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 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.

scholarly journals Double-Barrier Parisian Options

2011 ◽  
Vol 48 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Angelos Dassios ◽  
Shanle Wu
Keyword(s):  
Brownian Motion ◽  
Markov Model ◽  
Laplace Transform ◽  
Mathematical Finance ◽  
Important Application ◽  
Double Barrier ◽  
Parisian Options ◽  
Brownian Motion With Drift ◽  
The Laplace Transform ◽  
Excursion Time

In this paper we study the excursion time of a Brownian motion with drift outside a corridor by using a four-state semi-Markov model. In mathematical finance, these results have an important application in the valuation of double-barrier Parisian options. We subsequently obtain an explicit expression for the Laplace transform of its price.

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 Double-Barrier Parisian Options

2011 ◽  
Vol 48 (01) ◽  
pp. 1-20 ◽  
Author(s):  
Angelos Dassios ◽  
Shanle Wu
Keyword(s):  
Brownian Motion ◽  
Markov Model ◽  
Laplace Transform ◽  
Mathematical Finance ◽  
Important Application ◽  
Double Barrier ◽  
Parisian Options ◽  
Brownian Motion With Drift ◽  
The Laplace Transform ◽  
Excursion Time

In this paper we study the excursion time of a Brownian motion with drift outside a corridor by using a four-state semi-Markov model. In mathematical finance, these results have an important application in the valuation of double-barrier Parisian options. We subsequently obtain an explicit expression for the Laplace transform of its price.

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 integral of geometric Brownian motion

2001 ◽  
Vol 33 (1) ◽  
pp. 223-241 ◽  
Author(s):  
Daniel Dufresne
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Brownian Motion ◽  
Laplace Transform ◽  
Finite Time ◽  
Geometric Brownian Motion ◽  
Time Interval ◽  
Finite Time Interval ◽  
Special Cases ◽  
The Laplace Transform

This paper is about the probability law of the integral of geometric Brownian motion over a finite time interval. A partial differential equation is derived for the Laplace transform of the law of the reciprocal integral, and is shown to yield an expression for the density of the distribution. This expression has some advantages over the ones obtained previously, at least when the normalized drift of the Brownian motion is a non-negative integer. Bougerol's identity and a relationship between Brownian motions with opposite drifts may also be seen to be special cases of these results.

scholarly journals The Integral of the Supremum Process of Brownian Motion

2009 ◽  
Vol 46 (2) ◽  
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.

A result on the Laplace transform associated with the sticky Brownian motion on an interval

2020 ◽  
pp. 2150031
Author(s):  
Shiyu Song
Keyword(s):  
Brownian Motion ◽  
Boundary Conditions ◽  
Laplace Transform ◽  
Laplace Transforms ◽  
Perturbation Approach ◽  
Integrated Process ◽  
Modified Bessel Function ◽  
Airy Functions ◽  
Occupation Times ◽  
The Laplace Transform

In this paper, we study the joint Laplace transform of the sticky Brownian motion on an interval, its occupation time at zero and its integrated process. The perturbation approach of Li and Zhou [The joint Laplace transforms for diffusion occupation times, Adv. Appl. Probab. 45 (2013) 1049–1067] is adopted to convert the problem into the computation of three Laplace transforms, which is essentially equivalent to solving the associated differential equations with boundary conditions. We obtain the explicit expression for the joint Laplace transform in terms of the modified Bessel function and Airy functions.

scholarly journals On the Frequency of Drawdowns for Brownian Motion Processes

2015 ◽  
Vol 52 (1) ◽  
pp. 191-208 ◽  
Author(s):  
David Landriault ◽  
Bin Li ◽  
Hongzhong Zhang
Keyword(s):  
Risk Management ◽  
Brownian Motion ◽  
Laplace Transform ◽  
Insurance Policies ◽  
Frequency Rate ◽  
The Laplace Transform ◽  
Running Maximum ◽  
Time Sequences ◽  
Quantities Of Interest

Drawdowns measuring the decline in value from the historical running maxima over a given period of time are considered as extremal events from the standpoint of risk management. To date, research on the topic has mainly focused on the side of severity by studying the first drawdown over a certain prespecified size. In this paper we extend the discussion by investigating the frequency of drawdowns and some of their inherent characteristics. We consider two types of drawdown time sequences depending on whether a historical running maximum is reset or not. For each type we study the frequency rate of drawdowns, the Laplace transform of the nth drawdown time, the distribution of the running maximum, and the value process at the nth drawdown time, as well as some other quantities of interest. Interesting relationships between these two drawdown time sequences are also established. Finally, insurance policies protecting against the risk of frequent drawdowns are also proposed and priced.

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