Abstract nonlinear filtering theory in the presence of fractional Brownian motion

Non Linear Filtering Theory in Presence of Fractional Brownian Motion

We develop the ltering theory in the case where both the signal and the observation are solutions of some stochastic diierential equation driven by a multiidimensional fractional Brownian motion. We show that the classical approach fails to give a closed equation for the lter and we develop another approach using an auxiliary processsvalued semimartingale which solves theoretically this problem...

Fractional Brownian motion in presence of two fixed adsorbing boundaries

Abstract. We study the long-time asymptotics of the probability Pt that the Riemann-Liouville fractional Brownian motion with Hurst index H does not escape from a fixed interval [−L, L] up to time t. We show that for any H ∈]0, 1], for both subdiffusion and superdiffusion regimes, this probability obeys ln(Pt) ∼ −t2H/L2, i.e. may decay slower than exponential (subdiffusion) or faster than expon...

Perturbation theory for fractional Brownian motion in presence of absorbing boundaries.

Fractional Brownian motion is a Gaussian process x(t) with zero mean and two-time correlations (x(t(1))x(t(2)))=D(t(1)(2H)+t(2)(2H)-|t(1)-t(2)|(2H)), where H, with 0<H<1, is called the Hurst exponent. For H=1/2, x(t) is a Brownian motion, while for H≠1/2, x(t) is a non-Markovian process. Here we study x(t) in presence of an absorbing boundary at the origin and focus on the probability density P...

Filtering for Linear Systems Driven by Fractional Brownian Motion

Existence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion