Asymptotic Behavior of Weighted Power Variations of Fractional Brownian Motion in Brownian Time

We study the asymptotic behavior of weighted power variations of fractional Brownian motion in Brownian time \(Z_t:= X_{Y_t},t \geqslant 0\), where X is a fractional Brownian motion and Y is an independent Brownian motion.     

Viability for differential equations driven by fractional Brownian motion

In this paper we prove a viability result for multidimensional, time dependent, stochastic differential equations driven by fractional Brownian motion with Hurst parameter , using pathwise approach. The sufficient condition is also an alternative global existence result for the fractional differential equations with restrictions on the state.     

Lower Bounds for the Distribution of Suprema of Brownian Increments and Brownian Motion Normalized by the Corresponding Modulus Functions

The Lévy–Ciesielski construction of Brownian motion is used to determine non-asymptotic estimates for the maximal deviation of increments of a Brownian motion process \((W_{t})_{t\in \left[ 0,T\right] }\) normalized by the global modulus function, for all positive \(\varepsilon \) and \(\delta \). Additionally, uniform results over \(\delta \) are obtained. Using the same method, non-asymptotic estimates for the distribution function for the standard Brownian motion normalized by its local modulus of continuity are obtained. Similar results for the truncated Brownian motion are provided and play a crucial role in establishing the results for the standard Brownian motion case.     

Order preservation for multidimensional stochastic functional differential equations with jumps

Sufficient and necessary conditions are presented for the order preservation of stochastic functional differential equations on ${\mathbb{R}^d}$ with non-Lipschitzian coefficients driven by the Brownian motion and Poisson processes. The sufficiency of the conditions extends and improves some known comparison theorems derived recently for one-dimensional equations and multidimensional equations without delay, and the necessity is new even in these special situations.     

On the Weighted Local Time and the Tanaka Formula for the Multidimensional Fractional Brownian Motion

Abstract

We determine the weighted local time for the multidimensional fractional Brownian motion from the occupation time formula. We also discuss on the Itô and Tanaka formula for the multidimensional fractional Brownian motion. In these formulas the Skorohod integral is applicable if the Hurst parameter of fractional Brownian motion is greater than 1/2. If the Hurst parameter is less than 1/2, then we use the Skorohod type integral introduced by Nualart and Zakai for the stochastic integral and establish the Itô and Tanaka formulas.     

On Gaussian Processes Equivalent in Law to Fractional Brownian Motion

We consider Gaussian processes that are equivalent in law to the fractional Brownian motion and their canonical representations. We prove a Hitsuda type representation theorem for the fractional Brownian motion with Hurst index H1/2. For the case H>1/2 we show that such a representation cannot hold. We also consider briefly the connection between Hitsuda and Girsanov representations. Using the Hitsuda representation we consider a certain special kind of Gaussian stochastic equation with fractional Brownian motion as noise.     

Stochastic Differential Equations Driven by Fractional Brownian Motion and Standard Brownian Motion

Abstract

We prove an existence and uniqueness theorem for solutions of multidimensional, time dependent, stochastic differential equations driven simultaneously by a multidimensional fractional Brownian motion with Hurst parameter H > 1/2 and a multidimensional standard Brownian motion. The proof relies on some a priori estimates, which are obtained using the methods of fractional integration and the classical Itô stochastic calculus. The existence result is based on the Yamada–Watanabe theorem.     

A strong uniform approximation of fractional Brownian motion by means of transport processes

We construct a sequence of processes that converges strongly to fractional Brownian motion uniformly on bounded intervals for any Hurst parameter H$H$, and we derive a rate of convergence, which becomes better when H$H$ approaches 1/2$1/2$. The construction is based on the Mandelbrot–van Ness stochastic integral representation of fractional Brownian motion and on a strong transport process approximation of Brownian motion. The objective of this method is to facilitate simulation.     

Reflected Brownian Motion in a Convex Polyhedral Cone: Tail Estimates for the Stationary Distribution

Consider a multidimensional obliquely reflected Brownian motion in the positive orthant, or, more generally, in a convex polyhedral cone. We find sufficient conditions for existence of a stationary distribution and convergence to this distribution at an exponential rate, as time goes to infinity, complementing the results of Dupuis and Williams (Ann Probab 22(2):680–702, 1994) and Atar et al. (Ann Probab 29(2):979–1000, 2001). We also prove that certain exponential moments for this distribution are finite, thus providing a tail estimate for this distribution. Finally, we apply these results to systems of rank-based competing Brownian particles, introduced in Banner et al. (Ann Appl Probab 15(4):2296–2330, 2005).     

