Spike Variations for Stochastic Volterra Integral Equations
Spike Variations for Stochastic Volterra Integral Equations
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DOI：
10.1137/22m1522097
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发表时间：
2022-05
期刊：
影响因子：
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通讯作者：
Tianxiao Wang;J. Yong
中科院分区：
文献类型：
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作者：
Tianxiao Wang;J. Yong
当控制区域不是凸的时，尖峰变分技术在推导包括常微分方程组(ODE)、偏微分方程组(PDE)和随机微分方程组(SDE)在内的几类微分方程组的最优控制的Pontryagin型最大值原理中起着至关重要的作用。这一技术也适用于(确定性正演)Volterra积分方程(FVIE)。很自然地，这种技术可以扩展到(正向)随机Volterra积分方程组(FSVIE)的情况。然而，通过模仿SDE的情况，人们遇到了处理涉及的二次项的基本困难。为了克服这一困难，我们引入了一个可以使用它的公式的辅助过程，并采用了一个用于线性二次型随机最优控制问题的技巧。然后，得到了上述二次型的一个合适的表示式，并导出了二阶伴随方程。从而建立了庞特里亚金型的最大值原理。文中还研究了一些相关的扩展。
Spike variation technique plays a crucial role in deriving Pontryagin's type maximum principle of optimal controls for differential equations of several types, including ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differentia equations (SDEs), when the control domains are not assumed to be convex. This technique also applies to (deterministic forward) Volterra intrgral equations (FVIEs). 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 the difficulty, we introduce an auxiliary process for which one can use It\^o's formula, and adopt a trick used in linear-quadratic stochastic optimal control problems. Then a 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.