New approach to optimal control of stochastic Volterra integral equations

Abstract

This paper deals with optimal combined singular and regular control for stochastic Volterra integral equations, where the solution X u,ξ (t) = X(t) is given byHere ξ denotes the singular control and u denotes the regular control.Unless otherwise stated, b a h(s)dξ(s) means [a,b] h(s)dξ(s). Such systems may for example be used to model for harvesting of populations with memory, where X(t) represents the population density at time t, and the singular control process ξ represents the harvesting effort rate. The total income from the harvesting is represented byfor given functions f 0 , f 1 and g, where T > 0 is a constant denoting the terminal time of the harvesting.

Mean-field stochastic control with elephant memory in finite and infinite time horizon

“…We want to find a solution (p 0 ,q 0 ) of this infinite horizon BSDE such that the transversality condition holds, i.e. lim which is a stochastic Volterra equation of the type studied in [4] and [9].…”

“…See e.g. Agram et al [9]. Here we are interested in stochastic differential equations (SDEs) where the coefficients of the system depend upon the whole past.…”

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.…”