First passage times of two-dimensional correlated processes: Analytical results for the Wiener process and a numerical method for diffusion processes

Given a two-dimensional correlated diffusion process, we determine the joint density of the first passage times of the process to some constant boundaries. This quantity depends on the joint density of the first passage time of the first crossing component and of the position of the second crossing component before its crossing time. First we show that these densities are solutions of a system of Volterra-Fredholm first kind integral equations. Then we propose a numerical algorithm to solve it and we describe how to use the algorithm to approximate the joint density of the first passage times. The convergence of the method is theoretically proved for bivariate diffusion processes. We derive explicit expressions for these and other quantities of interest in the case of a bivariate Wiener process, correcting previous misprints appearing in the literature. Finally we illustrate the application of the method through a set of examples.

[1]  S. Redner.  A guide to first-passage processes , 2001 .

[2]  L. Sacerdote,et al.  Weak convergence of marked point processes generated by crossings of multivariate jump processes. Applications to neural network modeling , 2013, 1310.6933.

[3]  On the solution of the Fokker–Planck equation for a Feller process , 1990, Advances in Applied Probability.

[4]  Vadim Linetsky,et al.  Lookback options and diffusion hitting times: A spectral expansion approach , 2004, Finance Stochastics.

[5]  FIRST PASSAGE PROBABILITIES OF A TWO DIMENSIONAL BROWNIAN MOTION IN AN ANISOTROPIC MEDIUM , 2016 .

[6]  S. Kyllingsbæk,et al.  Gaussian counter models for visual identification of briefly presented, mutually confusable single stimuli in pure accuracy tasks , 2017 .

[7]  A. G. Nobile,et al.  A new integral equation for the evaluation of first-passage-time probability densities , 1987, Advances in Applied Probability.

[8]  Marco Dominé,et al.  First Passage Time Distribution of a Two-Dimensional Wiener Process with Drift , 1993, Probability in the Engineering and Informational Sciences.

[9]  Laura Sacerdote,et al.  A first passage problem for a bivariate diffusion process: Numerical solution with an application to neuroscience when the process is Gauss-Markov , 2012, J. Comput. Appl. Math..

[10]  Nagi Gebraeel,et al.  Computing and updating the first-passage time distribution for randomly evolving degradation signals , 2012 .

[11]  M. Lefebvre.  First-passage densities of a two-dimensional process , 1989 .

[12]  C. Gardiner.  Handbook of Stochastic Methods , 1983 .

[13]  Jacques Janssen,et al.  Applied Diffusion Processes from Engineering to Finance , 2013 .

[14]  M. A. Abdou.  Fredholm-Volterra integral equation of the first kind and contact problem , 2002, Appl. Math. Comput..

[15]  Chunsheng Zhou,et al.  An Analysis of Default Correlations and Multiple Defaults , 2001 .

[16]  E. Gobet.  Weak approximation of killed diffusion using Euler schemes , 2000 .

[17]  Volkmar Pieper,et al.  Level crossing problems and drift reliability , 1997, Math. Methods Oper. Res..

[18]  D. Williams.  STOCHASTIC DIFFERENTIAL EQUATIONS: THEORY AND APPLICATIONS , 1976 .

[19]  L. Sacerdote,et al.  Stochastic Integrate and Fire Models: a review on mathematical methods and their applications , 2011, 1101.5539.

[20]  Luigi M. Ricciardi,et al.  On the transformation of diffusion processes into the Feller process , 1976 .

[21]  C. W. Gardiner,et al.  Handbook of stochastic methods - for physics, chemistry and the natural sciences, Second Edition , 1986, Springer series in synergetics.

[22]  M. V. Tretyakov,et al.  Stochastic Numerics for Mathematical Physics , 2004, Scientific Computation.

[23]  Jinghai Shao,et al.  Estimates of the Exit Probability for Two Correlated Brownian Motions , 2013, Advances in Applied Probability.

[24]  Laura Sacerdote,et al.  On the inverse first-passage-time problem for a Wiener process , 2009, 0908.4213.

[25]  Philip Rabinowitz,et al.  Methods of Numerical Integration , 1985 .

[26]  L. Milne‐Thomson.  A Treatise on the Theory of Bessel Functions , 1945, Nature.

[27]  J. L. Pedersen,et al.  Representations of the First Hitting Time Density of an Ornstein-Uhlenbeck Process , 2005 .

[28]  H. D. Miller,et al.  The Theory Of Stochastic Processes , 1977, The Mathematical Gazette.

[29]  L. Arnold.  Stochastic Differential Equations: Theory and Applications , 1992 .

[30]  A. G. Nobile,et al.  ON A SYMMETRY-BASED CONSTRUCTIVE APPROACH TO PROBABILITY DENSITIES FOR TWO-DIMENSIONAL DIFFUSION PROCESSES , 1989 .

[31]  K. Vahala.  Handbook of stochastic methods for physics, chemistry and the natural sciences , 1986, IEEE Journal of Quantum Electronics.

[32]  One-Dimensional Homogeneous Diffusions , 2013 .

[33]  D. Navarro,et al.  Fast and accurate calculations for first-passage times in Wiener diffusion models , 2009 .

[34]  E. Messina,et al.  An adaptive method for Volterra-Fredholm integral equations on the half line , 2009 .

[35]  C. Zucca,et al.  Joint Densities of First Hitting Times of a Diffusion Process Through Two Time-Dependent Boundaries , 2014, Advances in Applied Probability.

[36]  Peter Linz,et al.  Analytical and numerical methods for Volterra equations , 1985, SIAM studies in applied and numerical mathematics.

[37]  M. Lefebvre.  First-passage problems for degenerate two-dimensional diffusion processes , 2003 .

[38]  A. Lachal.  Sur le premier instant de passage de l'intégrale du mouvement brownien , 1991 .

[39]  S. Iyengar.  Hitting Lines with Two-Dimensional Brownian Motion , 1985 .

[40]  Adam Metzler,et al.  On the first passage problem for correlated Brownian motion , 2010 .

[41]  Luigi M. Ricciardi.  On the transformation of diffusion processes into the Wiener process , 1976 .

[42]  Laura Sacerdote,et al.  Detecting dependencies between spike trains of pairs of neurons through copulas , 2012, Brain Research.

[43]  E. Tubaldi,et al.  New Analytical Solution of the First-Passage Reliability Problem for Linear Oscillators , 2012 .

[44]  Virginia Giorno,et al.  AN OUTLINE OF THEORETICAL AND ALGORITHMIC APPROACHES TO FIRST PASSAGE TIME PROBLEMS WITH APPLICATIONS TO BIOLOGICAL MODELING , 1999 .

[45]  Yongjin Wang,et al.  The Hitting Time Density for a Reflected Brownian Motion , 2010, Computational Economics.

[46]  J. Hammersley,et al.  Diffusion Processes and Related Topics in Biology , 1977 .