Estimation of change point for switching fractional diffusion processes

We study the asymptotic distribution of the maximum likelihood estimator (MLE) for the change point for fractional diffusion processes as the noise intensity tends to zero. It was shown that the rate of convergence here is higher than the rate of convergence of the distribution of the MLE in classical parametric models dealing with independent identically distributed observations with finite and positive Fisher information.     

Stochastic integration for tempered fractional Brownian motion

Tempered fractional Brownian motion is obtained when the power law kernel in the moving average representation of a fractional Brownian motion is multiplied by an exponential tempering factor. This paper develops the theory of stochastic integrals for tempered fractional Brownian motion. Along the way, we develop some basic results on tempered fractional calculus.     

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.     

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.     

On martingale characterizations for some generalized space fractional Poisson processes

We obtain martingale characterizations for the generalized space fractional Poisson process (GSFPP) and for counting processes with Bern?tein intertimes. These serve as extensions of the Watanabe's characterization for the classical homogenous Poisson process. The corresponding assertion for the space fractional Poisson process (SFPP) is obtained as a particular case of our results.     

Small deviations of weighted fractional processes and average non-linear approximation

We investigate the small deviation problem for weighted fractional Brownian motions in -norm, . Let be a fractional Brownian motion with Hurst index . If , then our main result asserts

provided the weight function satisfies a condition slightly stronger than the -integrability. Thus we extend earlier results for Brownian motion, i.e. , to the fractional case. Our basic tools are entropy estimates for fractional integration operators, a non-linear approximation technique for Gaussian processes as well as sharp entropy estimates for -sums of linear operators defined on a Hilbert space.

     

Conditions for singularity for measures generated by two fractional psuedo-diffusion processes

We derive sufficient conditions under which the probability measures generated by two fractional psuedo-diffusion processes are singular with respect to each other.     

On $L_p$-solution of fractional heat equation driven by fractional Brownian motion

In this paper, we study the fractional stochastic heat equation driven by fractional Brownian motions of the form $$ du(t,x)=\left(-(-\Delta)^{\alpha/2}u(t,x)+f(t,x)\right)dt +\sum\limits^{\infty}_{k=1} g^k(t,x)\delta\beta^k_t $$ with $u(0,x)=u_0$, $t\in[0,T]$ and $x\in\mathbb{R}^d$, where $\beta^k=\{\beta^k_t,t\in[0,T]\},k\geq1$ is a sequence of i.i.d. fractional Brownian motions with the same Hurst index $H>1/2$ and the integral with respect to fractional Brownian motion is Skorohod integral. By adopting the framework given by Krylov, we prove the existence and uniqueness of $L_p$-solution to such equation.     

Synchronization for fractional FitzHugh–Nagumo equations with fractional Brownian motion

This paper is devoted to dynamics of the Caputo-type fractional FitzHugh–Nagumo equations (FHN) driven by fractional Brownian motion (fBm). The existence and uniqueness of mild solution for of the Caputo-type fractional FHN are established, and the exponential synchronization and finite-time synchronization for the stochastic FHN are provided. Finally, the numerical simulation of the synchronization for time-fractional FHN perturbed by fBm is provided; the effects of the order of time fractional derivative and Hurst parameter $H$ on synchronization are also revealed.     

Approximation of Some Gaussian Processes

We study a class of processes which have a moving average representation with respect to a fixed driving martingale, and can be represented as a mixture of semi-martingale processes. When the driving martingale is Gaussian we obtain a numerically efficient approximation scheme and a central limit theorem (a typical process in this class is fractional Brownian motion).     

Some Processes Associated with Fractional Bessel Processes

Let be a d-dimensional fractional Brownian motion with Hurst parameter H and let be the fractional Bessel process. Itôs formula for the fractional Brownian motion leads to the equation . In the Brownian motion case is a Brownian motion. In this paper it is shown that X_t is not an -fractional Brownian motion if H 1/2. We will study some other properties of this stochastic process as well.     

A stochastic maximum principle for processes driven by fractional Brownian motion

We prove a stochastic maximum principle for controlled processes X(t)=X^(u)(t) of the form

dX(t)=b(t,X(t),u(t)) dt+σ(t,X(t),u(t)) dB^(H)(t),

where B^(H)(t) is m-dimensional fractional Brownian motion with Hurst parameter . As an application we solve a problem about minimal variance hedging in an incomplete market driven by fractional Brownian motion.     

Partitions of point processes: Multivariate poisson approximations

This study shows that when a point process is partitioned into certain uniformly sparse subprocesses, then the subprocesses are asymptotically multivariate Poisson or compound Poisson. Bounds are given for the total-variation distance between the subprocesses and their limits. Several partitioning rules are considered including independent, Markovian, and batch assignments of points.     

Local times of stochastic differential equations driven by fractional Brownian motions

In this paper, we study the existence and (Hölder) regularity of local times of stochastic differential equations driven by fractional Brownian motions. In particular, we show that in one dimension and in the rough case $H<1/2$, the Hölder exponent (in $t$) of the local time is $1?H$, where $H$ is the Hurst parameter of the driving fractional Brownian motion.     

Some Results on Fractional Brownian Sheets and Their Local Times

Let B0^H = {B0^H（t）,t ∈ R＋^N） be a real-valued fractional Brownian sheet. Define the （N,d）- Gaussian random field B^H by
B^H（t） = （B1^H（t）,...,Bd^H（t）） t ∈ R＋^N, where B1^H, ..., Bd^H are independent copies of B0^H. The existence and joint continuity of local times of B^H is proven in some given conditions in [22]. We then study further properties of the local times of B^H, such as the moments of increments of local times, the large increments and the maximum moduli of continuity of local times and as a result, we answer the questions posed in [22].     

Collision local times of two independent fractional Brownian motions

In this paper, the collision local times for two independent fractional Brownian motions are considered as generalized white noise functionals. Moreover, the collision local times exist in L ² under mild conditions and chaos expansions are also given.     

Packing measure of the sample paths of fractional Brownian motion

Let be a fractional Brownian motion of index in If , then there exists a positive finite constant such that with probability 1,

where and - is the -packing measure of .

     

Space-time properties of a storage process

In a recent paper, Stadje analyzed the space-time properties of some storage processes. We give a short probabilistic proof of these results. This revised version was published online in June 2006 with corrections to the Cover Date.     

Large deviation principle for enhanced Gaussian processes

We study large deviation principles for Gaussian processes lifted to the free nilpotent group of step N. We apply this to a large class of Gaussian processes lifted to geometric rough paths. A large deviation principle for enhanced (fractional) Brownian motion, in Hölder- or modulus topology, appears as special case.