Central limit theorem for the third moment in space of the Brownian local time increments

Hu, Yaozhong; Nualart, David

doi:10.1214/ecp.v15-1573

Cited by 8 publications

(11 citation statements)

References 12 publications

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“…This process has been further studied in [Mar08b,JM12,HN10] as well as some of their references. An FBM version of (1.3) was first considered in the works of [YYL08,YLY09].…”

Section: 3)mentioning

confidence: 99%

Hölder Continuity and Occupation-Time Formulas for fBm Self-Intersection Local Time and Its Derivative

Jung

Markowsky

2013

J Theor Probab

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We prove joint Hölder continuity and an occupation-time formula for the self-intersection local time of fractional Brownian motion. Motivated by an occupation-time formula, we also introduce a new version of the derivative of self-intersection local time for fractional Brownian motion and prove Hölder conditions for this process. This process is related to a different version of the derivative of self-intersection local time studied by the authors in a previous work.

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“…This process has been further studied in [Mar08b,JM12,HN10] as well as some of their references. An FBM version of (1.3) was first considered in the works of [YYL08,YLY09].…”

Section: 3)mentioning

confidence: 99%

Hölder Continuity and Occupation-Time Formulas for fBm Self-Intersection Local Time and Its Derivative

Jung

Markowsky

2013

J Theor Probab

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“…We will see later, in fact, that neither (1.4) nor (1.5) is correct, and that the deterministic term required is t 2 . Using (1.4) as motivation, Rosen [Ros05] showed the existence of a process α ′ t (y) now known as the derivative of self-intersection local time (DSLT) for B t which is formally defined as This process was later used in [HN09] and [HN10] to prove central limit theorems for the L 2 and L 3 moduli of continuity of Brownian local time.…”

Section: Introductionmentioning

confidence: 99%

On the Tanaka formula for the derivative of self-intersection local time of fractional Brownian motion

Jung

Markowsky

2014

Stochastic Processes and their Applications

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The derivative of self-intersection local time (DSLT) for Brownian motion was introduced by Rosen [Ros05] and subsequently used by others to study the L 2 and L 3 moduli of continuity of Brownian local time. A version of the DSLT for fractional Brownian motion (fBm) was introduced in [YYL08]; however, the definition given there presents difficulties, since it is motivated by an incorrect application of Itô's formula. To rectify this, we introduce a modified DSLT for fBm and prove existence using an explicit Wiener chaos expansion. We will then argue that our modification is the natural version of the DSLT by rigorously proving the corresponding Tanaka formula. This formula corrects a formal identity given in both [Ros05] and [YYL08]. In the course of this endeavor we prove a Fubini theorem for integrals with respect to fBm. The Fubini theorem may be of independent interest, as it generalizes (to Hida distributions) similar results previously seen in the literature. As a further byproduct of our investigation, we also provide a correction to an important technical secondmoment bound for fBm given in [Hu01].

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“…It is worthwhile noting that the equivalent of the main result of [7], namely the convergence in law of a suitably renormalized version of H h t (B), is not available in the two-dimensional case. Indeed, one can formally show that |h|…”

Section: Modulus Of Continuity Of Brownian Local Timesmentioning

confidence: 99%

“…Similar to the argument used in [6,7,13], our strategy toward ( 19) is now based on the martingale convergence criterion summed up in [4], Theorem A.1. Using the latter result, the proof of (19) reduces to showing that, as h → 0, we have simultaneously…”

Section: Study Of the Martingale Termmentioning

confidence: 99%

“…Let us quote, for instance, the use of Malliavin and stochastic calculus tools in order to get suitably normalized limits for L 2 modulus of continuity (see [6,13]) or third moment in space (cf. [7]) of Brownian local time. Malliavin calculus tools have also been essential in order to generalize the notion of self-intersection local time [5,8] and to obtain central limit theorems for additive functionals [9] of fractional Brownian motion.…”

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confidence: 99%

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On L2 modulus of continuity of Brownian local times and Riesz potentials

Deya¹,

Nualart²,

Tindel³

2015

Ann. Probab.

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This article is concerned with modulus of continuity of Brownian local times. Specifically, we focus on three closely related problems: (a) Limit theorem for a Brownian modulus of continuity involving Riesz potentials, where the limit law is an intricate Gaussian mixture. (b) Central limit theorems for the projections of L 2 modulus of continuity for a one-dimensional Brownian motion. (c) Extension of the second result to a two-dimensional Brownian motion. Our proofs rely on a combination of stochastic calculus and Malliavin calculus tools, plus a thorough analysis of singular integrals.1. Introduction. Let {B t , 0 ≤ t ≤ 1} be a standard linear Brownian motion defined on some complete probability space (Ω, F, P). In the sequel, we denote by L t (x) the local time of B at a given point x ∈ R, defined for t ∈ [0, 1]. A nice combination of stochastic calculus, stochastic analysis and evaluation of singularities associated with heat kernels have recently led to a number of interesting limit theorems for quantities related to the family {L t (x); t ∈ [0, 1], x ∈ R}. Let us quote, for instance, the use of Malliavin and stochastic calculus tools in order to get suitably normalized limits for L 2 modulus of continuity (see [6,13]) or third moment in space (cf. [7]) of Brownian local time. Malliavin calculus tools have also been essential in order to generalize the notion of self-intersection local time [5,8] and to obtain central limit theorems for additive functionals [9] of fractional Brownian motion.The current article proposes to take another step into the relationships between Brownian local time and stochastic analysis. Specifically, we shall handle the following problems:

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Central limit theorem for the third moment in space of the Brownian local time increments

Cited by 8 publications

References 12 publications

Hölder Continuity and Occupation-Time Formulas for fBm Self-Intersection Local Time and Its Derivative

Hölder Continuity and Occupation-Time Formulas for fBm Self-Intersection Local Time and Its Derivative

On the Tanaka formula for the derivative of self-intersection local time of fractional Brownian motion

On L2 modulus of continuity of Brownian local times and Riesz potentials

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