Saloua Labed | Searcharxiv

arXiv2024

Fokker-Planck equations for McKean-Vlasov SDEs driven by fractional Brownian motion

This paper investigates the probability distribution of solutions to McKean--Vlasov stochastic differential equations driven by fractional Brownian motion with Hurst parameter H>1/2. Our main contribution is the derivation of the associated Fokker--Planck equation, which characterizes the time evolution of the law of the solution in a suitable distributional framework. Under mild assumptions, we show that the law-valued process is absolutely continuous in time and provide an explicit weak formulation of the corresponding fractional McKean--Vlasov Fokker--Planck equation. In the case where the law admits a density, we obtain a more explicit partial differential equation with time-dependent diffusion coefficients induced by the fractional noise. We further establish a fractional Feynman--Kac representation, linking the forward Fokker--Planck equation with a backward Kolmogorov equation for functionals of the solution process. This result extends the classical Feynman--Kac framework to mean--field dynamics driven by fractional Brownian motion. To illustrate the theory, we analyze several explicit examples, including the law of fractional Brownian motion itself and linear McKean--Vlasov fractional SDEs. These examples highlight how fractional noise and mean--field interactions jointly affect the probabilistic and analytic structure of the system.

math.PR↗

arXiv2019

Singular optimal control of stochastic Volterra integral equations

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(t,s,X(s),u(s)) ds+\int_{0}^{t}\sigma(t,s,X(s),u(s))dB(s) +\int _{0}^{t}\int_{0}^{t}h(t,s) d\xi(s). Here dB(s) denotes the Brownian motion It\^o type differential and \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) =E[\int _{0}^{T}\int_{0}^{T} f_{0}(t,X(t),u(t))dt+\int _{0}^{T}\int_{0}^{T} f_{1}(t,X(t))d\xi(t)+g(X(T))], for 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.

math.OC↗