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.     

Efficient Monte Carlo simulation for integral functionals of Brownian motion

In the paper, we develop a variance reduction technique for Monte Carlo simulations of integral functionals of a Brownian motion. The procedure is based on a new method of sampling the process, which combines the Brownian bridge construction with conditioning on integrals along paths of the process. The key element in our method is the identification of a low-dimensional vector of variables that reduces the dimension of the integration problem more effectively than the Brownian bridge. We illustrate the method by applying it in conjunction with low-discrepancy sequences to the problem of pricing Asian options.     

Path Collapse for an Inhomogeneous Random Walk

On an open interval we follow the paths of a Brownian motion which returns to a fixed point as soon as it reaches the boundary and restarts afresh indefinitely. We determine that two paths starting at different points either cannot collapse or they do so almost surely. The problem can be modelled as a spatially inhomogeneous random walk on a group and contrasts sharply with the higher dimensional case in that if two paths may collapse they do so almost surely.     

A Set-indexed Fractional Brownian Motion

We define and prove the existence of a fractional Brownian motion indexed by a collection of closed subsets of a measure space. This process is a generalization of the set-indexed Brownian motion, when the condition of independance is relaxed. Relations with the Lévy fractional Brownian motion and with the fractional Brownian sheet are studied. We prove stationarity of the increments and a property of self-similarity with respect to the action of solid motions. Moreover, we show that there no “really nice” set indexed fractional Brownian motion other than set-indexed Brownian motion. Finally, behavior of the set-indexed fractional Brownian motion along increasing paths is analysed.      

Identification of the Hurst Index of a Step Fractional Brownian Motion

We propose a semi-parametric estimator for a piece-wise constant Hurst coefficient of a step fractional Brownian motion (SFBM). For the applications, we want to detect abrupt changes of the Hurst index (which represents long-range correlation) for a Gaussian process with a.s. continuous paths. The previous model of multifractional Brownian motion give a.s. discontinuous paths at change times of the Hurst index. Thus, we first propose a new kind of Fractional Brownian Motion, the SFBM and prove some (Hölder) continuity results. After, we propose an estimator of the piecewise constant Hurst parameter and prove its consistency.     

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.     

Random walks and Brownian motion on cubical complexes

Cubical complexes are metric spaces constructed by gluing together unit cubes in an analogous way to the construction of simplicial complexes. We construct Brownian motion on such spaces, define random walks, and prove that the transition kernels of the random walks converge to that for Brownian motion. The proof involves pulling back onto the complex the distribution of Brownian sample paths on a single cube, combined with a distribution on walks between cubes. The main application lies in analysing sets of evolutionary trees: several tree spaces are cubical complexes and we briefly describe our results and applications in this context.     

Brownian Motion Indexed by a Time Scale

In this article, we generalize Wiener's existence result for one-dimensional Brownian motion by constructing a suitable continuous stochastic process where the index set is a time scale. We construct a countable dense subset of a time scale and use it to prove a generalized version of the Kolmogorov–?entsov theorem. As a corollary, we obtain a local Hölder-continuity result for the sample paths of generalized Brownian motion indexed by a time scale.     

利用Ito公式求布朗运动和几何布朗运动的矩

利用Ito公式及Ito积分的性质求出了布朗运动和几何布朗运动的矩的一般形式,同时指出可以利用这种方法求其他扩散过程的矩.     

On Double Stratonovich Fractional Integrals and Some Strong and Weak Approximations

Abstract

Double Stratonovich integrals with respect to the odd part and even part of the fractional Brownian motion are constructed. The first and the second moments of such integrals are explicitly identified. As application of double Stratonovich integrals a strong law of large numbers for efBm and ofBm is derived.

Riemann–Stieltjes integral approximations to double Stratonovich fractional integrals are also considered. The strong convergence (almost surely and mean square) is obtained for approximations based on explicit series expansions of the fractional Brownian processes. The weak convergence is derived for approximations by processes with absolutely continuous paths which converge weakly to the considered fractional Brownian processes. The above-mentioned convergences are obtained for deterministic integrands which are given by bimeasures.     

