Diffusion Limits for a Markov Modulated Binomial Counting Process

Abstract

In this paper we study limit behavior for a Markov-modulated (MM) binomial counting process, also called a binomial counting process under regime switching. Such a process naturally appears in the context of credit risk when multiple obligors are present. Markovmodulation takes place when the failure/default rate of each individual obligor depends on an underlying Markov chain. The limit behavior under consideration occurs when the number of obligors increases unboundedly, and/or by accelerating the modulating Markov process, called rapid switching. We establish diffusion approximations, obtained by application of (semi)martingale central limit theorems. Depending on the specific circumstances, different approximations are found.The process we consider has various applications. In (software) reliability modelling early variants are Jelinski and Moranda [33], Koch and Spreij [37], Littlewood [42]. The value of modelling (software) failures within randomly changing environments, including Markovmodulation, has been acknowledged for some time now, see e.g.Özekici and Soyer [46,47], Ravishanker, Liu, and Ray [48]. In particular MM variants of Jelinski and Moranda [33] have been studied, i.e. in a Bayersian set-up in Landon,Özekici, and Soyer [38], with an estimation focus in Ando, Okamura, and Dohi [3], Hellmich [27] and with an added failure rate component in Subrahmaniam, Dewanji, and Roy [52]. A similar model to Jelinski and Moranda [33] has been used in epidemiology (see Andersson and Britton [2]) and a multivariate version of it in sampling design (see Berchenko, Rosenblatt, and Frost [9]), where the latter can also be used to model job switching behavior due to recruiters.An early application of Markov-modulation in economic modelling is Hamilton [26]. Since then Markov-modulation has been extensively used in various branches of mathematical finance. E.g. in optimal investment theory for pension funds (Chen and Delong [13]), interest rate modelling (Ang and Bekaert [4], Elliott and Mamon [20], Elliott and Siu [21]) and affine processes (van Beek, Mandjes, Spreij, and Winands [53]). Other financial applications concern option and bond valuation (Buffington and Elliott [12], Elliott, Kuen Siu, and Badescu [19], Jiang and Pistorius [36]), optimal dividend policies (Jiang [34], Jiang and Pistorius [35]), optimal portfolio and asset allocation (Elliott and Hinz [17], Elliott and Van der Hoek [22], Zhou and Yin [56]) and also most notably in the modelling of credit risk and credit derivatives (Banerjee [7], Banerjee, Ghosh, and Iyer [8], Choi and Marcozzi [14], Dunbar and Edwards [16], Giampieri, Davis, and Crowder [23], Hainaut and Colwell [25], Li and Ma [39], Liechty [40], Yin [55]). Markov-modulation has been used in insurance and risk theory as well (Asmussen and Albrecher [6]).Outside mathematical finance, a rich area of applications of regime switching is in operations research, where there is a sizeable body of work on Markov-modulated queues, see e.g. Asmussen [5] and Neuts [44]. Contributions in this field ...

A Diffusion-Based Analysis of a Multiclass Road Traffic Network

“…a network of Markov-modulated infinite-server queues). In other contexts, a similar methodology has been used, in, for example, Huang et al (2016) and Spreij and Storm (2019).…”

“…However, these models come with new complications, in that they do not automatically satisfy the anisotropic (informally meaning that information travels slower than the fastest class of vehicles) and even hyperbolic (informally meaning that any single vehicle has only local influence on the system dynamics) properties of conservation laws. As discussed in Daganzo (1995b) and van Wageningen-Kessels et al (2015), realistic traffic flow models should obey these properties.…”

A diffusion-based analysis of a multi-class road traffic network

“…a network of Markov-modulated infinite-server queues. In other contexts, a similar methodology has been used, in e.g., [12,30].…”