Rate of strong convergence to Markov-modulated Brownian motion

Occupation Times for Markov-Modulated Brownian Motion

We determine the distributions of occupation times of a Markov-modulated Brownian motion (MMBM) in separate intervals before a first passage time or an exit from an interval. They will be derived in terms of their Laplace transforms, distinguishing occupation times in different phases too. For MMBMs with strictly positive variation parameters we further propose scale functions.

The Strong Markov Property of the Brownian Motion

Markov-modulated Brownian Motion with Two Reflecting Barriers

We consider a Markov-modulated Brownian motion reflected to stay in a strip [0, B]. The stationary distribution of this process is known to have a simple form under some assumptions. We provide a short probabilistic argument leading to this result and explain its simplicity. Moreover, this argument allows for generalizations including the distribution of the reflected process at an independent,...

Strong Approximation of Brownian Motion

Simple random walk and Brownian motion are two strongly interconnected mathematical concepts. They are widely involved in not only pure math, but also in many other scientific fields. In this paper I will first introduce and define some basic concepts of discrete-time random walk. Then I will construct Brownian Motion with some basic properties, and use a method called the strong approximation ...

Convergence of Scaled Renewal Processes to Fractional Brownian Motion

The superposition process of independent counting renewal processes associated with a heavy-tailed interarrival time distribution is shown to converge weakly after rescaling in time and space to fractional Brownian motion, as the number of renewal processes tends to innnity. Corresponding results for continuous arrival uid processes are discussed.