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2 Probability and Mathematical Statistics

2 2020

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Weyl Multifractional Ornstein–Uhlenbeck Processes Mixed with A Gamma Distribution

Es-Sebaiy Khalifa, Farah Fatima-Ezzahra, Hilbert Astrid

Probability and Mathematical Statistics

2020

Vol. 40, Fasc. 2

269--295

The aim of this paper is to study the asymptotic behavior of aggregated Weyl multifractional Ornstein–Uhlenbeck processes mixed with Gamma random variables. This allows us to introduce a new class of processes, Gamma-mixed Weyl multifractional Ornstein–Uhlenbeck processes (GWmOU), and study their elementary properties such as Hausdorff dimension, local self-similarity and short-range dependence. We also prove that these processes approach the multifractional Brownian motion.

Mean-field optimal control problem of SDDES driven by fractional Brownian motion

Douissi Soukaina, Agram Nacira, Hilbert Astrid

Probability and Mathematical Statistics

2020

Vol. 40, Fasc. 1

139--158

We consider a mean-field optimal control problem for stochastic differential equations with delay driven by fractional Brownian motion with Hurst parameter greater than 1/2. Stochastic optimal control problems driven by fractional Brownian motion cannot be studied using classical methods, because the fractional Brownian motion is neither a Markov process nor a semi-martingale. However, using the fractional white noise calculus combined with some special tools related to differentiation for functions of measures, we establish necessary and sufficient stochastic maximum principles. To illustrate our study, we consider two applications: we solve a problem of optimal consumption from a cash flow with delay and a linear-quadratic (LQ) problem with delay.