We are concerned with the stationary distribution of a d-dimensional semimartingale reflecting Brownian motion on a nonnegative orthant, provided it is stable, and conjecture about the tail decay rate of its marginal distribution in an arbitrary direction. Due to recent studies, the conjecture is true for d = 2. We show its validity for the skew symmetric case for a general d .

We propose an energetic interpretation of stochastic processes described by Langevin equations with non-uniform temperature. In order to avoid Itô-Stratonovich dilemma, we start with a Kramers equation, and derive a Fokker-Plank equation by the renormalization group method. We give a proper definition of heat for the system. Based on our formulations, we analyze two examples, the Thomson effect...

the problem of steady magnetohydrodynamic boundary layer flow of an electrically conducting nanofluid due to an exponentially permeable stretching sheet with heat source/sink in presence of thermal radiation is numerically investigated. the effect of transverse brownian motion and thermophoresis on heat transfer and nano particle volume fraction considered. the governing partial differential eq...

Some probabilistic aspects of the number variance statistic are investigated. Infinite systems of independent Brownian motions and symmetric α-stable processes are used to construct explicit new examples of processes which exhibit both divergent and saturating number variance behaviour. We derive a general expression for the number variance for the spatial particle configurations arising from t...

We prove a comparison theorem, extending Li [6], and develop a complexanalytic approach to treat L2 small ball probabilities of Gaussian processes. We demonstrate the techniques for the m-times integrated Brownian motions, and in examples where one can not apply Li’s comparison theorem. ∗Department of Mathematics, University of Idaho, Moscow, ID 83844-1103, [email protected] †Department of Sta...

The distribution of a functional of two correlated vector Brownian motions is approximated by a Gamma distribution. This functional represents the limiting distribution for cointegration tests with stationary exogenous regressors, but also for cointegration tests based on a non-Gaussian likelihood. The approximation is accurate, fast, and easy to use in comparison to both tabulated critical val...

Arratia, and later Tóth and Werner, constructed random processes that formally correspond to coalescing one-dimensional Brownian motions starting from every space-time point. We extend their work by constructing and characterizing what we call the Brown-ian Web as a random variable taking values in an appropriate (metric) space whose points are (compact) sets of paths. This leads to general con...

We continue with the assumptions used in the previous lecture that price of a security can be modeled as an Ito process. Assume we have N price processes dXj = μj (t)dt + σj (t)dB(t), j = 1, 2, . . . , N . We assume that B is a vector of d independent Brownian motions and each σj is d­dimensional vector process as well (for simplicity we skip the · notation from here onwards). We recall that th...

The goal of this paper is to define and study a notion of fractional Brownian motion on a Lie group. We define it as at the solution of a stochastic differential equation driven by a linear fractional Brownian motion. We show that this process has stationary increments and satisfies a local self-similar property. Furthermore the Lie groups for which this self-similar property is global are char...