This paper deals with optimal combined singular and regular controls for stochas-tic Volterra integral equations,where the solution Xu,ξ(t)=X(t)is given by X(t)=φ(t)+∫t0b(t,s,X(s),u(s))ds+∫t0σ(t,s,X(s),u(s))dB(s)+∫t0h(t,s)dξ(s).Here dB(s)denotes the Brownian motion It? type differential,ξ denotes the singular control(singular in time t with respect to Lebesgue measure)and u denotes the regular control(absolutely continuous with respect to Lebesgue measure).Such systems may for example be used to model harvesting of populations with memory,where X(t)represents the population density at time t,and the singular control process ξrepresents the harvesting effort rate.The total income from the harvesting is represented by J(u,ξ)=E[∫T0 f0(t,X(t)u(t))dt+∫T0 f1(t,X(t))dξ(t)+g(X(T))for the given functions f0,f1 and g,where T＞0 is a constant denoting the terminal time of the harvesting.Note that it is important to allow the controls to be singular,because in some cases the optimal controls are of this type.Using Hida-Malliavin calculus,we prove sufficient conditions and necessary conditions of optimality of controls.As a consequence,we obtain a new type of backward stochastic Volterra integral equations with singular drift.Finally,to illustrate our results,we apply them to discuss optimal harvesting problems with possibly density dependent prices.

Nacira AGRAM;Saloua LABED;Bernt ?KSENDALt;Samia YAKHLEF

Department of Mathematics,KTH Royal Institute of Technology 10044,Stockholm,Sweden;University Mohamed Khider of Biskra,Algeria;Department of Mathematics,University of Oslo,PO Box 1053 Blindern,N-0316 Oslo,Norway

数学物理学报（英文版）

[1]Nacira AGRAM;Saloua LABED;Bernt ?KSENDALt;Samia YAKHLEF-.SINGULAR CONTROL OF STOCHASTIC VOLTERRA INTEGRAL EQUATIONS)[J].数学物理学报（英文版）,2022(03):1003-1017

SINGULAR,STOCHASTIC,VOLTERRA,INTEGRAL,EQUATIONS,t0b,ds+,t0h,dt+,Hida

CONTROL,OF,This,paper,deals,combined,singular,regular,controls,Volterra,integral,equations,where,solution,Xu,given,by,dB,Here,denotes,Brownian,motion,It,type,differential,respect,Lebesgue,measure,absolutely,continuous,Such,systems,may,example,used,model,harvesting,populations,memory,represents,density,process,effort,total,income,from,represented,T0,f0,f1,+g,functions,constant,denoting,terminal,Note,that,important,allow,because,some,cases,are,this,Using,Malliavin,calculus,we,prove,sufficient,conditions,necessary,optimality,consequence,obtain,new,backward,stochastic,drift,Finally,illustrate,our,results,apply,them,discuss,problems,possibly,dependent,prices

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