Constantin Tudor

Some properties of the sub-fractional Brownian motion

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.

Periodic and almost periodic solutions for semilinear stochastic equations

We discuss the problem of the existence of periodic and almost periodic solutions in distribution of semilinear stochastic equations on a separable Hilbert space. Under a dissipativity condition we prove that the translation of the mean square bounded solution is periodic or almost periodic. Similar results hold in the affine case under mean square stability of the linear part of the equation.

Almost periodic solutions of affine stochastic evolution equations

In this paper the problem of the existence of almost periodic in distribution solutions of almost periodic affine Ito equations of evolution type is studied. Under the hypotheses that the linear part of the equation is exponentially stable in mean square and the relative compactness of the one-dimensional distributions of the unique bounded solution of the equation, it is proved the almost periodicity of the one-dimensional distributions of the solution

Stationary and almost periodic solutions of almost periodic affine stochastic differential equations

We associate in a canonical way to an almost periodic affine SDE a stochastic equation driven by a dynamical system. We prove the existence of a unique stationary solution provided the corresponding linear SDE is hyperbolic. If the linear SDE is stable, the stationary solution is used to obtain a solution which is almost periodic in distribution.

Optimal control for semilinear stochastic evolution equations

Existence, uniqueness, continuous dependence with respect to controls and convergence in the probability of finite differences for controlled semilinear stochastic evolution equations, driven by continuous semimartingales, are considered under Lipschitz and monotone coefficients. The existence of discreteε-optimal feedback controls for an associated optimization problem is proved.

Almost periodic solutions of affine ito equations

We discuss the problem of the existence of almost periodic in distribution solutions of affine stochastic differential equations with almost periodic coefficients. We prove that if the linear part of the affine equation is exponentially stable in mean square then the unique continuous L2 -bounded solution of the affine system has the onedimensional distributions almost periodic. An analogous result is shown for the asymptotic almost periodic case

A CHAOS APPROACH TO THE ANTICIPATING CALCULUS FOR THE POISSON PROCESS

In this paper we use the Poisson-Ito chaos decomposition approach to define a variational derivative operator and its adjoint, which is an anticipating integral (i.e., it agrees with the martingale Poisson-Ito integral with respect to the compensated Poisson process for predictable integrands). Also an integration by parts formula and characterizations of these operators are given.Finally, we prove that, in the case where the basic probability space is the canonical Poisson space, our derivative operator is equal to the Carlen and Pardoux gradient operator that is defined by means of variation of the jump times

Large deviations for a class of chaos expansions

The purpose of this paper is to present a general extended contraction principle for large deviations and apply it to obtain large deviations for random variables having chaos developments of exponential type.

Multiple sub-fractional integrals and some approximations

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.

On stochastic evolution equations driven by continuous semimartingales

Existence and pathwise uniqueness of mild solutions for stochastic evolution equations driven by continuous semimartingales is proved under monotone coefficients. The convergence of finite differen...