Infinite horizon stochastic optimal control for Volterra equations with completely monotone kernels

The aim of the paper is to study an optimal control problem on infinite horizon for an infinite dimensional integro-differential equation with completely monotone kernels, where we assume that the noise enters the system when we introduce a control. We start by reformulating the state equation into a semilinear evolution equation which can be treated by semigroup methods. The application to optimal control provides other interesting results and requires a precise description of the properties of the generated semigroup. The main tools consist in studying the differentiability of the forward\u2013backward system with infinite horizon corresponding with the reformulated problem and the proof of existence and uniqueness of mild solutions to the corresponding Hamilton Jacobi Bellman (HJB) equation.The aim of the paper is to study an optimal control problem on infinite horizon for an infinite dimensional integro-differential equation with completely monotone kernels, where we assume that the noise enters the system when we introduce a control. We start by reformulating the state equation into a semilinear evolution equation which can be treated by semigroup methods. The application to optimal control provides other interesting results and requires a precise description of the properties of the generated semigroup. The main tools consist in studying the differentiability of the forward\u2013backward system with infinite horizon corresponding with the reformulated problem and the proof of existence and uniqueness of mild solutions to the corresponding Hamilton Jacobi Bellman (HJB) equation

Small noise asymptotic expansions for stochastic PDE's driven by dissipative nonlinearity and L\'evy noise

We study a reaction-diffusion evolution equation perturbed by a space-time L\'evy noise. The associated Kolmogorov operator is the sum of the infinitesimal generator of a $C_0$-semigroup of strictly negative type acting in a Hilbert space and a nonlinear term which has at most polynomial growth, is non necessarily Lipschitz and is such that the whole system is dissipative. The corresponding It\^o stochastic equation describes a process on a Hilbert space with dissi- pative nonlinear, non globally Lipschitz drift and a L\'evy noise. Under smoothness assumptions on the non-linearity, asymptotics to all orders in a small parameter in front of the noise are given, with detailed estimates on the remainders. Applications to nonlinear SPDEs with a linear term in the drift given by a Laplacian in a bounded domain are included. As a particular case we provide the small noise asymptotic expansions for the SPDE equations of FitzHugh Nagumo type in neurobiology with external impulsive noise.Comment: 29 page

Subgame-perfect equilibrium strategies for time-inconsistent recursive stochastic control problems

We study time-inconsistent recursive stochastic control problems. Since for this class of problems classical optimal controls may fail to exist or to be relevant in practice, we focus on subgame-perfect equilibrium policies. The approach followed in our work relies on the stochastic maximum principle: we adapt the classical spike variation technique to obtain a characterization of equilibrium strategies in terms of a generalized second-order Hamiltonian function defined through a pair of backward stochastic differential equations. The theoretical results are applied in the financial field to finite horizon investment-consumption policies with non-exponential actualization.Comment: arXiv admin note: substantial text overlap with arXiv:2105.0147

Equilibrium strategies in time-inconsistent stochastic control problems with constraints: necessary conditions

This paper is concerned with a time-inconsistent recursive stochastic control problems where the forward state process is constrained through an additional recursive utility system. By adapting the Ekeland variational principle, necessary conditions for equilibrium strategies are presented concerning a second-order Hamiltonian function defined by pairs of backward stochastic differential equations. At last, we consider a finite horizon state constrained investment-consumption problem with non-exponential actualisation as an example to show the application in finance. The class of constraints investigated here includes the possibility of imposing a risk bound on the terminal value of the wealth process

Analysis of the stochastic FitzHugh-Nagumo system

In this paper we study a system of stochastic differential equations with dissipative nonlinearity which arise in certain neurobiology models. Besides proving existence, uniqueness and continuous dependence on the initial datum, we shall be mainly concerned with the asymptotic behaviour of the solution. We prove the existence of an invariant ergodic measure ν associated with the transition semigroup Pt; further, we identify its infinitesimal generator in the space L2(H ; ν)