Acta mathematica scientia,Series A ›› 2022, Vol. 42 ›› Issue (3): 1003-1017.

• Articles •

### SINGULAR CONTROL OF STOCHASTIC VOLTERRA INTEGRAL EQUATIONS

Nacira AGRAM1, Saloua LABED2, Bernt ØKSENDAL3, Samia YAKHLEF2

1. 1. Department of Mathematics, KTH Royal Institute of Technology, 10044, Stockholm, Sweden;
2. University Mohamed Khider of Biskra, Biskra, Algeria;
3. Department of Mathematics, University of Oslo, PO Box 1053 Blindern, N-0316, Oslo, Norway
• Received:2020-09-18 Revised:2021-01-08 Online:2022-06-26 Published:2022-06-24
• Contact: Bernt ØKSENDAL,E-mail:oksendal@math.uio.no E-mail:oksendal@math.uio.no
• Supported by:
Nacira Agram and Bernt ?ksendal gratefully acknowledge the financial support provided by the Swedish Research Council grant (2020-04697) and the Norwegian Research Council grant (250768/F20), respectively. The order of the authors is purely alphabetical.

Abstract: This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations, where the solution $X^{u,\xi}(t)=X(t)$ is given by $$X(t) =\phi(t)+{ \int_{0}^{t}}b\left( t,s,X(s),u(s)\right){\rm d}s+% { \int_{0}^{t}} \sigma\left( t,s,X(s),u(s)\right) {\rm d}B(s) + { \int_{0}^{t}} h\left( t,s\right) {\rm d}\xi(s).$$ Here ${\rm d}B(s)$ denotes the Brownian motion Itô type differential, $\xi$ 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 $\xi$ represents the harvesting effort rate. The total income from the harvesting is represented by $$J(u,\xi) =\mathbb{E}\bigg[ \int_{0}^{T} f_{0}(t,X(t),u(t)){\rm d}t+ \int_{0}^{T} f_{1}(t,X(t)){\rm d}\xi(t)+g(X(T))\bigg]$$ for the given functions $f_{0},f_{1}$ 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.

CLC Number:

• 60H05
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