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Small deviations for admixture additive & multiplicative processes.

Mingjie Liang1, Bingyao Wu1

  • 1College of Mathematics and Informatics, Fujian Normal University, Fuzhou, P.R. China.

Journal of Inequalities and Applications
|March 7, 2019
PubMed
Summary
This summary is machine-generated.

This study defines admixture processes and derives their small deviation properties. Applications include limit theorems for stochastic processes, enhancing understanding of random system behavior.

Keywords:
Admixture additive processAdmixture multiplicative processLimit theoremSmall deviation

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Area of Science:

  • Stochastic Processes and Probability Theory
  • Mathematical Finance
  • Statistical Modeling

Background:

  • Stochastic processes are fundamental in modeling complex systems, but their behavior under additive and multiplicative noise requires rigorous definition.
  • Existing models often lack precise characterization of small deviation behavior for fractional and stable processes.

Purpose of the Study:

  • To formally define admixture additive and multiplicative processes.
  • To derive the small deviation for fractional integrated Brownian motion and the lower bound of small deviation for fractional Brownian motion.
  • To provide an application of these findings through a limit inferior type (LIL) result for a related stochastic process.

Main Methods:

  • Mathematical definitions for admixture additive and multiplicative processes involving standard Brownian motion, fractional integrated Brownian motion, fractional Brownian motion, and stable processes.
  • Derivation of small deviation properties and lower bounds using stochastic calculus and probability inequalities.
  • Application of derived properties to establish a limit inferior type (LIL) theorem.

Main Results:

  • Precise definitions of admixture additive and multiplicative processes are established.
  • The small deviation for fractional integrated Brownian motion and its lower bound for fractional Brownian motion are rigorously obtained.
  • A limit inferior type (LIL) result is presented as a direct application of the developed theory.

Conclusions:

  • The study provides a foundational framework for analyzing admixture processes with various stochastic components.
  • The derived small deviation results offer deeper insights into the behavior of these complex processes.
  • The LIL application demonstrates the practical utility of the theoretical advancements in stochastic analysis.