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# A proof of Itô’s formula using a discrete Itô’s formula

Studia Scientiarum Mathematicarum Hungarica
Authors: Takahiko Fujita and Yasuhiro Kawanishi

In this paper we will prove Itô’s formula for Brownian motion in the case of fC 2 (ℝ), using a discrete Itô’s formula.

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# Strong approximations of three-dimensional Wiener sausages

Acta Mathematica Hungarica
Authors: E. Csáki and Y. Hu

## Abstract

We prove that the centered three-dimensional Wiener sausage can be strongly approximated by a one-dimensional Brownian motion running at a suitable time clock. The strong approximation gives all possible laws of iterated logarithm as well as the convergence in law in terms of process for the normalized Wiener sausage. The proof relies on Le Gall [10]șs fine L 2-norm estimates between the Wiener sausage and the Brownian intersection local times.

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# A functional CLT for the L 2 modulus of continuity of local time

Periodica Mathematica Hungarica
Author: Jay Rosen

## Abstract

We show that as processes in (c, d, t) ∈ C(R 2 × R + 1)

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\frac{{\int_c^d {(L_t^{x + h} - L_t^x )^2 dx - 4h} \int_c^d {L_t^x dx} }} {{h^{3/2} }}\mathop \Rightarrow \limits^\mathcal{L} \left( {\frac{{64}} {3}} \right)^{1/2} \int_c^d {L_t^x d\eta (x)}$$ \end{document}
as h → 0 for Brownian local time L t x. Here η(x) is an independent two-sided Brownian motion.

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# On hitting times of affine boundaries by reflecting Brownian motion and Bessel processes

Periodica Mathematica Hungarica
Authors: Paavo Salminen and Marc Yor

## Abstract

Firstly, we compute the distribution function for the hitting time of a linear time-dependent boundary ta + bt, a ≥ 0, b ∈ ℝ, by a reflecting Brownian motion. The main tool hereby is Doob’s formula which gives the probability that Brownian motion started inside a wedge does not hit this wedge. Other key ingredients are the time inversion property of Brownian motion and the time reversal property of diffusion bridges. Secondly, this methodology can also be applied for the three-dimensional Bessel process. Thirdly, we consider Bessel bridges from 0 to 0 with dimension parameter δ > 0 and show that the probability that such a Bessel bridge crosses an affine boundary is equal to the probability that this Bessel bridge stays below some fixed value.

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# Some extensions of Pitman and Ray-Knight theorems for penalized Brownian motions and their local times, IV

Studia Scientiarum Mathematicarum Hungarica
Authors: Bernard Roynette, Pierre Vallois, and Marc Yor

We show that Pitman’s theorem relating Brownian motion and the BES (3) process, as well as the Ray-Knight theorems for Brownian local times remain valid, mutatis mutandis, under the limiting laws of Brownian motion penalized by a function of its one-sided maximum.

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# Penalizing a BES ( d ) process (0 < d < 2) with a function of its local time, V

Studia Scientiarum Mathematicarum Hungarica
Authors: Bernard Roynette, Pierre Vallois, and Marc Yor

We describe the limit laws, as t → ∞, of a Bessel process ( R s , st ) of dimension d ∈ (0, 2) penalized by an integrable function of its local time L t at 0, thus extending our previous work of this kind, relative to Brownian motion.

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# We like to walk on the comb

Periodica Mathematica Hungarica
Author: Antonia Földes

## Abstract

This is a brief account on how we have entertained ourselves in the last two years, that is, a summary of the results we have obtained in a joint work with E. Csáki, M. Csörgő and P. Révész on random walks on a comb.

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