A FILTERING PROBLEM FOR A LINEAR STOCHASTIC EVOLUTION EQUATION DRIVEN BY A FRACTIONAL BROWNIAN MOTION

2008 ◽  
Vol 08 (03) ◽  
pp. 397-412
Author(s):  
WILFRIED GRECKSCH ◽  
CONSTANTIN TUDOR
Keyword(s):  
Brownian Motion ◽  
Evolution Equation ◽  
Fractional Brownian Motion ◽  
Solution Process ◽  
Innovation Process ◽  
Optimal Estimation ◽  
Stochastic Evolution Equation ◽  
Stochastic Evolution ◽  
Finite Dimensional ◽  
Observation Process

A linear unbiased and square mean optimal estimation is obtained for the mild solution process of a stochastic evolution equation with an infinite-dimensional fractional Brownian motion as noise and the noise in the observation process is a finite-dimensional Brownian motion. An innovation process is introduced and the estimation is obtained as a solution of a stochastic differential equation with a finite-dimensional noise. By using an approach based on the equivalence with a deterministic control problem, the estimation for the Fourier coefficients of the signal process is also determined.

Moderate deviation for parameter estimator in the stochastic parabolic equations with additive fractional Brownian motion

2014 ◽  
Vol 14 (03) ◽  
pp. 1450002
Author(s):  
Jiang Hui
Keyword(s):  
Brownian Motion ◽  
Evolution Equation ◽  
Fractional Brownian Motion ◽  
Parabolic Equations ◽  
Moderate Deviation ◽  
Hurst Parameter ◽  
Stochastic Evolution Equation ◽  
Stochastic Evolution ◽  
Parameter Estimator ◽  
Asymptotic Behaviors

In this paper, we study the asymptotic behaviors of parameter estimator in a diagonalizable stochastic evolution equation driven by additive fractional Brownian motion with Hurst parameter H ∈ [½, 1). The moderate deviation for this estimator can be obtained.

scholarly journals Existence of densities for stochastic evolution equations driven by fractional Brownian motion

2020 ◽  
Vol 21 (02) ◽  
pp. 2150009
Author(s):  
Jorge A. de Nascimento ◽  
Alberto Ohashi
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Vector Fields ◽  
Evolution Equations ◽  
Trace Class ◽  
Stochastic Pdes ◽  
Stochastic Evolution ◽  
Finite Dimensional ◽  
Algebraic Constraints ◽  
Adapted Process

In this work, we prove a version of Hörmander’s theorem for a stochastic evolution equation driven by a trace-class fractional Brownian motion with Hurst exponent [Formula: see text] and an analytic semigroup on a given separable Hilbert space. In contrast to the classical finite-dimensional case, the Jacobian operator in typical solutions of parabolic stochastic PDEs is not invertible which causes a severe difficulty in expressing the Malliavin matrix in terms of an adapted process. Under a Hörmander’s bracket condition and some algebraic constraints on the vector fields combined with the range of the semigroup, we prove that the law of finite-dimensional projections of such solutions has a density with respect to Lebesgue measure. The argument is based on rough path techniques and a suitable analysis on the Gaussian space of the fractional Brownian motion.

scholarly journals Second-order limit laws for occupation times of fractional Brownian motion

2017 ◽  
Vol 54 (2) ◽  
pp. 444-461 ◽  
Author(s):  
Fangjun Xu
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Method Of Moments ◽  
Second Order ◽  
Hurst Index ◽  
Occupation Times ◽  
Limit Laws ◽  
Additive Functionals ◽  
Finite Dimensional ◽  
Functional Version

Abstract We prove a second-order limit law for additive functionals of a d-dimensional fractional Brownian motion with Hurst index H = 1 / d, using the method of moments and extending the Kallianpur–Robbins law, and then give a functional version of this result. That is, we generalize it to the convergence of the finite-dimensional distributions for corresponding stochastic processes.

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