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.
Nacira AGRAM
,
Saloua LABED
,
Bernt ØKSENDAL
,
Samia YAKHLEF
. SINGULAR CONTROL OF STOCHASTIC VOLTERRA INTEGRAL EQUATIONS[J]. Acta mathematica scientia, Series B, 2022
, 42(3)
: 1003
-1017
.
DOI: 10.1007/s10473-022-0311-9
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