scholarly journals Running supremum of Brownian motion in dimension 2: exact and asymptotic results

2021 ◽  
pp. 1-14
Author(s):  
Krzysztof Kȩpczyński
Keyword(s):  
Brownian Motion ◽  
Asymptotic Results ◽  
Dimension 2

Some asymptotic results for exponential functionals of Brownian motion

1995 ◽  
Vol 32 (04) ◽  
pp. 930-940 ◽  
Author(s):  
J.-C. Gruet ◽  
Z. Shi
Keyword(s):  
Brownian Motion ◽  
Upper Class ◽  
Financial Mathematics ◽  
Asymptotic Results ◽  
Integral Test ◽  
Iterated Logarithm ◽  
Planar Brownian Motion ◽  
The Law ◽  
Straightforward Consequence

The study of exponential functionals of Brownian motion has recently attracted much attention, partly motivated by several problems in financial mathematics. Let be a linear Brownian motion starting from 0. Following Dufresne (1989), (1990), De Schepper and Goovaerts (1992) and De Schepper et al. (1992), we are interested in the process (for δ > 0), which stands for the discounted values of a continuous perpetuity payment. We characterize the upper class (in the sense of Paul Lévy) of X, as δ tends to zero, by an integral test. The law of the iterated logarithm is obtained as a straightforward consequence. The process exp(W(u))du is studied as well. The class of upper functions of Z is provided. An application to the lim inf behaviour of the winding clock of planar Brownian motion is presented.

Some asymptotic results for exponential functionals of Brownian motion

1995 ◽  
Vol 32 (4) ◽  
pp. 930-940 ◽  
Author(s):  
J.-C. Gruet ◽  
Z. Shi
Keyword(s):  
Brownian Motion ◽  
Upper Class ◽  
Financial Mathematics ◽  
Asymptotic Results ◽  
Integral Test ◽  
Iterated Logarithm ◽  
Planar Brownian Motion ◽  
The Law ◽  
Straightforward Consequence

The study of exponential functionals of Brownian motion has recently attracted much attention, partly motivated by several problems in financial mathematics. Let be a linear Brownian motion starting from 0. Following Dufresne (1989), (1990), De Schepper and Goovaerts (1992) and De Schepper et al. (1992), we are interested in the process (for δ > 0), which stands for the discounted values of a continuous perpetuity payment. We characterize the upper class (in the sense of Paul Lévy) of X, as δ tends to zero, by an integral test. The law of the iterated logarithm is obtained as a straightforward consequence. The process exp(W(u))du is studied as well. The class of upper functions of Z is provided. An application to the lim inf behaviour of the winding clock of planar Brownian motion is presented.

Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion

2017 ◽  
Vol 17 (02) ◽  
pp. 1750013 ◽  
Author(s):  
Yong Xu ◽  
Bin Pei ◽  
Jiang-Lun Wu
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Stochastic Averaging ◽  
Hurst Parameter ◽  
Stochastic Integrals ◽  
Averaging Principle ◽  
Mean Square ◽  
Asymptotic Results ◽  
Different Types

In this paper, we are concerned with the stochastic averaging principle for stochastic differential equations (SDEs) with non-Lipschitz coefficients driven by fractional Brownian motion (fBm) of the Hurst parameter [Formula: see text]. We define the stochastic integrals with respect to the fBm in the integral formulation of the SDEs as pathwise integrals and we adopt the non-Lipschitz condition proposed by Taniguchi (1992) which is a much weaker condition with wider range of applications. The averaged SDEs are established. We then use their corresponding solutions to approximate the solutions of the original SDEs both in the sense of mean square and of probability. One can find that the similar asymptotic results are suitable for those non-Lipschitz SDEs with fBm under different types of stochastic integrals.

SUPER BROWNIAN MOTION IS A FRACTAL MEASURE FOR WHICH THE MULTIFRACTAL FORMALISM IS INVALID

Fractals ◽  
1995 ◽  
Vol 03 (04) ◽  
pp. 737-746 ◽  
Author(s):  
S. JAMES TAYLOR
Keyword(s):  
Brownian Motion ◽  
Hausdorff Dimension ◽  
Positive Constant ◽  
Packing Measure ◽  
Borel Set ◽  
Multifractal Formalism ◽  
Fractal Measure ◽  
Measure Function ◽  
Super Brownian Motion ◽  
Dimension 2

Whenever Xt is the measure value of super Brownian motion in Rd(d≥3), and [Formula: see text]St is the topological support of Xt it is known7 that there is a positive constant c, depending only on d, such that for every Borel set A, [Formula: see text] There is no such exact measure function for packing measure, but it follows from the precise results in Ref. 16 that the packing dimension as well as the Hausdorff dimension of St is 2. This means that Xt is dimension regular with exact dimension 2. We describe some of the key ideas, written up in Ref. 24, which show that a.s. [Formula: see text] while [Formula: see text] is not empty for 2<β<4. Further Aβ is not dimension regular since dim Aβ=[Formula: see text]. For this reason the multifractal formalism used in Ref. 11 or Ref. 9 is invalid because their function τ(q) cannot exist.

Integrated Fractional white Noise as an Alternative to Multifractional Brownian Motion

2007 ◽  
Vol 44 (02) ◽  
pp. 393-408 ◽  
Author(s):  
Allan Sly
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
White Noise ◽  
Gaussian Process ◽  
Discrete Data ◽  
Hölder Exponent ◽  
Scaling Properties ◽  
Multifractional Brownian Motion ◽  
Local Properties

Multifractional Brownian motion is a Gaussian process which has changing scaling properties generated by varying the local Hölder exponent. We show that multifractional Brownian motion is very sensitive to changes in the selected Hölder exponent and has extreme changes in magnitude. We suggest an alternative stochastic process, called integrated fractional white noise, which retains the important local properties but avoids the undesirable oscillations in magnitude. We also show how the Hölder exponent can be estimated locally from discrete data in this model.

Brownian motion with quadratic killing and some implications

1986 ◽  
Vol 23 (04) ◽  
pp. 893-903 ◽  
Author(s):  
Michael L. Wenocur
Keyword(s):  
Brownian Motion ◽  
Weibull Distribution ◽  
Special Function ◽  
Function Theory ◽  
Killing Rate

Brownian motion subject to a quadratic killing rate and its connection with the Weibull distribution is analyzed. The distribution obtained for the process killing time significantly generalizes the Weibull. The derivation involves the use of the Karhunen–Loève expansion for Brownian motion, special function theory, and the calculus of residues.

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