQ 离散化问题的收敛速率


我们考虑了随机最优控制问题在空间（参数 $h>0$）和时间（参数 $\tau>0$）上的新离散化，其中二次函数在具有线性噪声的线性随机热方程下最小化。它的构造基于广义差分 Riccati 方程的扰动，以近似于相关的反馈定律。我们证明了其解的收敛率接近 ${\mathcal O}（h^{2}+\tau ）$，并从中得出结论，对于最佳状态和加法或乘法噪声的最佳状态和控制的可计算近似值，收敛率接近 ${\mathcal O}（h^{2}+\tau ）$ 或 ${\mathcal O}（h^{2}+\tau ^{1/2}）$。