Excursion theory for Brownian motion indexed by the Brownian tree

In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area)(T) covered by the process in the time interval [0,T]. The Laplace transform of(T) follows as a consequence. The main proof involves a double Laplace transform of(T) and is based on excursion theory and local time for Brownian motion.

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Excursions height- and length-related stopping times, and application to finance

Advances in Applied Probability ◽

10.1239/aap/1037990956 ◽

2002 ◽

Vol 34 (4) ◽

pp. 846-868 ◽

Cited By ~ 12

Author(s):

Laurent Gauthier

Keyword(s):

Brownian Motion ◽

Laplace Transform ◽

Stopping Time ◽

Stopping Times ◽

Excursion Theory ◽

Delay Constraint ◽

Investment Projects ◽

The Laplace Transform

In this paper, we study the first instant when Brownian motion either spends consecutively more than a certain time above a certain level, or reaches another level. This stopping time generalizes the ‘Parisian’ stopping times that were introduced by Chesneyet al.(1997). Using excursion theory, we derive the Laplace transform of this stopping time. We apply this result to the valuation of investment projects with a delay constraint, but with an alternative: pay a higher cost and get the project started immediately

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The Integral of the Supremum Process of Brownian Motion

Journal of Applied Probability ◽

10.1017/s0021900200005672 ◽

2009 ◽

Vol 46 (02) ◽

pp. 593-600

Author(s):

Svante Janson ◽

Niclas Petersson

Keyword(s):

Brownian Motion ◽

Laplace Transform ◽

Explicit Formula ◽

Local Time ◽

Time Interval ◽

Standard Brownian Motion ◽

Excursion Theory ◽

Double Laplace Transform ◽

The Laplace Transform ◽

Supremum Process

In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area)(T) covered by the process in the time interval [0,T]. The Laplace transform of(T) follows as a consequence. The main proof involves a double Laplace transform of(T) and is based on excursion theory and local time for Brownian motion.

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Excursions height- and length-related stopping times, and application to finance

Advances in Applied Probability ◽

10.1017/s0001867800011940 ◽

2002 ◽

Vol 34 (04) ◽

pp. 846-868 ◽

Cited By ~ 2

Author(s):

Laurent Gauthier

Keyword(s):

Brownian Motion ◽

Laplace Transform ◽

Stopping Time ◽

Stopping Times ◽

Excursion Theory ◽

Delay Constraint ◽

Investment Projects ◽

The Laplace Transform

In this paper, we study the first instant when Brownian motion either spends consecutively more than a certain time above a certain level, or reaches another level. This stopping time generalizes the ‘Parisian’ stopping times that were introduced by Chesney et al. (1997). Using excursion theory, we derive the Laplace transform of this stopping time. We apply this result to the valuation of investment projects with a delay constraint, but with an alternative: pay a higher cost and get the project started immediately

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Integrated Fractional white Noise as an Alternative to Multifractional Brownian Motion

Journal of Applied Probability ◽

10.1017/s0021900200003041 ◽

2007 ◽

Vol 44 (02) ◽

pp. 393-408 ◽

Cited By ~ 1

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.

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Brownian motion with quadratic killing and some implications

Journal of Applied Probability ◽

10.1017/s0021900200116079 ◽

1986 ◽

Vol 23 (04) ◽

pp. 893-903 ◽

Cited By ~ 2

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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