Sur certaines fonctionnelles exponentielles du mouvement brownien réel

1992 ◽  
Vol 29 (1) ◽  
pp. 202-208 ◽  
Author(s):  
Marc Yor
Keyword(s):  
Brownian Motion ◽  
Bivariate Distribution ◽  
Exit Times ◽  
Bessel Processes ◽  
Gamma Variable ◽  
Negative Drift ◽  
Mouvement Brownien

Dufresne [1] recently showed that the integral of the exponential of Brownian motion with negative drift is distributed as the reciprocal of a gamma variable. In this paper, it is shown that this result is another formulation of the distribution of last exit times for transient Bessel processes. A bivariate distribution of such integrals of exponentials is also obtained explicitly.

Sur certaines fonctionnelles exponentielles du mouvement brownien réel

1992 ◽  
Vol 29 (01) ◽  
pp. 202-208 ◽  
Author(s):  
Marc Yor
Keyword(s):  
Brownian Motion ◽  
Bivariate Distribution ◽  
Exit Times ◽  
Bessel Processes ◽  
Gamma Variable ◽  
Negative Drift ◽  
Mouvement Brownien

Dufresne [1] recently showed that the integral of the exponential of Brownian motion with negative drift is distributed as the reciprocal of a gamma variable. In this paper, it is shown that this result is another formulation of the distribution of last exit times for transient Bessel processes. A bivariate distribution of such integrals of exponentials is also obtained explicitly.

Optimal Control of a Model for a System Subject to Continuous Wear

1988 ◽  
Vol 2 (3) ◽  
pp. 321-328 ◽  
Author(s):  
Laurence A. Baxter ◽  
Eui Yong Lee
Keyword(s):  
Optimal Control ◽  
Brownian Motion ◽  
Poisson Process ◽  
Average Cost ◽  
Infinite Horizon ◽  
Arrival Rate ◽  
The State ◽  
Negative Drift ◽  
Random Amount

The state of a system is modelled by Brownian motion with negative drift and an absorbing barrier at the origin. A repairman arrives according to a Poisson process of rate λ. If the state of the system at arrival of the repairman does not exceed a certain threshold, he/she increases it by a random amount, otherwise no action is taken. Costs are assigned to each visit of the repairman, to each repair, and to the system being in state 0. It is shown that there exists a unique arrival rate λ which minimizes the average cost per unit time over an infinite horizon.

scholarly journals On the time spent above a level by Brownian motion with negative drift

1986 ◽  
Vol 18 (04) ◽  
pp. 1017-1018 ◽  
Author(s):  
J.-P. Imhof
Keyword(s):  
Brownian Motion ◽  
Limit Theorems ◽  
Distribution Functions ◽  
Probability Densities ◽  
Negative Drift

Limit theorems of Berman involve the total time spent by Brownian motion with negative drift above a fixed or exponentially distributed negative level. We give explicitly the probability densities and distribution functions, obtained via an equivalence of laws.

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

2015 ◽  
Vol 47 (1) ◽  
pp. 210-230 ◽  
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.

L'intégrale du mouvement brownien

1993 ◽  
Vol 30 (01) ◽  
pp. 17-27
Author(s):  
Aimé Lachal
Keyword(s):  
Fourier Transform ◽  
Brownian Motion ◽  
Hitting Time ◽  
First Hitting Time ◽  
Brownian Motion Process ◽  
Classical Technique ◽  
Motion Process ◽  
Mouvement Brownien ◽  
The Law ◽  
The Right

Let be the Brownian motion process starting at the origin, its primitive and Ut = (Xt+x + ty, Bt + y), , the associated bidimensional process starting from a point . In this paper we present an elementary procedure for re-deriving the formula of Lefebvre (1989) giving the Laplace–Fourier transform of the distribution of the couple (σ α, Uσa ), as well as Lachal's (1991) formulae giving the explicit Laplace–Fourier transform of the law of the couple (σ ab, Uσab ), where σ α and σ ab denote respectively the first hitting time of from the right and the first hitting time of the double-sided barrier by the process . This method, which unifies and considerably simplifies the proofs of these results, is in fact a ‘vectorial' extension of the classical technique of Darling and Siegert (1953). It rests on an essential observation (Lachal (1992)) of the Markovian character of the bidimensional process . Using the same procedure, we subsequently determine the Laplace–Fourier transform of the conjoint law of the quadruplet (σ α, Uσa, σb, Uσb ).

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