Integral representation of generalized grey Brownian motion

ABSTRACT

In this paper, we investigate the representation of a class of non-Gaussian processes, namely generalized grey Brownian motion, in terms of a weighted integral of a stochastic process which is a solution of a certain stochastic differential equation. In particular, the underlying process can be seen as a non-Gaussian extension of the Ornstein–Uhlenbeck process, hence generalizing the representation results of Muravlev, Russian Math. Surveys 66 (2), 2011 as well as Harms and Stefanovits, Stochastic Process. Appl. 129, 2019 to the non-Gaussian case.     

The linear space of generalized Brownian motions with applications

In this paper, we define, motivated by recent works of Chang and Skoug, stochastic integrals for a generalized Brownian motion ( ) and then use it to study the representation problem on the linear space spanned by . We next establish a translation theorem for -functionals of , , and then use this translation to establish an integration by parts formula for -functionals of .

     

Intersections of Brownian motions

This article presents a survey of the theory of the intersections of Brownian motion paths. Among other things, we present a truly elementary proof of a classical theorem of A. Dvoretzky, P. Erdős and S. Kakutani. This proof is motivated by old ideas of P. Lévy that were originally used to investigate the curve of planar Brownian motion.     

Singularity of Subfractional Brownian Motions with Different Hurst Indices

We prove that the probability measures generated by two subfractional Brownian motions with different Hurst indices are singular with respect to each other.     

Solutions to BSDEs driven by both standard and fractional Brownian motions

The backward stochastic differential equations driven by both standard and fractional Brownian motions (or, in short, SFBSDE) are studied. A Wick-Itô stochastic integral for a fractional Brownian motion is adopted. The fractional Itô formula for the standard and fractional Brownian motions is provided. Introducing the concept of the quasi-conditional expectation, we study some its properties. Using the quasi-conditional expectation, we also discuss the existence and uniqueness of solutions to general SFBSDEs, where a fixed point principle is employed. Moreover, solutions to linear SFBSDEs are investigated. Finally, an explicit solution to a class of linear SFBSDEs is found.     

Singularity of Fractional Brownian Motions with Different Hurst Indices

Abstract

We prove that the probability measures generated by two fractional Brownian motions with different Hurst indices are singular with respect to each other.     

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.      

Lower classes for fractional Brownian motion

We characterize the lower classes of fractional Brownian motion by an integral test.Work partially supported by an NSF grant. Equipe d'Analyse, Tour 46, U.A. at C.N.R.S. n^o 754, Université Paris VI, 4 place Jussieu, 75230 Paris Cedex 05, and Department of Mathematics, 231 West 18th Avenue, Columbus, Ohio 43210.     

Two-state free Brownian motions

In a two-state free probability space (A,φ,ψ), we define an algebraic two-state free Brownian motion to be a process with two-state freely independent increments whose two-state free cumulant generating function Rφ,ψ(z) is quadratic. Note that a priori, the distribution of the process with respect to the second state ψ is arbitrary. We show, however, that if A is a von Neumann algebra, the states φ, ψ are normal, and φ is faithful, then there is only a one-parameter family of such processes. Moreover, with the exception of the actual free Brownian motion (corresponding to φ=ψ), these processes only exist for finite time.     

Brownian Motion on the Figure Eight

In an interval containing the origin we study a Brownian motion which returns to zero as soon as it reaches the boundary. We determine explicitly its transition probability, prove it is ergodic and calculate the decay rate to equilibrium. It is shown that the process solves the martingale problem for certain asymmetric boundary conditions and can be regarded as a diffusion on an eight shaped domain. In the case the origin is situated at a rationally commensurable distance from the two endpoints of the interval we give the complete characterization of the possibility of collapse of distinct paths.     

More on maximal inequalities for sub-fractional Brownian motion

Abstract

We derive some maximal inequalities for the sub-fractional Brownian motion using comparison theorems for Gaussian processes.     

