scholarly journals Measure pseudo almost periodic solution for a class of nonlinear delayed stochastic evolution equations driven by Brownian motion

Filomat ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 515-534
Author(s):  
Nadia Belmabrouk ◽  
Mondher Damak ◽  
Mohsen Miraoui
Keyword(s):  
Brownian Motion ◽  
Evolution Equations ◽  
Real Hilbert Space ◽  
Banach Fixed Point Theorem ◽  
Almost Periodic ◽  
Periodic Process ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Pseudo Almost Periodic Solution ◽  
Pseudo Almost Periodic

In this work, we present a new concept of measure-ergodic process to define the space of measure pseudo almost periodic process in the p-th mean sense. We show some results regarding the completeness, the composition theorems and the invariance of the space consisting in measure pseudo almost periodic process. Motivated by above mentioned results, the Banach fixed point theorem and the stochastic analysis techniques, we prove the existence, uniqueness and the global exponential stability of doubly measure pseudo almost periodic mild solution for a class of nonlinear delayed stochastic evolution equations driven by Brownian motion in a separable real Hilbert space. We provide an example to illustrate the effectiveness of our results.

Pseudo Almost Periodicity and Its Applications to Impulsive Nonautonomous Partial Functional Stochastic Evolution Equations

2018 ◽  
Vol 19 (5) ◽  
pp. 511-529
Author(s):  
Zuomao Yan ◽  
Xiumei Jia
Keyword(s):  
Evolution Equations ◽  
Mild Solutions ◽  
Almost Periodic ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Pseudo Almost Periodic Functions ◽  
Pseudo Almost Periodicity ◽  
Pseudo Almost Periodic ◽  
Lipschitz Conditions ◽  
Composition Theorem

AbstractIn this paper, we establish a new composition theorem for pseudo almost periodic functions under non-Lipschitz conditions. We apply this new composition theorem together with a fixed-point theorem for condensing maps to investigate the existence of$p$-mean piecewise pseudo almost periodic mild solutions for a class of impulsive nonautonomous partial functional stochastic evolution equations in Hilbert spaces, and then, the exponential stability of$p$-mean piecewise pseudo almost periodic mild solutions is studied. Finally, an example is given to illustrate our results.

scholarly journals On Almost Automorphic Mild Solutions for Nonautonomous Stochastic Evolution Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Jing Cui ◽  
Litan Yan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Banach Fixed Point Theorem ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Almost Automorphic ◽  
Composition Theorem

We consider a class of nonautonomous stochastic evolution equations in real separable Hilbert spaces. We establish a new composition theorem for square-mean almost automorphic functions under non-Lipschitz conditions. We apply this new composition theorem as well as intermediate space techniques, Krasnoselskii fixed point theorem, and Banach fixed point theorem to investigate the existence of square-mean almost automorphic mild solutions. Some known results are generalized and improved.

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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