scholarly journals Tutte’s invariant approach for Brownian motion reflected in the quadrant

2017 ◽  
Vol 21 ◽  
pp. 220-234 ◽  
Author(s):  
S. Franceschi ◽  
Kilian Raschel
Keyword(s):  
Functional Equation ◽  
Brownian Motion ◽  
Laplace Transform ◽  
Explicit Formula ◽  
Chebyshev Polynomials ◽  
Quarter Plane ◽  
The Laplace Transform ◽  
Negative Drift ◽  
Invariant Approach ◽  
Continuous Setting

We consider a Brownian motion with negative drift in the quarter plane with orthogonal reflection on the axes. The Laplace transform of its stationary distribution satisfies a functional equation, which is reminiscent from equations arising in the enumeration of (discrete) quadrant walks. We develop a Tutte’s invariant approach to this continuous setting, and we obtain an explicit formula for the Laplace transform in terms of generalized Chebyshev polynomials.

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 Diffusion Model for a System Subject to Continuous Wear

1987 ◽  
Vol 1 (4) ◽  
pp. 405-416 ◽  
Author(s):  
Laurence A. Baxter ◽  
Eui Yong Lee
Keyword(s):  
Distribution Function ◽  
Brownian Motion ◽  
Laplace Transform ◽  
Poisson Process ◽  
The State ◽  
Stationary Case ◽  
The Laplace Transform ◽  
Negative Drift ◽  
State Changes ◽  
Random Amount

A model for a system whose state changes continuously with time is introduced. It is assumed that the system is modeled by Brownian motion with negative drift and an absorbing barrier at the origin. A repairman arrives according to a Poisson process and increases the state of the system by a random amount if the state is below a threshold α. Explicit expressions are deduced for the distribution function of X(t), the state of the system at time 1, if X(t) ≤ α and for the Laplace transform of the density of X( t). The stationary case is examined in detail.

scholarly journals The Integral of the Supremum Process of Brownian Motion

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.

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 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.

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.

scholarly journals On the integral of geometric Brownian motion

2003 ◽  
Vol 35 (1) ◽  
pp. 159-183 ◽  
Author(s):  
Michael Schröder
Keyword(s):  
Brownian Motion ◽  
Integral Representation ◽  
Laplace Transform ◽  
Finite Time ◽  
Integral Representations ◽  
Geometric Brownian Motion ◽  
Bessel Processes ◽  
Time Intervals ◽  
The Laplace Transform ◽  
The Law

This paper studies the law of any real powers of the integral of geometric Brownian motion over finite time intervals. As its main results, an apparently new integral representation is derived and its interrelations with the integral representations for these laws originating by Yor and by Dufresne are established. In fact, our representation is found to furnish what seems to be a natural bridge between these other two representations. Our results are obtained by enhancing the Hartman-Watson Ansatz of Yor, based on Bessel processes and the Laplace transform, by complex analytic techniques. Systematizing this idea in order to overcome the limits of Yor's theory seems to be the main methodological contribution of the paper.

The Laplace Transform of Hitting Times of Integrated Geometric Brownian Motion

2013 ◽  
Vol 50 (1) ◽  
pp. 295-299 ◽  
Author(s):  
Adam Metzler
Keyword(s):  
Brownian Motion ◽  
Laplace Transform ◽  
Hypergeometric Functions ◽  
Diffusion Processes ◽  
Geometric Brownian Motion ◽  
Hitting Times ◽  
Confluent Hypergeometric Functions ◽  
Change Of Variables ◽  
The Laplace Transform ◽  
Kummer's Equation

In this note we compute the Laplace transform of hitting times, to fixed levels, of integrated geometric Brownian motion. The transform is expressed in terms of the gamma and confluent hypergeometric functions. Using a simple Itô transformation and standard results on hitting times of diffusion processes, the transform is characterized as the solution to a linear second-order ordinary differential equation which, modulo a change of variables, is equivalent to Kummer's equation.

Diffraction of elastic waves by a rigid 90° wedge Part I

1968 ◽  
Vol 58 (3) ◽  
pp. 1083-1096 ◽  
Author(s):  
Edgar A. Kraut
Keyword(s):  
Laplace Transform ◽  
Elastic Wave ◽  
Elastic Waves ◽  
Closed Form Solution ◽  
Compressional Wave ◽  
Form Solution ◽  
Complex Variables ◽  
Quarter Plane ◽  
The Laplace Transform ◽  
Diffraction Of Elastic Waves

abstract The problem of calculating the reflected and diffracted elastic waves generated when a plane compressional wave strikes a rigid quarter plane is formulated as a Wiener-Hopf problem in two complex variables. It is shown how a closed form solution of this two variable Wiener-Hopf problem can be obtained provided that the Laplace transform of the elastic wave Green's function can be factorized into factors having suitable analytic properties.

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