This paper is devoted to the analysis of an optimal control problem for stochastic integro-differential equations driven by a non-gaussian Lévy noise. The memory effect in the equation is driven by a completely monotone kernel (thus covering, for instance, the class of fractional time derivative of order less than 1). We suppose that the control acts on the jump rate of the noise. We show that this allows to tackle the problem through a backward stochastic differential equations approach, since the structure condition required by this approach is naturally satisfied. We solve the optimal control problem of minimizing a cost functional on a finite time horizon, with both running and final costs. We finally prove the existence of a weak solution of the closed-loop equation and we construct an optimal feedback control.

本文致力于分析由非高斯Lévy噪声驱动的随机积分微分方程的最优控制问题。等式中的记忆效应是由一个完全单调的内核驱动的（例如，覆盖了阶次小于1的分数时间导数的类）。我们假设控制作用于噪声的跳跃率。我们表明，由于可以自然地满足此方法所需的结构条件，因此可以通过后向随机微分方程方法解决该问题。我们解决了最佳控制问题，即在有限的时间范围内最小化具有运行成本和最终成本的成本函数。最后，我们证明了闭环方程的弱解的存在，并构造了最优反馈控制。