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.
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 ).
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.
Abstract. 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.
IntroductionFractional Brownian motion (fBm) with a Hurst parameter H ∈ (0, 1) is defined formally as a continuous centered Gaussian processit exhibits a property of long-range dependence, which makes it a popular model for long-range dependence in natural sciences, financial mathematics etc. For this reason, equations driven by fractional Brownian motion have been an object of intensive study during the last decade.There are two principal ways to define an integral with respect to fractional Brownian motion.One possibility is Skorokhod, or divergence integral introduced in the fractional Brownian setting in [3]. However this definition is not very practical: it is based on Wick rather than usual products, and unlike Brownian case, in the fractional Brownian case this makes difference when integrating non-anticipating functions because of dependence of increments. This makes this definition worthless for most applications (most notably, those in financial mathematics). Moreover, it is impossible to solve stochastic differential equations with such integral except the cases of additive or multiplicative noise; the latter case was considered in [9].Another approach is a pathwise integral, defined first in [16] for fBm with H > 1/2 as a Young integral. The papers [7,13,14] were the first to prove existence and uniqueness of stochastic differential equations involving such integrals. Later the pathwise approach was extended with the help of Lyons' rough path theory to the case of arbitrary H in [1] where also unique solvability of equations with H > 1/4 was proved. Numerical methods for pathwise stochastic differential equations with fBm were considered in [11,12,2,4].In this paper we focus on the following mixed stochastic differential equation involving Wiener process and fractional Brownian motion with Hurst index H ∈ 2010 Mathematics Subject Classification. 60G15; 60G22; 60H10; 60J65.
Abstract. This is an extended version of the lecture notes to a mini-course devoted to fractional Brownian motion and delivered to the participants of 7th Jagna International Workshop.
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