Let X be a (two-sided) fractional Brownian motion of Hurst parameter H ∈ (0, 1) and let Y be a standard Brownian motion independent of X. Fractional Brownian motion in Brownian motion time (of index H), recently studied in [17], is by definition the process Z = X ◦ Y . It is a continuous, non-Gaussian process with stationary increments, which is selfsimilar of index H/2. The main result of the ...

Fractional Brownian motion is a self-affine, non-Markovian, and translationally invariant generalization of Brownian motion, depending on the Hurst exponent H. Here we investigate fractional Brownian motion where both the starting and the end point are zero, commonly referred to as bridge processes. Observables are the time t_{+} the process is positive, the maximum m it achieves, and the time ...

Let X be a Brownian motion defined on the line (with X(0)=0) and let Y be an independent Brownian motion defined on the nonnegative real numbers. For all t ≥ 0, we define the iterated Brownian motion (IBM), Z, by setting Z t ∆ = X(Y t). In this paper we determine the exact uniform modulus of continuity of the process Z.

Purposeful motion of biological processes can be driven by Brownian motion of macromolecular complexes with one-sided binding biasing movement in one direction: a Brownian ratchet, now proposed to explain membrane motion during a phagocytosis-like process in bacteria.

We provide a simple algorithm for construction of Brownian paths approximating those Lévy process on finite time interval. It requires knowledge the trajectory chosen regular grid and law its endpoint, or ability to simulate from that. This is based reordering increments, it can be applied in recursive manner. establish an upper bound mean squared maximal distance between determine suitable mes...

in this paper, a numerical efficient method based on two-dimensional block-pulse functions (bpfs) is proposed to approximate a solution of the two-dimensional linear stochastic volterra-fredholm integral equation. finally the accuracy of this method will be shown by an example.

We supply two different descriptions of the pushing process driving the reflected Brownian motion in Weyl chambers, when the latter domains are simplexes. The first one shows that a simple root lies in one and only one orbit if and only if the pushing process in the direction of that simple root increases as the sum of all the Brownian local times in the directions of the orbit’s positive eleme...

1. Examples 1.1. Poisson Process 1.2. Markov Chain 1.3. Martingale 1.4. Brownian Motion 2. Definition of the Stochastic Process 3. Poisson Process 3.1. The Distribution 3.2. A Construction of the Poisson Process 3.3. Generalizations 4. Brownian Motion 4.1. Random Walk 4.2. Distributions 4.3. Path Properties 4.4. Local Time 4.5. Fourier Series Representation Glossary Bibliography Boigraphical Sk...