[Paper Review] The local principle of large deviations for compound Poisson process with catastrophes

A. V. Logachov, O. Logachova+1 more|arXiv (Cornell University)|Jun 19, 2018

Stochastic processes and statistical mechanics24 references3 citations

TL;DR

This paper establishes the local large deviation principle (LLDP) for a continuous-time Markov process modeling population dynamics with linear growth and catastrophes, where catastrophes occur as a Poisson process and eliminate a random fraction of the population. The key contribution is a rigorous asymptotic upper bound for the maximum value of the process and the proof of the LLDP under general conditions on the jump size distribution and catastrophe intensity.

ABSTRACT

The continuous time Markov process considered in this paper belongs to a class of population models with linear growth and catastrophes. There, the catastrophes happen at the arrival times of a Poisson process, and at each catastrophe time, a randomly selected portion of the population is eliminated. For this population process, we derive an asymptotic upper bound for the maximum value and prove the local large deviation principle.

Motivation & Objective

• To analyze the large deviation behavior of a population process with linear birth and Poisson-driven catastrophes.
• To establish a local large deviation principle (LLDP) for the process, which characterizes rare events involving large deviations in the population size.
• To derive an asymptotic upper bound for the maximum value of the process over a finite time horizon.
• To extend the theoretical framework of large deviations to processes with random resetting and catastrophes, particularly in the context of population models.
• To provide a foundation for studying optimal strategies in population survival under random catastrophic events using large deviation techniques.

Proposed method

• Model the population process as a continuous-time Markov process with linear drift and jumps at Poisson arrival times representing catastrophes.
• Define the process as a compound Poisson process with random jump sizes that represent the fraction of population eliminated at each catastrophe time.
• Use a conditioning argument on the number of catastrophes and the pre-catastrophe population size to analyze path behavior.
• Apply large deviation techniques, including exponential tilting and conditioning on rare events, to derive upper bounds on the probability of large deviations.
• Introduce auxiliary events such as $ A_k $, $ B_k $, $ G_{k_l} $, and $ H_{k_l} $ to control the trajectory behavior between catastrophes.
• Establish bounds on transition probabilities using the independence of jump size distributions from the pre-catastrophe state and the assumption of uniformity in jump sizes.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the maximum population size in a compound Poisson process with catastrophes over a finite time interval?
• RQ2 How can the local large deviation principle be rigorously established for a process with random jumps and resetting events?
• RQ3 What is the rate function governing the probability of rare events where the population size deviates significantly from its typical path?
• RQ4 How do the intensity of catastrophes and the distribution of jump sizes affect the likelihood of extreme population values?
• RQ5 Can an upper bound be derived for the maximum value of the process that holds uniformly over all time horizons?

Key findings

• An asymptotic upper bound is derived for the maximum value of the process over a finite time interval, showing that large values are exponentially unlikely.
• The local large deviation principle (LLDP) is proven for the process, establishing a rate function that characterizes the decay rate of probabilities of rare events.
• The upper bound on the maximum value is shown to decay exponentially with time, under the assumption that the jump size distribution satisfies a uniformity condition.
• The probability of observing a path that reaches a large value $ g^* $ is bounded above by an expression involving $ \varepsilon^3 $, $ \Delta $, and the number of catastrophes, indicating exponential decay.
• The proof relies on conditioning on the number of catastrophes and bounding the likelihood of trajectories that avoid extinction while reaching high population levels.
• The bound on the conditional probability of surviving multiple catastrophes without falling below a threshold is shown to decay exponentially with the number of events, confirming the rarity of such extreme paths.