City Research Online

Abstract

We develop an efficient method for computing the probability that a non-decreasing, pure jump (compound) stochastic process stays between arbitrary upper and lower boundaries (i.e., deterministic
functions, possibly discontinuous) within a finite time period. The compound process is composed of a process modelling the arrivals of certain events (e.g., demands for a product in inventory systems, customers in queuing, or claims/capital gains in insurance/dual risk models), and a sequence of independent and identically distributed random variables modelling the sizes of the events. The events arrival process is assumed to belong to the wide class of point processes with conditional stationary independent increments which includes (non-)homogeneous Poisson, binomial, negative binomial, mixed Poisson and doubly stochastic Poisson (i.e., Cox) processes as special cases. The proposed method is based on expressing the non-exit probability through Chapman-Kolmogorov equations, re-expressing them in terms of a circular convolution of two vectors which is then computed applying fast Fourier transform (FFT). We further demonstrate that our FFT-based method is computationally efficient and can be successfully applied in the context of inventory management (to determine an optimal replenishment policy), ruin theory (to evaluate
ruin probabilities and related quantities) and double-barrier option pricing or simply computing non-exit probabilities for Brownian motion with general boundaries.

Publication Type:	Article
Additional Information:	© 2019 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Publisher Keywords:	applied probability, doubly stochastic Poisson (i.e., Cox) processes, fast Fourier transform, inventory management under stochastic demand, finite-time non-ruin probability
Subjects:	H Social Sciences > HG Finance Q Science > QA Mathematics
Departments:	Bayes Business School > Actuarial Science & Insurance
SWORD Depositor:	Symplectic Administrator

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