A first passage problem for a bivariate diffusion process: Numerical solution with an application to neuroscience when the process is Gauss-Markov

AbstractWe consider a bivariate diffusion process and we study the fir st passage time ofone component through a boundary. We prove that its probability density is theunique solution of a new integral equation and we propose a numerical algorithmfor its solution. Convergence properties of this algorithm are discussed and themethod is applied to the study of the integrated Brownian Motion and to the in-tegrated Ornstein Uhlenbeck process. Finally a model of neuroscience interest isalso discussed.Keywords:First passage time, Bivariate diffusion, Integrated Brownian Motion, IntegratedOrnstein Uhlenbeck process, Two-compartment neuronal model1. IntroductionFirst passage time problems arise in a variety of applications ranging from fi-nance to biology, physics or psychology ([29, 20, 21] and examples cited therein).They have been largely studied (see [27] for a review on the subject): analytical[8, 9, 17, 19, 22, 26], numerical or approximate results [3, 4, 5, 6, 7, 23, 24, 30, 28]exist for specific classes of processes such as one dimension al diffusions or Gaus-sian processes. On the contrary, the case of bivariate processes has not beenwidely studied yet. Indeed, results are available only for specific problems suchas the first exit time of the considered two-dimensional process from a specific

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