First hitting time of Double Integral Processes to curved boundaries

Jonathan Touboul (1) , Olivier Faugeras (1)

1 ODYSSEE - Computer and biological vision

The problem of finding the first hitting time of a Double Integral Process (DIP) such as the Integrated Wiener Proces (IWP) has been a central and difficult endeavor in stochastic calculus and 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 of an integrate-and-fire neuron with exponentially decaying synaptic current). The only results available so far were an approximation of the stationnary mean crossing time and the distribution of the first hitting time of the IWP to a constant boundary. In this paper, we generalize those results and find an analytical formula for the first hitting time of the IWP to piecewise cubic boundaries. 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. This approximation formula is the first analytical description of the hitting time of a DIP to a curved boundary, and allows us to infer properties of this random variable and provides a way for computing accurately its law. The accuracy of the approximation is computed in the general case for the IWP and the calculation of crossing probability can be carried out through a Monte-Carlo method.

Mots clés: numerical computation; First hitting time; first passage time; curved boundary; integrated Wiener process; Double integral process; second-order stochastic differential equation; numerical computation.

Domaines: Probabilités [math.PR]

Jonathan Touboul, Olivier Faugeras. First hitting time of Double Integral Processes to curved boundaries. [Research Report] RR-6264, INRIA. 2008, pp.37. ⟨inria-00166335v3⟩