Volume 24, Issue 1

Let B be a bifractional Brownian motion with parameters H ∈ (0,1) and K ∈ (0,1]. For any n ≥ 1, set Zn=∑i=0n-1[n2HK(B(i+1)/n-Bi/n)2-𝔼((Bi+1-Bi)2)]${Z_n = \sum _{i=0 }^{n-1} [n^{2HK}(B_{(i+1)/n}-B_{i/n})^2-\mathbb {E} ((B_{i+1}-B_{i})^2 ) ]}$. We use Malliavin calculus and the so-called Stein's method on Wiener chaos introduced by Nourdin and Peccati [11] to derive, in the case when 0 < H K ≤ 3/4, Berry–Esseen-type bounds for the Kolmogorov distance between the law of the correct renormalization V n of Z n and the standard normal law. Finally, we study almost sure central limit theorems for the sequence V n .

We introduce a new notion of quantum white noise (QWN) first-order Wick differential operators with variable coefficients on the topological nuclear algebra ℒ(ℱ θ ( N '), ℱ θ * ( N ')) endowed with the Wick product and we give their chaos expansions in terms of the QWN-derivatives { D z - , D z + : z ∈ N }. We also study a generalized transport equation associated to a class of QWN-first-order Wick differential operators.

We study a version of the functional Hodrick–Prescott filter in the case when the associated operator is not necessarily compact but merely closed and densely defined with closed range. We show that the associated optimal smoothing operator preserves the structure obtained in the compact case when the underlying distribution of the data is Gaussian.

In this paper we apply the REFORM method for the deduction of the system of canonical equations K 62 , K 63 , K 64 and K 65 for normalized spectral functions of the matrices A + B ( U +γ H ) C .

In this paper, we present an SIRS (Susceptible, Infective, Recovered, Susceptible) epidemic model with a saturated incidence rate and disease causing death in a population of varying size. We define a parameter ℜ 0 * from which a study of stability is made. In the deterministic case, we discuss the existence of an endemic equilibria and prove the global stability of the disease-free equilibrium. By introducing a perturbation in the contact rate through a white noise, we consider a stochastic version. We prove the existence of a global positive solution which lives in a certain domain. Thereafter, we prove the global stability n th moment of the system as soon as the intensity of the white noise is below a certain threshold. Finally, we perform some numerical simulations to compare the dynamic behaviors of the deterministic system and the stochastic system.