scholarly journals On a Class of Measure-Dependent Stochastic Evolution Equations Driven by fBm

2007 ◽  
Vol 2007 ◽  
pp. 1-26 ◽  
Author(s):  
Eduardo Hernandez ◽  
David N. Keck ◽  
Mark A. McKibben
Keyword(s):  
Hilbert Space ◽  
Brownian Motion ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Evolution Equations ◽  
Separable Hilbert Space ◽  
Probability Measures ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Real Separable Hilbert Space

We investigate a class of abstract stochastic evolution equations driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separable Hilbert space. We establish the existence and uniqueness of a mild solution, a continuous dependence estimate, and various convergence and approximation results. Finally, the analysis of three examples is provided to illustrate the applicability of the general theory.

scholarly journals Second-order neutral stochastic evolution equations with heredity

2004 ◽  
Vol 2004 (2) ◽  
pp. 177-192 ◽  
Author(s):  
Mark A. McKibben
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Continuous Dependence ◽  
Evolution Equations ◽  
Separable Hilbert Space ◽  
Second Order ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Integro Differential Equation ◽  
Real Separable Hilbert Space

Existence, continuous dependence, and approximation results are established for a class of abstract second-order neutral stochastic evolution equations with heredity in a real separable Hilbert space. A related integro-differential equation is also mentioned, as well as an example illustrating the theory.

scholarly journals Abstract Functional Stochastic Evolution Equations Driven by Fractional Brownian Motion

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Mark A. McKibben ◽  
Micah Webster
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Evolution Equations ◽  
Separable Hilbert Space ◽  
Mild Solutions ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Convergence Results ◽  
Real Separable Hilbert Space ◽  
Wave Phenomena

We investigate a class of abstract functional stochastic evolution equations driven by a fractional Brownian motion in a real separable Hilbert space. Global existence results concerning mild solutions are formulated under various growth and compactness conditions. Continuous dependence estimates and convergence results are also established. Analysis of three stochastic partial differential equations, including a second-order stochastic evolution equation arising in the modeling of wave phenomena and a nonlinear diffusion equation, is provided to illustrate the applicability of the general theory.

scholarly journals Functional integro-differential stochastic evolution equations in Hilbert space

2003 ◽  
Vol 16 (2) ◽  
pp. 141-161 ◽  
Author(s):  
David N. Keck ◽  
Mark A. McKibben
Keyword(s):  
Mathematical Modeling ◽  
Hilbert Space ◽  
Evolution Equations ◽  
Separable Hilbert Space ◽  
Existence Results ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Abstract Theory ◽  
Convergence Results ◽  
Real Separable Hilbert Space

We investigate a class of abstract functional integro-differential stochastic evolution equations in a real separable Hilbert space. Global existence results concerning mild and periodic solutions are formulated under various growth and compactness conditions. Also, related convergence results are established and an example arising in the mathematical modeling of heat conduction is discussed to illustrate the abstract theory.

scholarly journals Transportation Inequalities for Coupled Fractional Stochastic Evolution Equations Driven by Fractional Brownian Motion

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Wentao Zhan ◽  
Yuanyuan Jing ◽  
Liping Xu ◽  
Zhi Li
Keyword(s):  
Fixed Point ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Mild Solution ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Hurst Parameter ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Uniform Distance

In this paper, we consider the existence and uniqueness of the mild solution for a class of coupled fractional stochastic evolution equations driven by the fractional Brownian motion with the Hurst parameter H∈1/4,1/2. Our approach is based on Perov’s fixed-point theorem. Furthermore, we establish the transportation inequalities, with respect to the uniform distance, for the law of the mild solution.

scholarly journals Existence and exponential stability of almost pseudo automorphic solution for neutral stochastic evolution equations driven by G-Brownian motion

Filomat ◽  
2020 ◽  
Vol 34 (4) ◽  
pp. 1075-1092
Author(s):  
Pengju Duan
Keyword(s):  
Fixed Point ◽  
Brownian Motion ◽  
Exponential Stability ◽  
Mild Solution ◽  
Existence And Uniqueness ◽  
Evolution Operator ◽  
Evolution Equations ◽  
Sufficient Conditions ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations

This paper mainly concerns the quasi sure exponential stability of square mean almost pseudo automorphic mild solution for a class of neutral stochastic evolution equations driven by G-Brownian motion. By means of evolution operator theorem and fixed point theorem, existence and uniqueness of square mean almost pseudo automorphic mild solution is obtained. Also, a series of sufficient conditions on exponential stability and quasi sure exponential stability are established.

scholarly journals Operator semistable probability measures on a Hilbert space

1980 ◽  
Vol 22 (3) ◽  
pp. 397-406 ◽  
Author(s):  
R.G. Laha ◽  
V.K. Rohatgi
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Probability Measures ◽  
Real Separable Hilbert Space ◽  
Semistable Probability

A characterization of the class of operator semistable probability measures on a real separable Hilbert space is given.

scholarly journals MULTIFRACTAL FLUCTUATIONS IN FINANCE

2000 ◽  
Vol 03 (03) ◽  
pp. 361-364 ◽  
Author(s):  
FRANCOIS SCHMITT ◽  
DANIEL SCHERTZER ◽  
SHAUN LOVEJOY
Keyword(s):  
Brownian Motion ◽  
Exchange Rate ◽  
Foreign Exchange ◽  
Stochastic Models ◽  
Evolution Equations ◽  
Foreign Exchange Rate ◽  
Scaling Exponent ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Multiplicative Processes

We consider the structure functions S(q)(τ), i.e. the moments of order q of the increments X(t + τ)-X(t) of the Foreign Exchange rate X(t) which give clear evidence of scaling (S(q)(τ)∝τζ(q)). We demonstrate that the nonlinearity of the observed scaling exponent ζ(q) is incompatible with monofractal additive stochastic models usually introduced in finance: Brownian motion, Lévy processes and their truncated versions. This nonlinearity correspond to multifractal intermittency yielded by multiplicative processes. The non-analyticity of ζ(q) corresponds to universal multifractals, which are furthermore able to produce "hyperbolic" pdf tails with an exponent qD > 2. We argue that it is necessary to introduce stochastic evolution equations which are compatible with this multifractal behaviour.

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