Georgiy Shevchenko

The rate of convergence for Euler approximations of 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 , where δ is the diameter of partition used for discretization. For discrete-time approximations of Skorohod-type quasilinear equation driven by fBm we prove that the rate of convergence is .

Existence and Uniqueness of the Solution of Stochastic Differential Equation Involving Wiener Process and Fractional Brownian Motion with Hurst Index H > 1/2

We consider a mixed stochastic differential equation driven by possibly dependent fractional Brownian motion and Brownian motion. Under mild regularity assumptions on the coefficients, it is proved that the equation has a unique solution.

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 Holder continuous process Z with Holder exponent @c>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.

Asymptotic Properties of Drift Parameter Estimator Based on Discrete Observations of Stochastic Differential Equation Driven by Fractional Brownian Motion

In this chapter, we consider a problem of statistical estimation of an unknown drift parameter for a stochastic differential equation driven by fractional Brownian motion. Two estimators based on discrete observations of solution to the stochastic differential equations are constructed. It is proved that the estimators converge almost surely to the parameter value, as the observation interval expands and the distance between observations vanishes. A bound for the rate of convergence is given and numerical simulations are presented. As an auxilliary result of independent interest we establish global estimates for fractional derivative of fractional Brownian motion.

MIXED STOCHASTIC DELAY DIFFERENTIAL EQUATIONS

We consider a stochastic delay differential equation driven by a Holder continuous process and a Wiener process. Under fairly general assumptions on its coefficients, we prove that this equation is uniquely solvable. We also give sufficient conditions for finiteness of its moments and establish a limit theorem.

Random variables as pathwise integrals with respect to fractional Brownian motion

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

Mixed fractional stochastic differential equations with jumps

In this paper, we consider a stochastic differential equation driven by a fractional Brownian motion and a Wiener process and having jumps. We prove that this equation has a unique solution and show that all moments of the solution are finite.

Real harmonizable multifractional stable process and its local properties

A real harmonizable multifractional stable process is defined, its Holder continuity and localizability are proved. The existence of local time is shown and its regularity is established.

Malliavin regularity of solutions to mixed stochastic differential equations

For a mixed stochastic differential equation driven by independent fractional Brownian motions and Wiener processes, the existence and integrability of the Malliavin derivative of the solution are established. It is also proved that the solution possesses exponential moments.

Adapted integral representations of random variables

We study integral representations of random variables with respect to general Holder continuous processes and with respect to two particular cases; fractional Brownian motion and mixed fractional Brownian motion. We prove that an arbitrary random variable can be represented as an improper integral, and that the stochastic integral can have any distribution. If in addition the random variable is a final value of an adapted Holder continuous process, then it can be represented as a proper integral. It is also shown that in the particular case of mixed fractional Brownian motion, any adapted random variable can be represented as a proper integral.