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

Abstract

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.

Global variational solutions to a class of fractional SPDE’s on unbounded domains

“…Moreover, there have also been several more recent articles dealing with the analysis of solutions to various types of semilinear parabolic stochastic partial differential equations driven either by a Brownian noise, or by a fractional noise with Hurst parameter H ∈ 1 2 , 1 (see, e.g., [3], [4], [5]- [7], and the plethora of references therein, particularly [11]). While these works have been primarily centered around questions of global existence, uniqueness and blowup in finite time, there have also been investigations essentially motivated by issues in financial mathematics devoted to the analysis of problems that involve a mixture of a Brownian noise with a fractional noise, within the realm of both ordinary and partial stochastic differential equations (see, e.g., [9], [13]- [15] and the references therein).…”

“…(48) Therefore, with (47) and (48) into (44) we obtain according to (13) and (49), with the obvious choice for r H α . But (50) follows from Relation (35) of Proposition A.1.…”

Convergence of solutions of mixed stochastic delay differential equations with applications

“…The existence and uniqueness of solution to equation (1) was established in [15], where also the finiteness of moments was shown under the additional assumption that the coefficient b is bounded. Mixed equations without delay were considered in articles [6,7,11,12,16,17].…”

Integrability of Solutions to Mixed Stochastic Differential Equations

“…Due to such twofold nature of the randomness, equation (1) is called a mixed stochastic differential equation. Existence and uniqueness of solution to a mixed stochastic diffential equation (1) were proved under different conditions in [2,4,5,6,7]. More generally, mixed equations with jumps were considered in [8] and mixed delay equations, in [9].…”