We consider the stochastic differential equations satisfied by the engenvalues of the operator Wishart and Laguerre processes. They are governed by a system of diffusions governed by Brownian motions that are not independent. It is shown that their traces are Bessel processes if and only if the corresponding operator processes are standard.

In this paper, we derive explicit bounds for the Kolmogorov distance in the CLT and we prove the almost sure CLT for the quadratic variation of the sub-fractional Brownian motion. We use recent results on the Stein method combined with the Malliavin calculus and an almost sure CLT for multiple integrals.

We prove the mean square convergence of multiple Riemann-Stieltjes integrals based on the integral process defined by the Gamma-mixed Ornstein-Uhlenbeck process to multiple fractional Stratonovich integrals.The integrands belong to the subclass of the Schwartz space S(R n ) of rapidly decreasing functions whose fractional integrals remain rapidly decreasing.In particular the result applies for integrands in the Lizorkin space, i.e., the subspace of S(R n ) which is orthogonal to all polynomials.

Almost sure and $L^k$-convergence of the traces of Laguerre processes to the family of dilations of the standard free Poisson distribution are established. We also prove that the fluctuations around the limiting process, converge weakly to a continuous centered Gaussian process. The almost sure convergence on compact time intervals of the largest and smallest eigenvalues processes is also established

The domain Λk,Tsf of the Wiener integral with respect to a sub-fractional Brownian motion (Stk)t∈[0,T], k∈(−12,12), k≠0, is characterized. The set Λk,Tsf is a Hilbert space which contains the class of elementary functions as a dense subset. If k∈(−12,0), any element of Λk,Tsf is a function and if k∈(0,12), the domain Λk,Tsf is a space of distributions.

We consider Wong-Zakai type approximations for a class of Ito-Volterra equations related to the fractional Brownian motion. The quadratic mean convergence, uniformly on compact time intervals, of the approximations to the solution of all Ito-Volterra equation with a modified drift is obtained.

Double Stratonovich integrals with respect to the odd part and even part of the fractional Brownian motion are constructed. The first and the second moments of such integrals are explicitly identified. As application of double Stratonovich integrals a strong law of large numbers for efBm and ofBm is derived. Riemann-Stieltjes integral approximations to double Stratonovich fractional integrals are also considered. The strong convergence (almost surely and mean square) is obtained for approximations based on explicit series expansions of the fractional Brownian processes. The weak convergence is derived for approximations by processes with absolutely continuous paths which converge weakly to the considered fractional Brownian processes. The above-mentioned convergences are obtained for deterministic integrands which are given by bimeasures.

We obtain a decomposition in distribution of the sub-fractional Brownian motion as a sum of independent fractional Brownian motion and a centered Gaussian process with absolutely continuous paths. Applications to the domain of the Wiener integral and the variation and strong variation of sub-fractional Brownian motion are given.

We characterize the domain of the Wiener integral with respect to a subfractional Brownian motion {SH(t)}t≥0,H∈(0,1),H≠12. The domain is a Hilbert space which contains the class of elementary functions as a dense subset. If 0<H<12, any element of the domain is a function and if 12<H<1, the domain is a space of distributions. The RKHS of SH is also determined.

We study multiple fractional integrals with respect to the even fractional Brownian motion (also called sub-fractional Brownian motion). The multiple integrals are introduced by using a representation formula for the even fractional Brownian motion as a Wiener integral with respect to a Brownian motion defined on the same probability space and a transfer principle. Then, Riemann-Stieltjes integral approximations to multiple Stratonovich fractional integrals are also considered. For two standard approximations (Wong-Zakai and mollifier approximations) and continuous integrands, the mean square convergence in the uniform norm of these approximations to the multiple Stratonovich sub-fractional integral is shown.

A linear unbiased and square mean optimal estimation is obtained for the mild solution process of a stochastic evolution equation with an infinite-dimensional fractional Brownian motion as noise and the noise in the observation process is a finite-dimensional Brownian motion. An innovation process is introduced and the estimation is obtained as a solution of a stochastic differential equation with a finite-dimensional noise. By using an approach based on the equivalence with a deterministic control problem, the estimation for the Fourier coefficients of the signal process is also determined.

