Brownian and fractional Brownian stochastic currents via Malliavin calculus

By using Malliavin calculus and multiple Wiener-Itô integrals, we study the existence and the regularity of stochastic currents defined as Skorohod (divergence) integrals with respect to the Brownian motion and to the fractional Brownian motion. We consider also the multidimensional multiparameter case and we compare the regularity of the current as a distribution in negative Sobolev spaces with its regularity in the Watanabe spaces.     

Approximation theorems for set-valued stochastic integrals

The article is devoted to new properties of Aumann, Lebesgue, and Itô set-valued stochastic integrals considered in papers [1 Kisielewicz, M. (2014). Properties of generalized set-valued stochastic integrals. Discuss. Math. (DICO) 34:131–147. [Google Scholar],2 Kisielewicz, M., Michta, M. (2017). Integrably bounded set-valued stochastic integrals. J. Math. Anal. Appl. 449:1893–1910.[Crossref], [Web of Science ®] , [Google Scholar]]. In particular, it contains some approximation theorems for Aumann and Itô set-valued stochastic integrals. Hence, in particular, it follows that Aumann and Lebesgue set-valued stochastic integrals cover a.s., both for measurable and IF-nonanticipative integrably bounded set-valued stochastic processes.     

Some linear fractional stochastic equations

Using the multiple stochastic integrals, we prove an existence and uniqueness result for a linear stochastic equation driven by the fractional Brownian motion with any Hurst parameter. We study both the one- and two-parameter cases. When the drift is zero, we show that in the one-parameter case the solution is an exponential—thus positive—function while in the two-parameter setting the solution is negative on a non-negligible set.     

Fractional Bilinear Stochastic Equations with the Drift in the First Fractional Chaos

Abstract

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.     

Frechet-valued martingales and stochastic integrals

Stochastic processes with values in a separable Frechet space whose a itinuous linear functional are real-valued square integrable martingales are investigated. The coordinate measures on the Fréchet space are obtained from cylinder set measures on a Hilbert space that is dense in the Fréchet space. Real-valued stochastic integrals are defined from the Fréchet-valued martingales using integrands from the topological dual of the aforementioned Hilbert space. An increasing process with values in the self adjoint operators on the Hilbert space plays a fundamental role in the definition of stochastic integrals. For Banach-valued Brownian motion the change of variables formula of K. Itô is generalized. A converse to the construction of the measures on the Fréchet space from cylinder set measures on a Hilbert space is also obtained.     

A change of variable formula for the 2D fractional Brownian motion of Hurst index bigger or equal to 1/4

We prove a change of variable formula for the 2D fractional Brownian motion of index H bigger or equal to 1/4. For H strictly bigger than 1/4, our formula coincides with that obtained by using the rough paths theory. For H=1/4 (the more interesting case), there is an additional term that is a classical Wiener integral against an independent standard Brownian motion.     

Berry-Esséen Bounds for Long Memory Moving Averages via Stein's Method and Malliavin Calculus

Using the Stein method on Wiener chaos introduced in Nourdin and Peccati [10], we prove Berry-Esséen bounds for long memory moving averages.     

Maximal Inequalities for Fractional Brownian Motion: An Overview

We give an overview of some maximal inequalities and limit theorems for the tail probabilities for the supremum of a fractional Brownian motion.     

Weak convergence of the scaled median of independent Brownian motions

We consider the median of n independent Brownian motions, denoted by M _n (t), and show that $\sqrt{n}\,M_nWe consider the median of n independent Brownian motions, denoted by M _n (t), and show that converges weakly to a centered Gaussian process. The chief difficulty is establishing tightness, which is proved through direct estimates on the increments of the median process. An explicit formula is given for the covariance function of the limit process. The limit process is also shown to be H?lder continuous with exponent γ for all γ < 1/4.      

