Abstract
In this paper, we consider the stochastic Lotka–Volterra model with additive jump noises. We show some desired properties of the solution such as existence and uniqueness of positive strong solution, unique stationary distribution, and exponential ergodicity. After that, we investigate the maximum likelihood estimation for the drift coefficients based on continuous time observations. The likelihood function and explicit estimator are derived by using semimartingale theory. In addition, consistency and asymptotic normality of the estimator are proved. Finally, computer simulations are presented to illustrate our results.
Highlights
The following famous population dynamics dXt = Xt(a – bXt) dt is often used to model population growth of a single species, where Xt represents its population size at time t, a > 0 is the rate of growth, and b > 0 represents the effect of intraspecies interaction
We will focus on the maximum likelihood estimation (MLE) of the parameter θ = (a, b) ∈ R2++ based on the continuous time observations of the path XT := (Xt)0≤t≤T
Zhang et al [8] considered a stochastic Lotka–Volterra model driven by α-stable noise, they got a unique positive strong solution of their model
Summary
Zhang et al [8] considered a stochastic Lotka–Volterra model driven by α-stable noise, they got a unique positive strong solution of their model They proved stationary property and exponential ergodicity under relatively small noise and extinction under large enough noise. 2, we firstly prove the existence of a unique strong positive solution of equation (1.1). Our aim is to show that under assumption (A1) equation (2.3) has a unique stationary distribution. We assume that, for each φ ∈ {ψ, ψ }, local uniqueness holds for the martingale problem on the canonical space corresponding to the triple (Bφ, Cφ, μφ) with the given initial value x0, and Pφ is the unique solution. Note that the Poisson process with intensity 1 is a subordinator with Lévy measure ν(dz) = δ1(dz) It follows from Proposition 2.5 there is a unique stationary distribution. The trend of normality of each element of the estimator βT can be seen from Fig. 4, where the histogram of each element is given
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