5,561 research outputs found
On the maximum principle for optimal control problems of stochastic Volterra integral equations with delay
In this paper, we prove both necessary and sufficient maximum principles for
infinite horizon discounted control problems of stochastic Volterra integral
equations with finite delay and a convex control domain. The corresponding
adjoint equation is a novel class of infinite horizon anticipated backward
stochastic Volterra integral equations. Our results can be applied to
discounted control problems of stochastic delay differential equations and
fractional stochastic delay differential equations. As an example, we consider
a stochastic linear-quadratic regulator problem for a delayed fractional
system. Based on the maximum principle, we prove the existence and uniqueness
of the optimal control for this concrete example and obtain a new type of
explicit Gaussian state-feedback representation formula for the optimal
control.Comment: 28 page
Long time behavior of a mean-field model of interacting neurons
We study the long time behavior of the solution to some McKean-Vlasov
stochastic differential equation (SDE) driven by a Poisson process. In
neuroscience, this SDE models the asymptotic dynamic of the membrane potential
of a spiking neuron in a large network. We prove that for a small enough
interaction parameter, any solution converges to the unique (in this case)
invariant measure. To this aim, we first obtain global bounds on the jump rate
and derive a Volterra type integral equation satisfied by this rate. We then
replace temporary the interaction part of the equation by a deterministic
external quantity (we call it the external current). For constant current, we
obtain the convergence to the invariant measure. Using a perturbation method,
we extend this result to more general external currents. Finally, we prove the
result for the non-linear McKean-Vlasov equation
Malliavin calculus and optimal control of stochastic Volterra equations
Solutions of stochastic Volterra (integral) equations are not Markov
processes, and therefore classical methods, like dynamic programming, cannot be
used to study optimal control problems for such equations. However, we show
that by using {\em Malliavin calculus} it is possible to formulate a modified
functional type of {\em maximum principle} suitable for such systems. This
principle also applies to situations where the controller has only partial
information available to base her decisions upon. We present both a sufficient
and a necessary maximum principle of this type, and then we use the results to
study some specific examples. In particular, we solve an optimal portfolio
problem in a financial market model with memory.Comment: 18 page
New approach to optimal control of stochastic Volterra integral equations
We study optimal control of stochastic Volterra integral equations (SVIE)
with jumps by using Hida-Malliavin calculus.
- We give conditions under which there exists unique solutions of such
equations.
- Then we prove both a sufficient maximum principle (a verification theorem)
and a necessary maximum principle via Hida-Malliavin calculus.
- As an application we solve a problem of optimal consumption from a cash
flow modelled by an SVIE
- …