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首页> 外文期刊>SIAM Journal on Control and Optimization >Stochastic minimum principle for partially observed systems subject to continuous and jump diffusion processes and driven by relaxed controls
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Stochastic minimum principle for partially observed systems subject to continuous and jump diffusion processes and driven by relaxed controls

机译：局部最小系统的随机最小原理，其受连续和跳跃扩散过程的影响，并由放松的控制来驱动

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摘要

In this paper, we consider nonconvex control problems of stochastic differential equations driven by relaxed controls adapted, in the weak star sense, to a current of sigma algebras generated by observable processes. We cover in a unified way both continuous diffusion and jump processes. We present existence of optimal controls before we construct the necessary conditions of optimality (unlike some papers in this area) using only functional analysis. We develop a stochastic Hamiltonian system of equations on a rigorous basis using the semimartingale representation theory and the Riesz representation theorem, leading naturally to the existence of the adjoint process which satisfies a backward stochastic differential equation. In other words, our approach predicts the existence of the adjoint process as a natural consequence of Riesz representation theory ensuring at the same time the (weak star) measurability. This is unlike other papers, where the adjoint process is introduced before its existence is proved. We believe this is one of our major contributions in this paper. We also discuss the realizability of relaxed controls by regular controls using the Krein- Millman theorem. We believe this is another major contribution of this paper. We also believe that our approach is direct and easy to understand following simply the precise logic of functional analysis.
机译：在本文中，我们考虑了由松弛控制驱动的随机微分方程的非凸控制问题，该控制在弱星形意义上适应了由可观测过程产生的西格玛代数的潮流。我们以统一的方式涵盖了连续扩散和跳跃过程。在仅使用功能分析构建最优条件的必要条件之前（与该领域的某些论文不同），我们介绍了最优控制的存在。我们使用半mart表示理论和Riesz表示定理在严格的基础上开发了随机汉密尔顿方程组，自然导致伴随过程的存在，该过程满足后向随机微分方程。换句话说，我们的方法预测伴随过程的存在是Riesz表示理论的自然结果，可同时确保（弱星）可测量性。这与其他论文不同，后者在证明伴随过程之前就引入了伴随过程。我们相信这是本文的主要贡献之一。我们还将讨论使用Krein-Millman定理通过常规控件实现松弛控件的可实现性。我们认为这是本文的另一个主要贡献。我们还相信，只要遵循功能分析的精确逻辑，我们的方法便是直接且易于理解的。

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