scholarly journals Level sets and drift estimation for reflected Brownian motion with drift

2020 ◽  
Author(s):  
Alejandro Cholaquidis ◽  
Ricardo Fraiman ◽  
Ernesto Mordecki ◽  
Cecilia Papalardo
Keyword(s):  
Brownian Motion ◽  
Level Sets ◽  
Reflected Brownian Motion ◽  
Brownian Motion With Drift ◽  
Drift Estimation

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

1992 ◽  
Vol 29 (04) ◽  
pp. 996-1002 ◽  
Author(s):  
R. J. Williams
Keyword(s):  
Brownian Motion ◽  
Asymptotic Variance ◽  
Hitting Time ◽  
Local Times ◽  
Compact Interval ◽  
Reflected Brownian Motion ◽  
Direct Derivation ◽  
One Dimensional ◽  
Brownian Motion With Drift ◽  
Variance Parameters

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.

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

2004 ◽  
Vol 41 (04) ◽  
pp. 1059-1070 ◽  
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.

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

2004 ◽  
Vol 41 (4) ◽  
pp. 1059-1070 ◽  
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.

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

Risks ◽  
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.

scholarly journals HEAVY TRAFFIC LIMITS VIA BROWNIAN EMBEDDINGS

2006 ◽  
Vol 20 (4) ◽  
pp. 595-598 ◽  
Author(s):  
Erol A. Peköz ◽  
Jose Blanchet
Keyword(s):  
Brownian Motion ◽  
Heavy Traffic ◽  
Continuous Mapping ◽  
Queue Size ◽  
Reflected Brownian Motion ◽  
Probabilistic Argument ◽  
Brownian Motion With Drift ◽  
Queuing Models ◽  
Topological Argument ◽  
Critical Model

For theGI/GI/1 queue we show that the scaled queue size converges to reflected Brownian motion in a critical queue and converges to reflected Brownian motion with drift for a sequence of subcritical queuing models that approach a critical model. Instead of invoking the topological argument of the usual continuous-mapping approach, we give a probabilistic argument using Skorokhod embeddings in Brownian motion.

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

1992 ◽  
Vol 29 (4) ◽  
pp. 996-1002 ◽  
Author(s):  
R. J. Williams
Keyword(s):  
Brownian Motion ◽  
Asymptotic Variance ◽  
Hitting Time ◽  
Local Times ◽  
Compact Interval ◽  
Reflected Brownian Motion ◽  
Direct Derivation ◽  
One Dimensional ◽  
Brownian Motion With Drift ◽  
Variance Parameters

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.

scholarly journals Parisian Time of Reflected Brownian Motion With Drift on Rays

2020 ◽  
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 is 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 with the existing literature.

Moderate deviation principle for multivalued stochastic differential equations

2019 ◽  
Vol 20 (03) ◽  
pp. 2050015 ◽  
Author(s):  
Hua Zhang
Keyword(s):  
Brownian Motion ◽  
Weak Convergence ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Moderate Deviation ◽  
Reflected Brownian Motion ◽  
Moderate Deviation Principle ◽  
Deviation Principle ◽  
Weak Convergence Approach

In this paper, we prove a moderate deviation principle for the multivalued stochastic differential equations whose proof are based on recently well-developed weak convergence approach. As an application, we obtain the moderate deviation principle for reflected Brownian motion.

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