Stochastic differential equations driven by fractional Brownian motions

Abstract

In this paper, we study the existence and uniqueness of a class of stochastic differential equations driven by fractional Brownian motions with arbitrary Hurst parameter $H\in (0,1)$. In particular, the stochastic integrals appearing in the equations are defined in the Skorokhod sense on fractional Wiener spaces, and the coefficients are allowed to be random and even anticipating. The main technique used in this work is an adaptation of the anticipating Girsanov transformation of Buckdahn [Mem. Amer. Math. Soc. 111 (1994)] for the Brownian motion case. By extending a fundamental theorem of Kusuoka [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982) 567--597] using fractional calculus, we are able to prove that the anticipating Girsanov transformation holds for the fractional Brownian motion case as well. We then use this result to prove the well-posedness of the SDE.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ169 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Semilinear Backward Doubly Stochastic Differential Equations and SPDEs Driven by Fractional Brownian Motion with Hurst Parameter in (0,1/2)

“…In this subsection we discuss the existence and uniqueness of solutions to anticipating semilinear equations driven by a fractional Brownian motion B with Hurst parameter H ∈ (0, 1/2). This type of equation was studied by Jien and Ma [10], and since it motivates the approach in our work, we give it in details. We consider the fractional anticipating equation…”

“…Nevertheless, there are many papers considering stochastic differential equations driven by fractional Brownian motion with Hurst parameter H > 1/2 ( [2], [15] and references therein) or H < 1/2 ( [13]), or covering both cases ( [10]). For the case H < 1/2, one of the main difficulties is how to properly define the stochastic integral with respect to the fBm.…”

“…and Y t , Z t , t ∈ [0, T ] = Y t (T t )ε −1 t (T t ), Z t (T t )ε −1 t (T t ) , t ∈ [0, T ] , where A t and T t are Girsanov transformations. It is worth noting that such kind of method was also used by Jien and Ma [10] to deal with fractional stochastic differential equations.…”

Increasing Domain Infill Asymptotics for Stochastic Differential Equations Driven by Fractional Brownian Motion

“…Longrange dependence results when H > 1/2. Detailed of fractional Brownian motion and that of SDEs driven by fractional Brownian motion can be found in Norros et al (1999), Biagini et al (2008), Mishura (2008), Jien and Ma (2009), Xu and Luo (2018). Statistical inference, including asymptotic theory in such SDEs, assuming that H > 1/2 and known, is detailed in Rao (2010); see also Mishura and Ralchenko (2014), Neuenkirch and Tindel (2014), Hu et al (2017), where again H is assumed to be known.…”

Anticipating linear stochastic differential equations driven by a Lévy process

“…In particular, he showed that Itô formula is not useful in this case. This approach has been also useful to deal with fractional stochastic differential equations (see [12,13]).…”