Optimal control of forward-backward stochastic Volterra equations

Abstract

We study the problem of optimal control of a coupled system of forward-backward stochastic Volterra equations. We use Hida-Malliavin calculus to prove a sufficient and a necessary maximum principle for the optimal control of such systems. Existence and uniqueness of backward stochastic Volterra integral equations are proved. As an application of our methods, we solve a recursive utility optimisation problem in a financial model with memory.

Linear Volterra backward stochastic integral equations

“…It is of interest to study when the solution components Z(t, s), K(t, s, ζ) are smooth (C 1 ) with respect to t. Such smoothness properties are important in the study of optimal control (see e.g. [3]). It is also important in the numerical solutions (see e.g.…”

A Hida–Malliavin white noise calculus approach to optimal control

“…These statements are made more precise in the following brief review, where we recall the basic definition and properties of Hida-Malliavin calculus for Lévy processes. The summary is partly based on Agram and Øksendal [2] and Agram et al [3], [4]. General references for this presentation are Aase et al [1], Benth [6], Lindstrøm et al [9], and the books Hida et al [8] and Di Nunno et al [7].…”

A unified approach to well-posedness of type-I backward stochastic Volterra integral equations

“…Kromer and Overbeck [35] also studied the question of dynamic capital allocations via BSVIEs. Wang and Shi [60] dealt with a risk minimisation problem by means of the maximum principle for FBSVIEs, while the optimal control of SVIEs and BSVIEs via the maximum principle has been studied in Chen and Yong [15], Wang [59], Agram, Øksendal, and Yakhlef [3,4], Shi, Wang, and Yong [49], Shi, Wen, and Xiong [50], see also Wei and Xiao [67] for the case with state constraints.…”