This paper provides a an introduction to some basic properties of Brownian motion. In particular, it shows that Brownian motion exists, that Brownian motion is nowhere differentiability, and that Brownian motion has finite quadratic variation.

We find the exact winding number distribution of Riemann-Liouville fractional Brownian motion for large times in two dimensions using the propagator of a free particle. The distribution is similar to the Brownian motion case and it is of Cauchy type. In addition we find the winding number distribution of fractal time process, i.e., time fractional Fokker-Planck equation, in the presence of fini...

(To Appear) Stochastic Calculus for Brownian Motion on a Brownian Fracture By Davar Khoshnevisan* & Thomas M. Lewis University of Utah & Furman University Abstract. The impetus behind this work is a pathwise development of stochastic integrals with respect to iterated Brownian motion. We also provide a detailed analysis of the variations of iterated Brownian motion. These variations are linked ...

In this paper we introduce a new method for the simulation of the exit time and exit position of a δ-dimensional Brownian motion from a domain. The main interest of our method is that it avoids splitting time schemes as well as inversion of complicated series. The method, called walk on moving spheres algorithm, was first introduced for hitting times of Bessel processes. In this study this meth...

In this paper we nd a nonexponential Lundberg approximation of the ruin probability in a Cox model, in which a governing process has a regenerative structure and the claims are light-tailed or have an intermediate regularly varying distribution. Examples include an intensity process being reeected Brownian motion, square functions of the Ornstein-Uhlenbeck process and splitting reeected Brownia...

We introduce a two-parameter family of diffusion processes (B r,s(t))t≥0, r, s > 0, on the general linear group GLN that are Brownian motions with respect to certain natural metrics on the group. At the same time, we introduce a two-parameter family of free Itô processes (br,s(t))t≥0 in a faithful, tracial W ∗-probability space, and we prove that the full process (B r,s(t))t≥0 converges to (br,...

Burke’s theorem is a well-known fundamental result in queueing theory, stating that a stationary M/M/1 queue has a departure process that is identical in law to the arrival process and, moreover, for each time t , the following three random objects are independent: the queue length at time t , the arrival process after t and the departure process before t . Burke’s theorem also holds for a stat...

Some stochastic control systems that are described by stochastic differential equations with a fractional Brownian motion are considered. The solutions of these systems are defined by weak solutions. These weak solutions are obtained by the transformation of the measure for a fractional Brownian motion by a Radon-Nikodym derivative. This weak solution approach is used to solve a control problem...

We prove that reflecting Brownian motion on a bounded Lipschitz domain is a semimartingale. We also extend the well-known Skorokhod equation to this case. In this note we study the semimartingale property and the Skorokhod equation of reflecting Brownian motion on a bounded Euclidean domain. A R d_ valued continuous stochastic process X = {Xt ; t > 0} is said to be a semimartingale if it can be...

We investigate a functional limit theorem (homogenization) for Reflected Stochastic Differential Equations on a half-plane with stationary coefficients when it is necessary to analyze both the effective Brownian motion and the effective local time. We prove that the limiting process is a reflected non-standard Brownian motion. Beyond the result, this problem is known as a prototype of non-trans...