Weak convergence to the fractional Brownian sheet in Besov spaces

In this paper we study the problem of the approximation in law of the fractional Brownian sheet in the topology of the anisotropic Besov spaces. We prove the convergence in law of two families of processes to the fractional Brownian sheet: the first family is constructed from a Poisson procces in the plane and the second family is defined by the partial sums of two sequences of real independent fractional brownian motions.     

A Fractional Donsker Theorem

We prove a Donsker-type approximation of the fractional Brownian motion which extends a result by Sottinen for the case H > 1/2 to the full range of Hurst parameters H ∈ (0, 1). The convergence is established by a Donsker-type theorem for Volterra Gaussian processes. The approximation is applied to weak convergence of fractional Wiener integrals.     

Weak convergence to the fractional Brownian sheet using martingale differences

We establish an invariance principle for the fractional Brownian sheet, starting from discrete random fields constructed from two-parameter strong martingales. This is an approximation in law of the fractional Brownian sheet in Skorohord space in the plane.     

Various types of stochastic integrals with respect to fractional Brownian sheet and their applications

In this paper, we develop a stochastic calculus related to a fractional Brownian sheet as in the case of the standard Brownian sheet. Let be a fractional Brownian sheet with Hurst parameters H=(H₁,H₂), and (²[0,1],B(²[0,1]),μ) a measure space. By using the techniques of stochastic calculus of variations, we introduce stochastic line integrals along all sufficiently smooth curves γ in ²[0,1], and four types of stochastic surface integrals: , i=1,2, , , , . As an application of these stochastic integrals, we prove an Itô formula for fractional Brownian sheet with Hurst parameters H₁,H₂∈(1/4,1). Our proof is based on the repeated applications of Itô formula for one-parameter Gaussian process.     

Stochastic elastic equation driven by fractional Brownian motion

In this paper, we consider the stochastic elastic equation driven by a cylindrical fractional Brownian motion. The regularities of the solution to the linear stochastic problem corresponding to the stochastic elastic equation are proved. Then, we obtain the existence of the solution using the Picard iteration.     

Weak solutions for stochastic differential equations with additive fractional noise

In this paper, we show the existence of a weak solution for a stochastic differential equation driven by an additive fractional Brownian motion with Hurst parameter , and a discontinuous drift. The proof of this result is based on the Girsanov theorem for the fractional Brownian motion.     

Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations

Stock exchange dynamics of fractional order are usually modeled as a non-random exponential growth process driven by a fractional Brownian motion. Here we propose to use rather a non-random fractional growth driven by a (standard) Brownian motion. The key is the Taylor’s series of fractional order where E_α(.) denotes the Mittag-Leffler function, and is the so-called modified Riemann-Liouville fractional derivative which we introduced recently to remove the effects of the non-zero initial value of the function under consideration. Various models of fractional dynamics for stock exchange are proposed, and their solutions are obtained. Mainly, the Itô’s lemma of fractional order is illustrated in the special case of a fractional growth with white noise. Prospects for the Merton’s optimal portfolio are outlined, the path probability density of fractional stock exchange dynamics is obtained, and two fractional Black-Scholes equations are derived. This approach avoids using fractional Brownian motion and thus is of some help to circumvent the mathematical difficulties so involved.     

A time fractional functional differential equation driven by the fractional Brownian motion

Let $B^H$ be a fractional Brownian motion with Hurst index $H>\frac12$. In this paper, we prove the global existence and uniqueness of the equation $$ \begin{cases} ^CD_t^{\gamma}x(t)=f(x_t)+G(x_t)\frac{d}{dt}B^H(t),\ \ \ \ &t\in(0,T], \x(t)=\eta(t), \ \ \ \ \ &t\in[-r,0], \end{cases} $$ where $\max\{H,2-2H\}<\gamma<1$, $^CD_t^{\gamma}$ is the Caputo derivative, and $x_t\in \mathcal{C}_r=\mathcal{C}([-r,0],\mathbb{R})$ with $x_t(u)=x(t+u),u\in[-r,0]$. We also study the dependence of the solution on the initial condition.