L'intégrale du mouvement brownien

1993 ◽  
Vol 30 (01) ◽  
pp. 17-27
Author(s):  
Aimé Lachal

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 ).

1993 ◽  
Vol 30 (1) ◽  
pp. 17-27 ◽  
Author(s):  
Aimé Lachal

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).


2000 ◽  
Vol 23 (11) ◽  
pp. 759-776 ◽  
Author(s):  
Seung Jun Chang ◽  
Soon Ja Kang ◽  
David Skoug

We use a generalized Brownian motion process to define a generalized Feynman integral and a conditional generalized Feynman integral. We then establish the existence of these integrals for various functionals. Finally we use the conditional generalized Feynman integral to derive a Schrödinger integral equation.


2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
ZaiTang Huang ◽  
ChunTao Chen

We study the stability, attractors, and bifurcation of stochastic Rayleigh-van der Pol equations with jumps. We first established the stochastic stability and the large deviations results for the stochastic Rayleigh-van der Pol equations. We then examine the existence limit circle and obtain some new random attractors. We further establish stochastic bifurcation of random attractors. Interestingly, this shows the effect of the Poisson noise which can stabilize or unstabilize the system which is significantly different from the classical Brownian motion process.


1974 ◽  
Vol 6 (4) ◽  
pp. 651-665 ◽  
Author(s):  
David C. Heath ◽  
William D. Sudderth

An abstract gambler's problem is formulated in a continuous-time setting and analogues are proved for some of the discrete-time results of Dubins and Savage in their book How to Gamble if You Must. Applications are made to problems of controlling a Brownian motion process.


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