scholarly journals The Combined Estimator for Stochastic Equations on Graphs with Fractional Noise

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1766
Author(s):  
Pavel Kříž ◽  
Leszek Szała
Keyword(s):  
Strong Consistency ◽  
Evolution Equations ◽  
Stochastic Equations ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Fractional Brownian Motions ◽  
Fractional Noise ◽  
Long Span ◽  
Finite Samples ◽  
Drift Parameter

In the present paper, we study the problem of estimating a drift parameter in stochastic evolution equations on graphs. We focus on equations driven by fractional Brownian motions, which are particularly useful e.g., in biology or neuroscience. We derive a novel estimator (the combined estimator) and prove its strong consistency in the long-span asymptotic regime with a discrete-time sampling scheme. The promising performance of the combined estimator for finite samples is examined under various scenarios by Monte Carlo simulations.

scholarly journals A space-consistent version of the minimum-contrast estimator for linear stochastic evolution equations

2019 ◽  
Vol 20 (03) ◽  
pp. 2050019
Author(s):  
Pavel Kříž
Keyword(s):  
Evolution Equations ◽  
Index Formula ◽  
Spectral Approach ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Self Similarity ◽  
Fractional Noise ◽  
Minimum Contrast Estimator ◽  
Minimum Contrast ◽  
Drift Parameter

A new modification of the minimum-contrast estimator (the weighted MCE) of drift parameter in a linear stochastic evolution equation with additive fractional noise is introduced in the setting of the spectral approach (Fourier coordinates of the solution are observed). The reweighing technique, which utilizes the self-similarity property, achieves strong consistency and asymptotic normality of the estimator as number of coordinates increases and time horizon is fixed (the space consistency). In this respect, this modification outperforms the standard (non-weighted) MCE. Compared to other drift estimators studied within spectral approach (e.g., maximum likelihood, trajectory fitting), the weighted MCE is rather universal. It covers discrete time as well as continuous-time observations and it is applicable to processes with any value of Hurst index [Formula: see text]. To the author’s best knowledge, this is so far the first space-consistent estimator studied for [Formula: see text].

scholarly journals Almost Automorphic Solutions to Nonautonomous Stochastic Functional Integrodifferential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-13
Author(s):  
Li Xi-liang ◽  
Han Yu-liang
Keyword(s):  
Hilbert Spaces ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Stochastic Equations ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Almost Automorphic ◽  
Banach Contraction ◽  
Stability Results ◽  
Stochastic Functional

This paper concerns the square-mean almost automorphic solutions to a class of abstract semilinear nonautonomous functional integrodifferential stochastic evolution equations in real separable Hilbert spaces. Using the so-called “Acquistapace-Terreni” conditions and Banach contraction principle, the existence, uniqueness, and asymptotical stability results of square-mean almost automorphic mild solutions to such stochastic equations are established. As an application, square-mean almost automorphic solution to a concrete nonautonomous integro-differential stochastic evolution equation is analyzed to illustrate our abstract results.

Uniform attractors for a class of stochastic evolution equations with multiplicative fractional noise

2020 ◽  
pp. 2150020
Author(s):  
Caibin Zeng ◽  
Xiaofang Lin ◽  
Hongyong Cui
Keyword(s):  
Evolution Equations ◽  
Stopping Time ◽  
Random Dynamical System ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Uniform Attractor ◽  
Existence And Uniqueness Results ◽  
Fractional Noise ◽  
Uniqueness Results ◽  
Time Periodic

This paper studies the (random) uniform attractor for a class of non-autonomous stochastic evolution equations driven by a time-periodic forcing and multiplicative fractional noise with Hurst parameter bigger than 1/2. We first establish the existence and uniqueness results for the solution to the considered equation and show that the solution generates a jointly continuous non-autonomous random dynamical system (NRDS). Moreover, we prove the existence of the uniform attractor for this NRDS through stopping time technique. Particularly, a compact uniformly absorbing set is constructed under a smallness condition imposed on the fractional noise.

scholarly journals ON STOCHASTIC EVOLUTION EQUATIONS WITH NON-LIPSCHITZ COEFFICIENTS

2009 ◽  
Vol 09 (04) ◽  
pp. 549-595 ◽  
Author(s):  
XICHENG ZHANG
Keyword(s):  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Reaction Diffusion ◽  
Stochastic Equations ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Uniqueness Of Solutions ◽  
Volterra Type ◽  
Porous Medium Equations ◽  
Stochastic Functional

In this paper, we study the existence and uniqueness of solutions for several classes of stochastic evolution equations with non-Lipschitz coefficients, that contains backward stochastic evolution equations, stochastic Volterra type evolution equations and stochastic functional evolution equations. In particular, the results can be used to treat a large class of quasi-linear stochastic equations, which includes the reaction diffusion and porous medium equations.

scholarly journals Smoothness of solutions of stochastic evolution equations and the existence of a filtering transition density

1981 ◽  
Vol 84 ◽  
pp. 195-208 ◽  
Author(s):  
B. L. Rozovskii ◽  
A. Shimizu
Keyword(s):  
Evolution Equations ◽  
Transition Density ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Smoothness Of Solutions

In this paper, we shall discuss the smoothness of solutions of stochastic evolution equations, which has been investigated in N. V. Krylov and B. L. Rozovskii [2] [3], to establish the existence of a filtering transition density.

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.

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