Asymptotic variance parameters for the boundary local times of reflected Brownian motion on a compact interval

A direct derivation is given of a formula for the normalized asymptotic variance parameters of the boundary local times of reflected Brownian motion (with drift) on a compact interval. This formula was previously obtained by Berger and Whitt using an M/M/1/C queue approximation to the reflected Brownian motion. The bivariate Laplace transform of the hitting time of a level and the boundary local time up to that hitting time, for a one-dimensional reflected Brownian motion with drift, is obtained as part of the derivation.

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Occupation Times, Drawdowns, and Drawups for One-Dimensional Regular Diffusions

Advances in Applied Probability ◽

10.1239/aap/1427814588 ◽

2015 ◽

Vol 47 (1) ◽

pp. 210-230 ◽

Cited By ~ 18

Author(s):

Hongzhong Zhang

Keyword(s):

Brownian Motion ◽

Diffusion Process ◽

Time Horizon ◽

Three Dimensional ◽

Bessel Processes ◽

Occupation Times ◽

Exponential Time ◽

One Dimensional ◽

Brownian Motion With Drift ◽

Running Maximum

The drawdown process of a one-dimensional regular diffusion process X is given by X reflected at its running maximum. The drawup process is given by X reflected at its running minimum. We calculate the probability that a drawdown precedes a drawup in an exponential time-horizon. We then study the law of the occupation times of the drawdown process and the drawup process. These results are applied to address problems in risk analysis and for option pricing of the drawdown process. Finally, we present examples of Brownian motion with drift and three-dimensional Bessel processes, where we prove an identity in law.

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The Hitting Time Density for a Reflected Brownian Motion

Computational Economics ◽

10.1007/s10614-011-9264-0 ◽

2011 ◽

Vol 40 (1) ◽

pp. 1-18 ◽

Cited By ~ 15

Author(s):

Qin Hu ◽

Yongjin Wang ◽

Xuewei Yang

Keyword(s):

Brownian Motion ◽

Hitting Time ◽

Reflected Brownian Motion ◽

Time Density

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The first rendezvous time of Brownian motion and compound Poisson-type processes

Journal of Applied Probability ◽

10.1017/s0021900200020829 ◽

2004 ◽

Vol 41 (04) ◽

pp. 1059-1070 ◽

Cited By ~ 4

Author(s):

D. Perry ◽

W. Stadje ◽

S. Zacks

Keyword(s):

Brownian Motion ◽

Stochastic Processes ◽

Poisson Process ◽

Compound Poisson Process ◽

Reflected Brownian Motion ◽

Compound Poisson ◽

Brownian Motion With Drift ◽

First Time ◽

Random Jump ◽

Poisson Type

The ‘rendezvous time’ of two stochastic processes is the first time at which they cross or hit each other. We consider such times for a Brownian motion with drift, starting at some positive level, and a compound Poisson process or a process with one random jump at some random time. We also ask whether a rendezvous takes place before the Brownian motion hits zero and, if so, at what time. These questions are answered in terms of Laplace transforms for the underlying distributions. The analogous problem for reflected Brownian motion is also studied.

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Passage-time moments for continuous non-negative stochastic processes and applications

Advances in Applied Probability ◽

10.1017/s0001867800046474 ◽

1996 ◽

Vol 28 (03) ◽

pp. 747-762

Author(s):

M. Menshikov ◽

R. J. Williams

Keyword(s):

Brownian Motion ◽

Stochastic Processes ◽

Discrete Time ◽

Passage Time ◽

Reflected Brownian Motion ◽

One Dimensional

We give criteria for the finiteness or infiniteness of the passage-time moments for continuous non-negative stochastic processes in terms of sub/supermartingale inequalities for powers of these processes. We apply these results to one-dimensional diffusions and also reflected Brownian motion in a wedge. The discrete-time analogue of this problem was studied previously by Lamperti and more recently by Aspandiiarov, Iasnogorodski and Menshikov [2]. Our results are continuous analogues of those in [2], but our proofs are direct and do not rely on approximation by discrete-time processes.

