[Paper Review] Diffusion limits for a Markov modulated counting process

Peter Spreij, Jaap Storm|arXiv (Cornell University)|Jan 11, 2018

Mathematical Dynamics and Fractals3 citations

TL;DR

This paper establishes weak limit theorems for a Markov-modulated binomial counting process under rapid switching and large population limits, deriving functional central limit approximations that yield diffusion processes. The key contribution is identifying distinct diffusion limits depending on whether the number of obligors grows or the modulating Markov chain accelerates, with applications in credit risk modeling under regime switching.

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. The concept of Markov-modulation has become increasingly popular in many branches of science. For example, one can model asset prices with stochastic processes, or model the 'state of the economy' by a Markov. The binomial counting process naturally appears in the context of credit risk when multiple obligors are present. Markov modulation 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. The interest is in finding weak limits in each of these cases, more specifically of a functional central limit type. In other words we will study diffusion approximations. Depending on the specific circumstances, different approximations are found.

Motivation & Objective

• To analyze the asymptotic behavior of a binomial counting process when the number of obligors grows large.
• To study the impact of rapid switching in an underlying Markov modulating process on the limiting distribution.
• To derive weak limits of functional central limit type for the counting process under different scaling regimes.
• To provide diffusion approximations suitable for modeling credit risk with regime-dependent default intensities.
• To clarify the conditions under which distinct diffusion limits emerge depending on the scaling of the process.

Proposed method

• Model the counting process as a binomial process where default rates are governed by an underlying continuous-time Markov chain.
• Apply weak convergence techniques to analyze the limit behavior as the number of obligors tends to infinity.
• Introduce a time-scale separation by accelerating the modulating Markov chain to study rapid switching regimes.
• Derive functional central limit theorems for the normalized counting process under joint scaling of population size and switching speed.
• Use martingale functional central limit theory to establish convergence to diffusion processes in the limit.
• Distinguish between two asymptotic regimes: large population with fixed switching, and fixed population with rapid switching, each yielding different diffusion limits.

Experimental results

Research questions

• RQ1 What is the weak limit of a Markov-modulated binomial counting process when the number of obligors increases without bound?
• RQ2 How does rapid switching in the underlying Markov modulating process affect the limiting distribution of the counting process?
• RQ3 Under what conditions does the normalized counting process converge to a diffusion process in distribution?
• RQ4 What are the distinct diffusion approximations that arise in different scaling regimes—large population versus rapid switching?
• RQ5 How do the limiting diffusion processes reflect the regime-switching dynamics in credit risk applications?

Key findings

• When the number of obligors grows large while the modulating Markov chain remains fixed, the normalized counting process converges weakly to a diffusion process.
• Under rapid switching of the modulating Markov chain, the limit process is again a diffusion, but with a different effective intensity due to averaging over the fast-switching states.
• The limiting diffusion processes are characterized by time-changed Brownian motion components, reflecting the regime-switching dynamics.
• The functional central limit theorem applies under joint scaling, yielding a diffusion approximation that captures both population size and switching speed effects.
• The results provide a rigorous foundation for using diffusion processes to approximate credit risk models with state-dependent default intensities.
• The paper identifies two distinct limiting regimes—large population and rapid switching—each leading to a different diffusion approximation.