The classical maximum principle for optimal stochastic control states that if a control [Formula: see text] is optimal, then the corresponding Hamiltonian has a maximum at [Formula: see text]. The first proofs for this result assumed that the control did not enter the diffusion coefficient. Moreover, it was assumed that there were no jumps in the system. Subsequently, it was discovered by Shige Peng (still assuming no jumps) that one could also allow the diffusion coefficient to depend on the control, provided that the corresponding adjoint backward stochastic differential equation (BSDE) for the first-order derivative was extended to include an extra BSDE for the second-order derivatives. In this paper, we present an alternative approach based on Hida–Malliavin calculus and white noise theory. This enables us to handle the general case with jumps, allowing both the diffusion coefficient and the jump coefficient to depend on the control, and we do not need the extra BSDE with second-order derivatives. The result is illustrated by an example of a constrained linear-quadratic optimal control.

中文翻译：

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最优控制的 Hida-Malliavin 白噪声演算方法

最优随机控制的经典最大值原理指出，如果控制 [公式：参见文本] 是最优的，则相应的哈密顿量在 [公式：参见文本] 处具有最大值。这个结果的第一个证明假设控制没有进入扩散系数。此外，假设系统中没有跳跃。随后，Shige Peng（仍然假设没有跳跃）发现，如果将一阶导数的相应伴随反向随机微分方程（BSDE）扩展到包括二阶导数的额外 BSDE。在本文中，我们提出了一种基于 Hida-Malliavin 演算和白噪声理论的替代方法。这使我们能够处理带有跳转的一般情况，允许扩散系数和跳跃系数都取决于控制，并且我们不需要具有二阶导数的额外 BSDE。结果通过一个受约束的线性二次最优控制的例子来说明。