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Passage-time moments for continuous non-negative stochastic processes and applications

Advances in Applied Probability ◽

10.2307/1428179 ◽

1996 ◽

Vol 28 (3) ◽

pp. 747-762 ◽

Cited By ~ 9

Author(s):

M. Menshikov ◽

R. J. Williams

Keyword(s):

Brownian Motion ◽

Stochastic Processes ◽

Discrete Time ◽

Passage Time ◽

Reflected Brownian Motion ◽

One Dimensional

We give criteria for the finiteness or infiniteness of the passage-time moments for continuous non-negative stochastic processes in terms of sub/supermartingale inequalities for powers of these processes. We apply these results to one-dimensional diffusions and also reflected Brownian motion in a wedge. The discrete-time analogue of this problem was studied previously by Lamperti and more recently by Aspandiiarov, Iasnogorodski and Menshikov [2]. Our results are continuous analogues of those in [2], but our proofs are direct and do not rely on approximation by discrete-time processes.

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The first rendezvous time of Brownian motion and compound Poisson-type processes

Journal of Applied Probability ◽

10.1239/jap/1101840551 ◽

2004 ◽

Vol 41 (4) ◽

pp. 1059-1070 ◽

Cited By ~ 6

Author(s):

D. Perry ◽

W. Stadje ◽

S. Zacks

Keyword(s):

Brownian Motion ◽

Stochastic Processes ◽

Poisson Process ◽

Compound Poisson Process ◽

Reflected Brownian Motion ◽

Compound Poisson ◽

Brownian Motion With Drift ◽

First Time ◽

Random Jump ◽

Poisson Type

The ‘rendezvous time’ of two stochastic processes is the first time at which they cross or hit each other. We consider such times for a Brownian motion with drift, starting at some positive level, and a compound Poisson process or a process with one random jump at some random time. We also ask whether a rendezvous takes place before the Brownian motion hits zero and, if so, at what time. These questions are answered in terms of Laplace transforms for the underlying distributions. The analogous problem for reflected Brownian motion is also studied.

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The joint distribution of the hitting time and place to a sphere or spherical shell for Brownian motion with drift

Statistics & Probability Letters ◽

10.1016/s0167-7152(98)00231-4 ◽

1999 ◽

Vol 42 (4) ◽

pp. 367-373 ◽

Cited By ~ 5

Author(s):

Chuancun Yin

Keyword(s):

Brownian Motion ◽

Spherical Shell ◽

Joint Distribution ◽

Hitting Time ◽

Brownian Motion With Drift

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Hitting Time and Place of Brownian Motion with Drift

The Open Statistics & Probability Journal ◽

10.2174/1876527000901010038 ◽

2009 ◽

Vol 1 (1) ◽

pp. 38-42

Author(s):

Chuancun Yin ◽

Chunwei Wang

Keyword(s):

Brownian Motion ◽

Hitting Time ◽

Brownian Motion With Drift

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Parisian Time of Reflected Brownian Motion with Drift on Rays and Its Application in Banking

Risks ◽

10.3390/risks8040127 ◽

2020 ◽

Vol 8 (4) ◽

pp. 127

Author(s):

Angelos Dassios ◽

Junyi Zhang

Keyword(s):

Brownian Motion ◽

Laplace Transform ◽

Liquidity Risk ◽

Finite Collection ◽

Recursive Method ◽

Simulation Algorithm ◽

Reflected Brownian Motion ◽

Brownian Motion With Drift ◽

The Laplace Transform ◽

First Time

In this paper, we study the Parisian time of a reflected Brownian motion with drift on a finite collection of rays. We derive the Laplace transform of the Parisian time using a recursive method, and provide an exact simulation algorithm to sample from the distribution of the Parisian time. The paper was motivated by the settlement delay in the real-time gross settlement (RTGS) system. Both the central bank and the participating banks in the system are concerned about the liquidity risk, and are interested in the first time that the duration of settlement delay exceeds a predefined limit. We reduce this problem to the calculation of the Parisian time. The Parisian time is also crucial in the pricing of Parisian type options; to this end, we will compare our results to the existing literature.

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