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Brownian Motion
Brownian motion owes its name to the botanist Robert Brown who observed the chaotic motion of pollen grains in a liquid. From the mathematical point...
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Faking Brownian motion with continuous Markov martingales
Hamza and Klebaner
(2007) [10] posed the problem of constructing martingales with one-dimensional Brownian marginals that differ from Brownian... -
On the Theory of Relativistic Brownian Motion
AbstractThe approach to the theory of a relativistic random process is considered by the path integral method as Brownian motion taking into...
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Path Regularity of the Brownian Motion and the Brownian Sheet
By the work of P. Lévy, the sample paths of the Brownian motion are known to satisfy a certain Hölder regularity condition almost surely. This was...
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Brownian Motion and Theta Functions
We introduce the Brownian motion on a real line \(\mathbb {R}\) . First we...
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The Moduli of Continuity for Operator Fractional Brownian Motion
The almost-sure sample path behavior of the operator fractional Brownian motion with exponent D , including multivariate fractional Brownian motion,...
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Distributions of Functionals of a Skew Brownian motion with Discontinuous Drift
A skew Brownian motion with piecewise constant drift is considered. This diffusion includes a skew Brownian motion with linear drift with equal...
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Arbitrage problems with reflected geometric Brownian motion
Contrary to the claims made by several authors, a financial market model in which the price of a risky security follows a reflected geometric...
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Brownian Motion
This chapter is devoted to the study of Brownian motion, which, together with the Poisson process studied in Chapter 9, is one of the most important...
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Towards a Better Understanding of Fractional Brownian Motion and Its Application to Finance
The aim of this work is to first build the underlying theory behind fractional Brownian motion and applying fractional Brownian motion to financial...
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Distribution of Functionals of Brownian Motion with Linear Drift and Elastically Killed at Zero
Brownian motion with linear drift on positive half-line and killed elastically at zero is considered. A goal is to get a result that allows us to...
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Brownian motion approximation by parametrized and deformed neural networks
The first author recently derived several approximation results by neural network operators see the new monograph (Anastassiou GA, Parametrized,...
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First Passage Density of Brownian Motion with Two-sided Piecewise Linear Boundaries
The first passage time has many applications in fields like finance, econometrics, statistics, and biology. However, explicit formulas for the first...
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When can we reconstruct the ancestral state? Beyond Brownian motion
Reconstructing the ancestral state of a group of species helps answer many important questions in evolutionary biology. Therefore, it is crucial to...
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Distributions of Functionals of the Local Time of Brownian Motion with Discontinuous Drift
Diffusion with piecewise constant drift and diffusion coefficient 1 is considered. Such a process is called the Brownian motion with discontinuous...
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Local law and rigidity for unitary Brownian motion
We establish high probability estimates on the eigenvalue locations of Brownian motion on the N -dimensional unitary group, as well as estimates on...
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Stochastic Averaging Principle for Two-Time-Scale SDEs with Distribution-Dependent Coefficients Driven by Fractional Brownian Motion
In this paper, we derive an averaging principle for a fast–slow system of stochastic differential equations (SDEs) involving distribution-dependent...
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Moderate Deviations for Two-Time Scale Systems with Mixed Fractional Brownian Motion
This work focuses on moderate deviations for two-time scale systems with mixed fractional Brownian motion. Our proof uses the weak convergence method...
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Local Time for Brownian Motion
Brownian paths are continuous but otherwise quite irregular and chaotic, as is indicated by the fact that when a path hits a point x, it visits every...
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Harnack Type Inequalities for SDEs Driven by Fractional Brownian Motion with Markovian Switching
In this paper, by constructing a coupling equation, we establish the Harnack type inequalities for stochastic differential equations driven by...