In this paper we use the chaos expansion method to define a derivative operator and the corresponding Skorokhod integral for Levy processes with no drift and Brownian part. The main tool in establishing the derivation property is the product formula for two multiple integrals.

We study multiple Riemann-Stieltjes integral approximations to multiple Stratonovich fractional integrals. Two standard approximations (Wong-Zakai and Mollifier approximations) are considered and we show the convergence in the mean square sense and uniformly on compact time intervals of these approximations to the multiple Stratonovich fractional integral.

AbstractWe study the convergence in probability of the normalized q-variation of the multiple fractional multiparameter integral processes $$\begin{gathered} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _r = (t_1 ,...,t_r ) \to I_r^H (f_r )_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _r } : = \int_{[0,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _r ]} {f_r (s_1 ,...,s_r )dB_{s_1 }^H ...dB_{s_r }^H } , \hfill \\ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _r = (t_1 ,...,t_r ) \to I_r^{H, - } (f_r )_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _r } : = \int_{[0,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _r ]} {f_r (s_1 ,...,s_r )dS_{s_1 }^H ...dS_{s_r }^H } , \hfill \\ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _2 = (t_1 ,t_2 ) \to I_r^H (g)_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _2 } : = \int_{[0,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _2 ]} {g(s_1 ,s_2 )dB_{s_1 }^{H,1} dB_{s_2 }^{H,2} } , \hfill \\ \end{gathered} $$ where fr, g are continuous deterministic functions, BH (resp. SH) is a fractional (resp. a sub-fractional) Brownian motion with Hurst parameter H > 1/2 and BH,1, BH,1 are independent fractional Brownian motions with Hurst parameter H.

We study several properties of the sub-fractional Brownian motion (fBm) introduced by Bojdecki et al. related to those of the fBm. This process is a self-similar Gaussian process depending on a parameter H ∈ (0, 2) with non stationary increments and is a generalization of the Brownian motion (Bm). The strong variation of the indefinite stochastic integral with respect to sub-fBm is also discussed.

In this paper we study functional asymptotic behavior of p-trace processes of n × n Hermitian matrix valued Brownian motions, when n goes to infinity.For each p ≥ 1 we establish uniform a.s. and L q laws of large numbers and study the a.s.convergence of the supremum (respectively infimum) over a compact interval of the largest (respectively smallest) eigenvalue process.We also prove that the fluctuations around the limiting process, converge weakly to a one-dimensional centered Gaussian process Zp, given as a Wiener integral with a deterministic Volterra kernel.This process depends on Z p-1 , ..., Z 1 and a Gaussian martingale of independent interest whose increasing process is explicitly derived.Our approach is based on stochastic analysis and semimartingales tools.

For an additive process (X-t) t >= 0 with values in the dual of a nuclear Frechet space Phi' = (infinity)(p=0) Phi(-p) and for each finite timeT > 0, the existence of an equivalent additive process ((x) over tilde (t)) 0 <= t <= T which takes values in a Hilbert space Phi(-pT) is shown. The additive process takes values in a cone C' subset of Phi' if and only if it has a special Levy-Khintchine representation and in this case for eachT > 0 there exists a pathwise version ((y) over tilde (t) )(0 <= t <= T) in some Hilbert space Phi(-qT).

In the paper we compute the explicit form of the fractional chaos decomposition of the solution of a fractional stochastic bilinear equation with the drift in the fractional chaos of order one and initial condition in a finite fractional chaos. The large deviations principle is also obtained for the one-dimensional distributions of the solution of the equation perturbed by a small noise.

We give the explicit chaos representation of the solution of a class of one dimensional stochastic bilinear heat equations with drift in the first haos and driven by a space-time white noise. The solution is a stochastic process with finite moments of all orders. The L2-Lyapunov exponents of the solution are also estimated.