An Excursion characterization of the first hitting time of Brownian motion in a smooth boundary

2007 ◽  
Vol 15 (1) ◽  
Author(s):  
Doncho S. Donchev
Keyword(s):  
Brownian Motion ◽  
Smooth Boundary ◽  
Hitting Time ◽  
First Hitting Time

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 ).

L'intégrale du mouvement brownien

1993 ◽  
Vol 30 (1) ◽  
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).

scholarly journals A characterization of the first hitting time of double integral processes to curved boundaries

2008 ◽  
Vol 40 (02) ◽  
pp. 501-528 ◽  
Author(s):  
Jonathan Touboul ◽  
Olivier Faugeras
Keyword(s):  
Exit Time ◽  
Analytical Formula ◽  
Hitting Time ◽  
First Hitting Time ◽  
Double Integral ◽  
First Exit Time ◽  
Physics First ◽  
Crossing Probability ◽  
Integral Process

The problem of finding the probability distribution of the first hitting time of a double integral process (DIP) such as the integrated Wiener process (IWP) has been an important and difficult endeavor in stochastic calculus. It has applications in many fields of physics (first exit time of a particle in a noisy force field) or in biology and neuroscience (spike time distribution of an integrate-and-fire neuron with exponentially decaying synaptic current). The only results available are an approximation of the stationary mean crossing time and the distribution of the first hitting time of the IWP to a constant boundary. We generalize these results and find an analytical formula for the first hitting time of the IWP to a continuous piecewise-cubic boundary. We use this formula to approximate the law of the first hitting time of a general DIP to a smooth curved boundary, and we provide an estimation of the convergence of this method. The accuracy of the approximation is computed in the general case for the IWP and the effective calculation of the crossing probability can be carried out through a Monte Carlo method.

A Numerical Scheme for Expectations with First Hitting Time to Smooth Boundary

2019 ◽  
Vol 26 (4) ◽  
pp. 553-565 ◽  
Author(s):  
Yuji Hishida ◽  
Yuta Ishigaki ◽  
Toshiki Okumura
Keyword(s):  
Numerical Scheme ◽  
Smooth Boundary ◽  
Hitting Time ◽  
First Hitting Time

scholarly journals A characterization of the first hitting time of double integral processes to curved boundaries

2008 ◽  
Vol 40 (2) ◽  
pp. 501-528 ◽  
Author(s):  
Jonathan Touboul ◽  
Olivier Faugeras
Keyword(s):  
Exit Time ◽  
Analytical Formula ◽  
Hitting Time ◽  
First Hitting Time ◽  
Double Integral ◽  
First Exit Time ◽  
Physics First ◽  
Crossing Probability ◽  
Integral Process

The problem of finding the probability distribution of the first hitting time of a double integral process (DIP) such as the integrated Wiener process (IWP) has been an important and difficult endeavor in stochastic calculus. It has applications in many fields of physics (first exit time of a particle in a noisy force field) or in biology and neuroscience (spike time distribution of an integrate-and-fire neuron with exponentially decaying synaptic current). The only results available are an approximation of the stationary mean crossing time and the distribution of the first hitting time of the IWP to a constant boundary. We generalize these results and find an analytical formula for the first hitting time of the IWP to a continuous piecewise-cubic boundary. We use this formula to approximate the law of the first hitting time of a general DIP to a smooth curved boundary, and we provide an estimation of the convergence of this method. The accuracy of the approximation is computed in the general case for the IWP and the effective calculation of the crossing probability can be carried out through a Monte Carlo method.

scholarly journals First Hitting Time of Brownian Motion on Simple Graph with Skew Semiaxes

2021 ◽  
Author(s):  
Angelos Dassios ◽  
Junyi Zhang
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Laplace Transform ◽  
Simple Graph ◽  
Distribution Functions ◽  
Hitting Time ◽  
Common Origin ◽  
First Hitting Time ◽  
Numerical Examples ◽  
The Laplace Transform

AbstractConsider a stochastic process that lives on n-semiaxes emanating from a common origin. On each semiaxis it behaves as a Brownian motion and at the origin it chooses a semiaxis randomly. In this paper we study the first hitting time of the process. We derive the Laplace transform of the first hitting time, and provide the explicit expressions for its density and distribution functions. Numerical examples are presented to illustrate the application of our results.

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