Large deviations in a population dynamics with catastrophes

Abstract

The large deviation principle on phase space is proved for a class of Markov processes known as random population dynamics with catastrophes. In the paper we study the process which corresponds to the random population dynamics with linear growth and uniform catastrophes, where an eliminating portion of the population is chosen uniformly. The large deviation result provides an optimal trajectory of large fluctuation: it shows how the large fluctuations occur for this class of processes.

The local principle of large deviations for compound Poisson process with catastrophes

“…In this work, we study the limiting properties (asymptotic upper bound for the maximum value and the local large deviation principle) of an integer-valued non-decreasing compound Poisson process with almost-uniform catastrophes catastrophes. In our earlier work [19], we proved the large deviation principle in the phase space for the homogeneous Poisson processes with uniform catastrophes. Thus, providing the optimal trajectory for large fluctuation.…”

Order Book Dynamics with Liquidity Fluctuations: Limit Theorems and Large Deviations

“…Large deviations for the Poisson processes with uniform (almost uniform) catastrophes was considered in the papers Logachov et al (2019) and Logachov et al (2018). Large deviation can be considered as a finishing step in a sequence of limit theorems for the processes.…”

“…Let T be the length of the time interval [0, T ] we observed our process. We consider the following scaled process This principle was established in the paper Logachov et al (2019), in which the logarithmic asymptotic for the probability P(S T (1) > x) was calculated. Note, that the principle was proved for the state x of the spread at the time T , it is not the principle on the functional space.…”

“…It provides the existence of an optimal trajectory for the large deviations. The existence of the optimal trajectories of large deviations S T (1) > x were established in Logachov et al (2019). If x < γ + then there exists the moment t x = 1 − x γ+ ∈ (0, 1) such that the spread process S T (•) stays near the zero up to the time t x and after that S T (t), t t x increases according to the straight line which starts at point (t x , 0) and grows up to the point (1, x) with the slope γ + , see the function f 2 on Fig.…”