The Master Equation approach to model anomalous diffusion is considered. Anomalous diffusion in complex media can be described as the result of a superposition mechanism reflecting inhomogeneity and nonstationarity properties of the medium. For instance, when this superposition is applied to the time-fractional diffusion process, the resulting Master Equation emerges to be the governing equatio...

In this work we introduce correlated random walks on Z. When picking suitably at random the coefficient of correlation, and taking the average over a large number of walks, we obtain a discrete Gaussian process, whose scaling limit is the fractional Brownian motion. We have to use two radically different models for both cases 1 2 ≤ H < 1 and 0 < H < 1 2 . This result provides an algorithm for t...

Fractional Brownian motion has been suggested as a model for the movement of log share prices which would allow long-range dependence between returns on different days. While this is true, it also allows arbitrage opportunities, which we demonstrate both indirectly and by constructing such an arbitrage. Nonetheless, it is possible by looking at a process similar to the fractional Brownian motio...

Assuming that the underlying stock follows Fractional Brownian motion and that stochastic interest rate meets the Vasicek model of interest rates, this paper establishes pricing model of Warrant Bonds and deduces the pricing formula of Warrant Bonds by utilizing risk-neutral valuation theory. Finally, this paper analyzes influence of concerned parameters of pricing model on the value of Warrant...

A mathematical model to price convertible bonds involving mixed fractional Brownian motion with jumps is presented. We obtain a general pricing formula using the risk neutral pricing principle and quasi-conditional expectation. The sensitivity of the price to changing various parameters is discussed. Theoretical prices from our jump mixed fractional Brownian motion model are compared with the p...

In this paper we develop the spectral theory of the fractional Brownian motion (fBm) using the ideas of Krein’s work on continuous analogous of orthogonal polynomials on the unit circle. We exhibit the functions which are orthogonal with respect to the spectral measure of the fBm and obtain an explicit reproducing kernel in the frequency domain. We use these results to derive an extension of th...

The goal of this paper is to define and study a notion of fractional Brownian motion on a Lie group. We define it as at the solution of a stochastic differential equation driven by a linear fractional Brownian motion. We show that this process has stationary increments and satisfies a local self-similar property. Furthermore the Lie groups for which this self-similar property is global are char...

We analyze the fractal behavior of the high frequency part of the Fourier spectrum of fBm using multifractal analysis and show that it is not consistent with what is measured on real traac traces. We propose two extensions of fBm which come closer to actual traac traces multifractal properties.