On the Simulation of a Special Class of Time-Inhomogeneous Diffusion Processes

General methods to simulate probability density functions and first passage time densities are provided for time-inhomogeneous stochastic diffusion processes obtained via a composition of two Gauss–Markov processes conditioned on the same initial state. Many diffusion processes with time-dependent infinitesimal drift and infinitesimal variance are included in the considered class. For these processes, the transition probability density function is explicitly determined. Moreover, simulation procedures are applied to the diffusion processes obtained starting from Wiener and Ornstein–Uhlenbeck processes. Specific examples in which the infinitesimal moments include periodic functions are discussed.

[1]  Aniello Buonocore,et al.  Generating random variates from PDF of Gauss-Markov processes with a reflecting boundary , 2018, Comput. Stat. Data Anal..

[2]  J. J. Serrano-Pérez,et al.  Inference on an heteroscedastic Gompertz tumor growth model. , 2020, Mathematical biosciences.

[3]  Powers of the Stochastic Gompertz and Lognormal Diffusion Processes, Statistical Inference and Simulation , 2020 .

[4]  Virginia Giorno,et al.  On the Construction of Densities for Time Non-homogeneous Diffusion Processes , 2011, EUROCAST.

[5]  R. Ratcliff,et al.  A comparison of four methods for simulating the diffusion process , 2001, Behavior research methods, instruments, & computers : a journal of the Psychonomic Society, Inc.

[6]  Ralf Metzler,et al.  Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous diffusion. , 2014, Physical chemistry chemical physics : PCCP.

[7]  S. C. Lim,et al.  Self-similar Gaussian processes for modeling anomalous diffusion. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Aniello Buonocore,et al.  Restricted Ornstein-Uhlenbeck process and applications in neuronal models with periodic input signals , 2015, J. Comput. Appl. Math..

[9]  Todd C. Headrick,et al.  On simulating multivariate non-normal distributions from the generalized lambda distribution , 2006, Comput. Stat. Data Anal..

[10]  A. G. Nobile,et al.  Analysis of Reflected Diffusions via an Exponential Time-Based Transformation , 2016 .

[11]  C. Zucca,et al.  Exact simulation of first exit times for one-dimensional diffusion processes , 2019, ESAIM: Mathematical Modelling and Numerical Analysis.

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

[13]  D. Veestraeten.  On Transition and First Hitting Time Densities and Moments of the Ornstein–Uhlenbeck Process , 2014 .

[14]  C. Zucca,et al.  A Monte Carlo Method for the Simulation of First Passage Times of Diffusion Processes , 2001 .

[15]  M. Abundo.  On the first-passage times of certain Gaussian processes, and related asymptotics , 2020 .

[16]  M. Magnasco,et al.  A Fast Algorithm for the First-Passage Times of Gauss-Markov Processes with Hölder Continuous Boundaries , 2010 .

[17]  Aniello Buonocore,et al.  Gauss-Markov processes in the presence of a reflecting boundary and applications in neuronal models , 2014, Appl. Math. Comput..

[18]  J. McFadden,et al.  Certain Properties of Gaussian Processes and Their First‐Passage Times , 1965 .

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

[20]  Virginia Giorno,et al.  Inferring time non-homogeneous Ornstein Uhlenbeck type stochastic process , 2020, Comput. Stat. Data Anal..

[21]  V. Linetsky.  On the transition densities for reflected diffusions , 2005, Advances in Applied Probability.

[22]  Luigi M. Ricciardi,et al.  Simulation of First-Passage Times for Alternating Brownian Motions , 2005, 2101.11578.

[23]  A. G. Nobile,et al.  Simulation of sample paths for Gauss-Markov processes in the presence of a reflecting boundary , 2017 .

[24]  V. Giorno,et al.  On the return process with refractoriness for a non-homogeneous Ornstein-Uhlenbeck neuronal model. , 2013, Mathematical biosciences and engineering : MBE.

[25]  A. G. Nobile,et al.  On the Construction of a Special Class of Time-Inhomogeneous Diffusion Processes , 2019, Journal of Statistical Physics.

[26]  Amilcare Porporato,et al.  First passage time statistics of Brownian motion with purely time dependent drift and diffusion , 2010, 1009.0557.

[27]  Virginia Giorno,et al.  A Time-Non-Homogeneous Double-Ended Queue with Failures and Repairs and Its Continuous Approximation , 2018 .

[28]  Asymptotics of Two-boundary First-exit-time Densities for Gauss-Markov Processes , 2019 .