Brownian Motion: The Link Between Probability and Mathematical Analysis

The probability that Brownian motion almost contains a line

Let R be the Wiener sausage of radius about a planar Brownian motion run to time 1. Let P be the probability that R contains the line segment [0, 1] on the x-axis. Upper and lower estimates are given for P . The upper estimate implies that the range of a planar Brownian motion contains no line segment.

Existence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion

Stochastic Analysis of the Fractional Brownian Motion

Since the fractional Brownian motion is not a semi–martingale, the usual Ito calculus cannot be used to define a full stochastic calculus. However, in this work, we obtain the Itô formula, the Itô–Clark representation formula and the Girsanov theorem for the functionals of a fractional Brownian motion using the stochastic calculus of variations.

Ruin Probability for Generalized Φ-sub-gaussian Fractional Brownian Motion

for various types of risk process X = (X(t), t ≥ 0) and functions f(t). The similar problem of finding the buffer overflow probability appears in the queuing theory for different communication network models. The tasks of such type were solved for many types of processes, including Gaussian ones and aforementioned FBM (see, for example, Norros [1], Michna [2], Baldi and Pacchiarotti [3], etc.)....

Nonstandard Analysis, Fractal Properties and Brownian Motion

In this paper I explore a nonstandard formulation of Hausdorff dimension. By considering an adapted form of the counting measure formulation of Lebesgue measure, I prove a nonstandard version of Frostman’s lemma and show that Hausdorff dimension can be computed through a counting argument rather than by taking the infimum of a sum of certain covers. This formulation is then applied to obtain a ...

Brownian Brownian Motion – I

A classical model of Brownian motion consists of a heavy molecule submerged into a gas of light atoms in a closed container. In this work we study a 2D version of this model, where the molecule is a heavy disk of mass M ≫ 1 and the gas is represented by just one point particle of mass m = 1, which interacts with the disk and the walls of the container via elastic collisions. Chaotic behavior of...