发表机构
School of Mathematics, Shandong University（山东大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究内部信息下的随机最优控制问题，采用前向积分处理非适配情形，通过Donsker delta函数变换系统，推导高斯假设下的最优性条件，构建带随机系数的LQ问题并给出数值例子验证理论。
AI 中文摘要
本文研究内部信息下的随机最优控制问题。控制过程依赖于一个$\boldsymbol{\text{\mathcal{F}}_{T_0}}$-可测随机变量$Y$，该变量代表静态内部信息，且适配于由基础布朗运动和随机变量$Y$生成的增广滤子。相应地，传统随机积分在这种非适配情形下无法良好定义；我们采用前向积分来构建系统中的随机积分项。借助Donsker delta函数，原受控系统被转换为一个带$y$参数的系统。我们进一步建立了前向随机微分方程（SDE）解的存在性，并通过流变换技术证明了解的唯一性。在$Y$服从高斯假设下，我们推导了上述控制问题的最优性必要条件和充分条件。随后，我们构建了内部信息下的线性二次（LQ）最优控制问题。通过$y$参数化变换，原问题被转化为带随机系数的LQ控制问题。文末给出了LQ情形的数值例子，以验证我们的理论结果。
英文摘要
This paper is concerned with a stochastic optimal control problem under inside information. The control process depends on an $\mathcal{F}_{T_0}$-measurable random variable $Y$, representing the static inside information, and is adapted to the enlarged filtration generated by the underlying Brownian motion and the random variable $Y$. Accordingly, the traditional stochastic integral fails to be well-defined in this non-adapted setting; we adopt forward integrals to formulate the stochastic integral terms in the system. By means of the Donsker delta function, the original controlled system is transformed into a $y$-parameterized system. We further establish the existence of solutions to forward \textit{stochastic differential equations} (SDEs), and prove the uniqueness of solutions via flow transformation techniques. Under a Gaussian assumption on $Y$, we derive both necessary and sufficient optimality conditions for the aforementioned control problem. Subsequently, we formulate the \textit{linear-quadratic} (LQ) optimal control problem under inside information. Through the $y$-parameterized transformation, the original problem is converted into an LQ control problem with random coefficients. A numerical example for the LQ case is provided at the end to validate our theoretical findings.
Comments25 pages, 3 figures
