Abstract
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.
Highlights
Time-inhomogeneous stochastic diffusion processes are often used to model real dynamic systems in various scientific areas as neurosciences, population dynamics, queueing systems, finance, and economics, Di Crescenzo et al [7], Linetsky [8], Glasserman [9])
The histogram of first passage times is compared with the closed form first-passage time (FPT) density through special boundaries
Starting from two Gauss–Markov processes conditioned on the same initial state, we have constructed a continuous process Y (t) via the composition method
Summary
Time-inhomogeneous stochastic diffusion processes are often used to model real dynamic systems in various scientific areas as neurosciences, population dynamics, queueing systems, finance, and economics (cf., for instance, Buonocore et al [1,2], Albano and Giorno [3], Giorno and Spina [4], Ricciardi et al [5], Renshaw [6]), Di Crescenzo et al [7], Linetsky [8], Glasserman [9]). In Di. Crescenzo et al [10], and in Giorno et al [11,12], new procedures for constructing transition probability density functions and first passage time densities through constant boundaries are proposed for generally time-inhomogeneous diffusion processes. In both these cases, for fixed time instants, we simulate the random variable describing Y (t), and we show that the transition pdf of the related diffusion process Z (t) can be superimposed over the histogram of the process Y (t) obtained via the simulation method. The histogram of first passage times is compared with the closed form FPT density through special boundaries
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