Delay Equations with Non-negativity Constraints Driven by a Hölder Continuous Function of Order \beta \in \left(\frac13,\frac12\right)

In this note we prove an existence and uniqueness result of solution for multidimensional delay differential equations with normal reflection and driven by a Hölder continuous function of order \(\beta \in (\frac13,\frac12)\) . We also obtain a bound for the supremum norm of this solution. As an application, we get these results for stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H \(\in (\frac13,\frac12)\) .     

Stratonovich’s signatures of Brownian motion determine Brownian sample paths

The signature of Brownian motion in $\mathbb R ^{d}$ over a running time interval $[0,T]$ is the collection of all iterated Stratonovich path integrals along the Brownian motion. We show that, in dimension $d\ge 2$ , almost all Brownian motion sample paths (running up to time $T$ ) are determined by their signature over $[0,T]$ .     

Heavy traffic limits associated with >M/G/∞ input processes

We study the heavy traffic regime of a discrete-time queue driven by correlated inputs, namely the M/G/ input processes of Cox. We distinguish between M/G/ processes with short- and long-range dependence, identifying in each case the appropriate heavy traffic scaling that results in a nondegenerate limit. As expected, the limits we obtain for short-range dependent inputs involve the standard Brownian motion. Of particular interest are the conclusions for the long-range dependent case: the normalized queue length can be expressed as a function not of a fractional Brownian motion, but of an -stable, 1/ self-similar independent increment Lévy process. The resulting buffer content distribution in heavy traffic is expressed through a Mittag–Leffler special function and displays a hyperbolic decay of power 1-. Thus, M/G/ processes already demonstrate that under long-range dependence, fractional Brownian motion does not necessarily assume the ubiquitous role that standard Brownian motion plays in the short-range dependence setup.     

Backward stochastic differential equations with jumps and related non-linear expectations

In this paper, we are interested in real-valued backward stochastic differential equations with jumps together with their applications to non-linear expectations. The notion of non-linear expectations has been studied only when the underlying filtration is given by a Brownian motion and in this work the filtration will be generated by both a Brownian motion and a Poisson random measure. We study at first backward stochastic differential equations driven by a Brownian motion and a Poisson random measure and then introduce the notions of f$f$-expectations and of non-linear expectations in this set-up.     

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 nonstandard construction of Lévy Brownian motion

Summary A nonstandard construction of Lévy Brownian motion on ^d is presented, which extends R.M. Anderson's nonstandard representation of Brownian motion. It involves a nonstandard construction of white noise and gives as a classical corollary a new white noise integral representation of Lévy Brownian motion. Moreover, a new invariance principle can be deduced in a similar way as Donsker's invariance principles follows from Anderson's construction.     

Joint convergence along different subsequences of the signed cubic variation of fractional Brownian motion

The purpose of this paper is to study the convergence in distribution of two subsequences of the signed cubic variation of the fractional Brownian motion with Hurst parameter $H=1/6$ . We prove that, under some conditions on both subsequences, the limit is a two-dimensional Brownian motion whose components may be correlated and we find explicit formulae for its covariance function.     

A Brownian Motion on the Diffeomorphism Group of the Circle

Let Diff(S ¹) be the group of orientation preserving C ^?∞? diffeomorphisms of S ¹. In 1999, P. Malliavin and then in 2002, S. Fang constructed a canonical Brownian motion associated with the H ^3/2 metric on the Lie algebra diff(S ¹). The canonical Brownian motion they constructed lives in the group Homeo(S ¹) of Hölderian homeomorphisms of S ¹, which is larger than the group Diff(S ¹). In this paper, we present another way to construct a Brownian motion that lives in the group Diff(S ¹), rather than in the larger group Homeo(S ¹).     

A first-order limit law for functionals of two independent fractional Brownian motions in the critical case

We prove a first-order limit law for functionals of two independent \(d\)-dimensional fractional Brownian motions with the same Hurst index \(H=2/d\,(d\ge 4)\), using the method of moments and extending a result by LeGall in the case of Brownian motion.     

Forward Brownian motion

We consider processes which have the distribution of standard Brownian motion (in the forward direction of time) starting from random points on the trajectory which accumulate at \(-\infty \) . We show that these processes do not have to have the distribution of standard Brownian motion in the backward direction of time, no matter which random time we take as the origin. We study the maximum and minimum rates of growth for these processes in the backward direction. We also address the question of which extra assumptions make one of these processes a two-sided Brownian motion.     

Pathwise uniqueness for perturbed versions of Brownian motion and reflected Brownian motion

Any solution of the functional equation