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<p>We study the optimal control problem of stochastic delay differential equations driven by fractional Brownian motions with Hurst parameter <i>H</i> &gt; <i>1/2</i> and the underlying standard Brownian motions, where the state and the control variables both have delays. With the help of duality approach and the anticipated backward stochastic differential equations driven by fractional Brownian motions and the underlying standard Brownian motions, we obtain a Pontryagin’s type stochastic maximum principle involving Malliavin derivatives. Moreover, as illustrations, one kind of linear quadratic stochastic control problem and an optimal consumption problem are solved.</p>

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Maximum Principle for Stochastic Delay System Driven by Fractional and Standard Brownian Motions

  • 期刊论文

  • 发表日期：

    2025年5月29日

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  • Qiu-ling Hua,
  • Xin-rui Jing

摘要

We study the optimal control problem of stochastic delay differential equations driven by fractional Brownian motions with Hurst parameter H > 1/2 and the underlying standard Brownian motions, where the state and the control variables both have delays. With the help of duality approach and the anticipated backward stochastic differential equations driven by fractional Brownian motions and the underlying standard Brownian motions, we obtain a Pontryagin’s type stochastic maximum principle involving Malliavin derivatives. Moreover, as illustrations, one kind of linear quadratic stochastic control problem and an optimal consumption problem are solved.