SIAM Journal on Control and Optimization, Volume 61, Issue 6, Page 3608-3634, December 2023.
Abstract. The spike variation technique plays a crucial role in deriving Pontryagin’s type maximum principle of optimal controls for ordinary differential equations (ODEs), partial differential equations (PDEs), stochastic differential equations (SDEs), and (deterministic forward) Volterra integral equations (FVIEs), when the control domains are not assumed to be convex. It is natural to expect that such a technique could be extended to the case of (forward) stochastic Volterra integral equations (FSVIEs). However, by mimicking the case of SDEs, one encounters an essential difficulty of handling an involved quadratic term. To overcome this difficulty, we introduce an auxiliary process for which one can use Itô’s formula, and develop new technologies inspired by stochastic linear-quadratic optimal control problems. Then the suitable representation of the above-mentioned quadratic form is obtained, and the second-order adjoint equations are derived. Consequently, the maximum principle of Pontryagin type is established. Some relevant extensions are investigated as well.

中文翻译：

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随机 Volterra 积分方程的尖峰变体

SIAM 控制与优化杂志，第 61 卷，第 6 期，第 3608-3634 页，2023 年 12 月。
摘要。尖峰变异技术在推导常微分方程 (ODE)、偏微分方程 (PDE)、随机微分方程 (SDE) 和（确定性正向）Volterra 积分方程 (FVIE) 最优控制的 Pontryagin 型最大原理方面发挥着至关重要的作用，当控制域不假设为凸时。很自然地期望这种技术可以扩展到（正向）随机 Volterra 积分方程（FSVIE）的情况。然而，通过模仿 SDE 的情况，我们会遇到处理所涉及的二次项的本质困难。为了克服这一困难，我们引入了一种可以使用伊藤公式的辅助过程，并受随机线性二次最优控制问题的启发开发新​​技术。然后得到上述二次形式的适当表示，并推导出二阶伴随方程。由此，庞特里亚金型极大值原理成立。还研究了一些相关的扩展。