Stochastic calculus with respect to fractional Brownian motion

Nualart, David

doi:10.5802/afst.1113

Cited by 52 publications

(54 citation statements)

References 24 publications

(12 reference statements)

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“…By plugging inequality (28)- (29), for δ > 0 small enough, into (30) and by recalling that δX 2,ε = δ X ε ⊗ δ X ε and inequality (27), we obtain easily the second part of (21).…”

Section: Tightness Inmentioning

confidence: 98%

Weak approximation of a fractional SDE

Bardina

Nourdin²,

Rovira³

et al. 2010

Stochastic Processes and their Applications

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32 pages; this is a major revision, with two additional co-authors (X. Bardina and C. Rovira)International audienceIn this note, a diffusion approximation result is shown for stochastic differential equations driven by a (Liouville) fractional Brownian motion B with Hurst parameter H in (1/3,1/2). More precisely, we resort to the Kac-Stroock type approximation using a Poisson process studied in Bardina, Jolis and Tudor (2003) and Delgado and Jolis (2000), and our method of proof relies on the algebraic integration theory introduced by Gubinelli (2004)

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“…By plugging inequality (28)- (29), for δ > 0 small enough, into (30) and by recalling that δX 2,ε = δ X ε ⊗ δ X ε and inequality (27), we obtain easily the second part of (21).…”

Section: Tightness Inmentioning

confidence: 98%

Weak approximation of a fractional SDE

Bardina

Nourdin²,

Rovira³

et al. 2010

Stochastic Processes and their Applications

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show abstract

“…Another formula, similar to (52), also holds for standard fBm, with, instead of the quantity √ 2H (t − u) H −1/2 , a more complicated kernel K(t, u); this is often called the kernel representation of fBm. See [18] for details. It is well known that B f H has stationary increments, that is, for any t, h ∈ R + , …”

Section: Appendix a Riemann-liouville Fractional Brownian Motionmentioning

confidence: 98%

Almost sure exponential behavior of a directed polymer in a fractional Brownian environment

Viens

Zhang

2008

Journal of Functional Analysis

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This paper studies the asymptotic behavior of a one-dimensional directed polymer in a random medium. The latter is represented by a Gaussian field B H on R + × R with fractional Brownian behavior in time (Hurst parameter H ) and arbitrary function-valued behavior in space. The partition function of such a polymer isHere b is a continuous-time nearest neighbor random walk on Z with fixed intensity 2κ, defined on a complete probability space P b independent of B H . The spatial covariance structure of B H is assumed to be homogeneous and periodic with period 2π . For H < 1 2 , we prove existence and positivity of the Lyapunov exponent defined as the almost sure limit lim t→∞ t −1 log u(t). For H > 1 2 , we prove that the upper and lower almost sure limits lim sup t→∞ t −2H log u(t) and lim inf t→∞ (t −2H log t) log u(t) are non-trivial in the sense that they are bounded respectively above and below by finite, strictly positive constants. Thus, as H passes through 1 2 , the exponential behavior of u(t) changes abruptly. This can be considered as a phase transition phenomenon. Novel tools used in this paper include sub-Gaussian concentration theory via the Malliavin calculus, detailed analyses of the long-range memory of fractional Brownian motion, and an almost-superadditivity property.

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“…A possible ansatz to overcome this fact is to use the Skorohod integral for non-adapted integrands and Malliavin calculus [145]. Another possibility is to exploit the integration by parts formula in order to define (2.18) in a meaningful way.…”

Section: The Quasilinear Casementioning

confidence: 99%

“…The key feature which is required to comprehend the next steps is the Hölder continuity of its trajectories for an exponent α < H. The entire solution theory relies only on this regularity result and is developed within the framework of the rough path approach [134,94,98,76]. Similar to Skorohod integrals [145], rough path techniques also go beyond the semi-martingale case, where Itô calculus and Walsh theory no longer apply. The major difference is that these provide a completely pathwise construction of stochastic integrals and implicitly of the solution of SPDEs.…”

Section: Mild Solutions For Rough Spdesmentioning

confidence: 99%