Abstract
Algorithms for the simulation of sample paths of Gauss–Markov processes, restricted from below by particular time-dependent reflecting boundaries, are proposed. These algorithms are used to build the histograms of first passage time density through specified boundaries and for the estimation of related moments. Particular attention is dedicated to restricted Wiener and Ornstein–Uhlenbeck processes due to their central role in the class of Gauss–Markov processes.
Highlights
Diffusion and Gauss–Markov processes in the presence of a reflecting boundary are widely used for modeling continuous time phenomena in many scientific fields, such as neurosciences, mathematical biology, finance, and queueing systems
Their research activity has been intense within the following themes: (i) the study of the first-passage time densities for diffusion processes and Gauss-Markov processes and their asymptotic behavior; (ii) the development of mathematical methods and tools mainly of a probabilistic and computational nature to analyze biological dynamics, among them the input-output behavior of neurons subject to random perturbations
We remark that the Algorithm 4.2 is a generalization to restricted Gauss–Markov processes of the simulation algorithm given in Asmussen et al (1995) and Kroese et al (2011) for a Brownian motion with a negative drift and unit infinitesimal variance in the presence of a reflecting boundary in zero state
Summary
Diffusion and Gauss–Markov processes in the presence of a reflecting boundary are widely used for modeling continuous time phenomena in many scientific fields, such as neurosciences, mathematical biology, finance, and queueing systems. Making use of the Algorithm 4.1, we include in the Figure 1 a simulated sample path (red) of the related restricted process X(t) in [ (t), +∞), with (t) = t + 1.0 reflecting boundary In this case, in (17), we set A = 0 and B = 1.0. By virtue of the Algorithm 4.1, we include in Figure 2 a simulated sample path (red) of the related restricted process X(t) in [ (t), +∞), with (t) = (e t − 1)∕ reflecting boundary, obtained by setting A = B = 0 in (18). We remark that the Algorithm 4.2 is a generalization to restricted Gauss–Markov processes of the simulation algorithm given in Asmussen et al (1995) and Kroese et al (2011) for a Brownian motion with a negative drift and unit infinitesimal variance in the presence of a reflecting boundary in zero state. An estimation of the FPT pdf can be achieved by the histogram of such first passage times
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