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Yuliya Mishura

Publications and source records attributed to Yuliya Mishura.

100 records · Page 6Linked to original sources
arXiv2013

On the distribution of local times and integral functionals of a homogeneous diffusion process

In this article we study a homogeneous transient diffusion process $X$. We combine the theories of differential equations and of stochastic processes to obtain new results for homogeneous diffusion processes, generalizing the results of Salminen and Yor. The distribution of local time of $X$ is found in a closed form. To this end, a second order differential equation corresponding to the generator of $X$ is considered, and properties of its monotone solutions as functions of a parameter are established using their probabilistic representations. We also provide expressions and upper bounds for moments, exponential moments, and potentials of integral functionals of $X$.

math.PR↗
arXiv2012

Mixed stochastic differential equations with long-range dependence: existence, uniqueness and convergence of solutions

For a mixed stochastic differential equation involving standard Brownian motion and an almost surely Hölder continuous process $Z$ with Hölder exponent $γ>1/2$, we establish a new result on its unique solvability. We also establish an estimate for difference of solutions to such equations with different processes $Z$ and deduce a corresponding limit theorem. As a by-product, we obtain a result on existence of moments of a solution to a mixed equation under an assumption that $Z$ has certain exponential moments.

math.PR↗
arXiv2012

Random variables as pathwise integrals with respect to fractional Brownian motion

We show that a pathwise stochastic integral with respect to fractional Brownian motion with an adapted integrand $g$ can have any prescribed distribution, moreover, we give both necessary and sufficient conditions when random variables can be represented in this form. We also prove that any random variable is a value of such integral in some improper sense. We discuss some applications of these results, in particular, to fractional Black--Scholes model of financial market.

math.PR↗
arXiv2012

Approximation of fractional Brownian motion by martingales

We study the problem of optimal approximation of a fractional Brownian motion by martingales. We prove that there exist a unique martingale closest to fractional Brownian motion in a specific sense. It shown that this martingale has a specific form. Numerical results concerning the approximation problem are given.

math.PR↗
arXiv2011

On drift parameter estimation in models with fractional Brownian motion

We consider a stochastic differential equation involving standard and fractional Brownian motion with unknown drift parameter to be estimated. We investigate the standard maximum likelihood estimate of the drift parameter, two non-standard estimates and three estimates for the sequential estimation. Model strong consistency and some other properties are proved. The linear model and Ornstein-Uhlenbeck model are studied in detail. As an auxiliary result, an asymptotic behavior of the fractional derivative of the fractional Brownian motion is established.

math.PR↗
arXiv2011

An extension of the Lévy characterization to fractional Brownian motion

Assume that $X$ is a continuous square integrable process with zero mean, defined on some probability space $(Ω,\mathrm {F},\mathrm {P})$. The classical characterization due to P. Lévy says that $X$ is a Brownian motion if and only if $X$ and $X_t^2-t$, $t\ge0,$ are martingales with respect to the intrinsic filtration $\mathrm {F}^X$. We extend this result to fractional Brownian motion.

math.PR↗
arXiv2010

Fractional Lévy processes as a result of compact interval integral transformation

Fractional Brownian motion can be represented as an integral of a deterministic kernel w.r.t. an ordinary Brownian motion either on infinite or compact interval. In previous literature fractional Lévy processes are defined by integrating the infinite interval kernel w.r.t. a general Lévy process. In this article we define fractional Lévy processes using the compact interval representation. We prove that the fractional Lévy processes presented via different integral transformations have the same finite dimensional distributions if and only if they are fractional Brownian motions. Also, we present relations between different fractional Lévy processes and analyze the properties of such processes. A financial example is introduced as well.

math.PR↗
arXiv2008

The rate of convergence of Euler approximations for solutions of stochastic differential equations driven by fractional Brownian motion

The paper focuses on discrete-type approximations of solutions to non-homogeneous stochastic differential equations (SDEs) involving fractional Brownian motion (fBm). We prove that the rate of convergence for Euler approximations of solutions of pathwise SDEs driven by fBm with Hurst index $H>1/2$ can be estimated by $O(δ^{2H-1})$ ($δ$ is the diameter of partition). For discrete-time approximations of Skorohod-type quasilinear equation driven by fBm we prove that the rate of convergence is $O(δ^H)$.

math.PR↗