The transversal homoclinic solutions and chaos for stochastic ordinary differential equations

We consider the persistence of a transversal homoclinic solution and chaotic motion for ordinary differential equations with a homoclinic solution to a hyperbolic equilibrium under an unbounded random forcing driven by a Brownian force. By Lyapunov–Schmidt reduction, the persistence of transversal homoclinic solution is reduced to find the zeros of some bifurcation functions defined between two finite spaces. It is shown that, for almost all sample paths of the Brownian motion, the perturbed system exhibits chaos.     

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.     

Integration by parts on the Brownian Meander

We prove infinite-dimensional integration by parts formulae for the laws of the Brownian Meander, of the Bessel Bridge of dimension 3 between and of the Brownian Motion on the set of all paths taking values greater than or equal to a nonpositive constant. We give applications to SPDEs with reflection.

     

Brownian Optimal Stopping and Random Walks

One way to compute the value function of an optimal stopping problem along Brownian paths consists of approximating Brownian motion by a random walk. We derive error estimates for this type of approximation under various assumptions on the distribution of the approximating random walk.     

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.     

On the connected components of the support of super Brownian motion and of its exit measure

Tribe proved in a previous paper that a typical point of the support of super Brownian motion considered at a fixed time is a.s. disconnected from the others when the space dimension is greater than or equal to 3. We give here a simpler proof of this result based on Le Gall's Brownian snake. This proof can then be adapted in order to obtain an analogous result for the support of the exit measure of the super Brownian motion from a smooth domain of ^d when d is greater than or equal to 4.     

Brownian Optimal Stopping and Random Walks

One way to compute the value function of an optimal stopping problem along Brownian paths consists of approximating Brownian motion by a random walk. We derive error estimates for this type of approximation under various assumptions on the distribution of the approximating random walk.     

Non-independence of excursions of the Brownian sheet and of additive Brownian motion

A classical and important property of Brownian motion is that given its zero set, distinct excursions away from zero are independent. In this paper, we examine the analogous question for the Brownian sheet, and also for additive Brownian motion. Our main result is that given the level set of the Brownian sheet at level zero, distinct excursions of the sheet away from zero are not independent. In fact, given the zero set of the Brownian sheet in the entire non-negative quadrant, and the sign of all but a finite number of excursions away from zero, the signs of the remaining excursions are determined. For additive Brownian motion, we prove the following definitive result: given the zero set of additive Brownian motion and the sign of a single excursion, the signs of all other excursions are determined. In an appendix by John B. Walsh, it is shown that given the absolute value of the sheet in the entire quadrant and, in addition, the sign of the sheet at a fixed, non-random time point, then the whole sheet can be recovered.

     

An approach to generate deterministic Brownian motion

We propose an approach for generation of deterministic Brownian motion. By adding an additional degree of freedom to the Langevin equation and transforming it into a system of three linear differential equations, we determine the position of switching surfaces, which act as a multi-well potential with a short fluctuation escape time. Although the model is based on the Langevin equation, the final system does not contain a stochastic term, and therefore the obtained motion is deterministic. Nevertheless, the system behavior exhibits important characteristic properties of Brownian motion, namely, a linear growth in time of the mean square displacement, a Gaussian distribution, and a −2 power law of the frequency spectrum. Furthermore, we use the detrended fluctuation analysis to prove the Brownian character of this motion.     

A Factorisation of Diffusion Measure and Finite Sample Path Constructions

In this paper we introduce decompositions of diffusion measure which are used to construct an algorithm for the exact simulation of diffusion sample paths and of diffusion hitting times of smooth boundaries. We consider general classes of scalar time-inhomogeneous diffusions and certain classes of multivariate diffusions. The methodology presented in this paper is based on a novel construction of the Brownian bridge with known range for its extrema, which is of interest on its own right.