A trivariate law for certain processes related to perturbed Brownian motions

D. Williams' path decomposition and Pitman's representation theorem for BES(3) are expressions of some deep relations between reflecting Brownian motion and the 3-dimensional Bessel process.In [Ph. Carmona et al., Stochastic Process. Appl. 7 (1999) 323–333], we presented an attempt to relate better reflecting Brownian motion and the 2-dimensional Bessel process, using space and time changes related to the Ray–Knight theorems on local times, in the manner of Jeulin [Lect. Notes Math., vol. 1118, Springer, Berlin, 1985] and Biane–Yor [Bull. Sci. Math. 2ème Sér. 111 (1987) 23–101].Here, we characterize the law of a triplet linked to the perturbed Brownian motion which naturally arises in [Ph. Carmona et al., Stochastic Proc. Appl. 7 (1999) 323–333], and we point out its relations with Bessel processes of several dimensions.The results provide some new understanding of the generalizations of Lévy's arc sine law for perturbed Brownian motions previously obtained by the second author.     

Fractional Brownian motion and packing dimension

LetX(t) (tR ^N ) be a fractional Brownian motion of index inR ^d . For any compact setER ^N , we compute the packing dimension ofX(E).Partially supported by an NSF grant.     

On Fractional Brownian Processes

We use Liouville spaces in order to prove the existence of some different fractional -Brownian motion ( 0 < 1 ), or fractional ( , )-Brownian sheets. There are also applications to the Wiener stochastic integral with respect to these -Brownian.     

Reinsurance control in a model with liabilities of the fractional Brownian motion type

We propose a model for reinsurance control for an insurance firm in the case where the liabilities are driven by fractional Brownian motion, a stochastic process exhibiting long-range dependence. The problem is transformed to a nonlinear programming problem, the solution of which provides the optimal reinsurance policy. The effect of various parameters of the model, such as the safety loading of the reinsurer and the insurer, the Hurst parameter, etc. on the optimal reinsurance program is studied in some detail. Copyright © 2007 John Wiley & Sons, Ltd.     

Stochastic delay fractional evolution equations driven by fractional Brownian motion

In this paper, we consider a class of stochastic delay fractional evolution equations driven by fractional Brownian motion in a Hilbert space. Sufficient conditions for the existence and uniqueness of mild solutions are obtained. An application to the stochastic fractional heat equation is presented to illustrate the theory. Copyright © 2014 John Wiley & Sons, Ltd.     

Markov-modulated Brownian motions perturbed by catastrophes

ABSTRACT

We consider a one-sided Markov-modulated Brownian motion perturbed by catastrophes that occur at some rates depending on the modulating process. When a catastrophe occurs, the level drops to zero for a random recovery period. Then the process evolves normally until the next catastrophe. We use a semi-regenerative approach to obtain the stationary distribution of this perturbed MMBM. Next, we determine the stationary distribution of two extensions: we consider the case of a temporary change of regime after each recovery period and the case where the catastrophes can only happen above a fixed threshold. We provide some simple numerical illustrations.     

A note on large-buffer asymptotics for generalized processor sharing with Gaussian inputs

In a previous study by Dębicki and van Uitert (Queueing Syst. 54, 111–120, 2006) logarithmic large-buffer asymptotics were derived for a two-class generalized processor sharing system with Gaussian inputs, for three of the four possible scenarios. In this note we show how the large-buffer asymptotics for the remaining fourth regime follow from a recently derived result for tandem systems. We also provide a heuristic interpretation of the result. M.M. is also affiliated to CWI, P.O. Box 94079, 1090 GB Amsterdam, the Netherlands, and EURANDOM, Eindhoven, the Netherlands.     

Operators associated with a stochastic differential equation driven by fractional Brownian motions

In this paper, by using a Taylor type development, we show how it is possible to associate differential operators with stochastic differential equations driven by fractional Brownian motions. As an application, we deduce that invariant measures for such SDE’s must satisfy an infinite dimensional system of partial differential equations.