# Least Squares Estimation for Discretely Observed Stochastic Lotka–Volterra Model Driven by Smallα-Stable Noises

Authors: Chao Wei; Yan Wei; Yingying Zhou
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

Stochastic Lotka–Volterra model driven by smallα-stable noises is used to describe population dynamics perturbed by random environment. However, parameters in the model are always unknown. The contrast function is given to obtain least squares estimators. The consistency and the rate of convergence of the least squares estimators are proved, and the asymptotic distribution of the estimators are derived by Markov inequality, Cauchy–Schwarz inequality, and Gronwall’s inequality. Some numerical examples are provided to verify the effectiveness of the estimators.


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## Body

## 1. Introduction

Stochastic differential equations are basic tools for modeling random phenomena in the financial field. Due to the widespread application of the stochastic differential equations in the field of financial economics, it has attracted a large number of scholars to devote themselves to research in this field [1–3]. However, the parameters in stochastic model are always unknown. In the past few decades, the parameter estimation problem for economical models has been studied by many authors. For example, Yu and Phillips [4] utilized the Gaussian method to estimate the parameters of continuous time short-term interest rate models. Faff and Gray [5] discussed the estimation and comparison of short-rate models by the generalised method of moments. Rossi [6] applied particle filters and maximum likelihood estimation to solve the parameter estimation for Cox–Ingersoll–Ross model. Wei et al. [7] utilized the Gaussian estimation method to investigate the parameter estimation for discretely observed Cox–Ingersoll–Ross model. However, it is well known that many financial processes exhibit discontinuous sample paths and heavy tailed properties (e.g., certain moments are infinite). These features cannot be captured by Brownian motion [8, 9]. Therefore, it is natural to replace the driving Brownian motion by Lévy noises. In recent years, with the development of Lévy process theory and its application in the fields of engineering systems, economic systems, and management systems, it has attracted great attention from scholars. Therefore, some authors considered parameter estimation for stochastic differential equations driven by Lévy noises. For example, Li and Ma [10] discussed the asymptotic properties of estimators in a stable Cox–Ingersoll–Ross model. Long [11] analyzed the least squares estimator for discretely observed 2Ornstein–Uhlenbeck processes with small Lévy noises. Then, Long et al. [8] tackled the least squares estimators for discretely observed stochastic processes driven by small Lévy noises. Singh et al. [12] utilized a randomized response model under Poisson distribution to estimate a rare sensitive attribute in two-stage sampling. Wei [13] used least squares estimation to discuss the discretely observed CoxCIngersollCRoss model driven by small symmetrical stable noises and studied the consistency and asymptotic distribution of the estimators. There have been many applications of small noise asymptotics to mathematical finance [14, 15]. From a practical point of view in parametric inference, it is more realistic and interesting to consider asymptotic estimation for diffusion processes with small noise based on discrete observations.Lotka–Volterra model is often used to model population growth of a single species. However, systems are more or less influenced by random factors. Thus, stochastic Lotka–Volterra equation, being a reasonable and popular approach to model population dynamics perturbed by random environment, has recently been studied by many authors both from a mathematical perspective and in the context of real biological dynamics [16–19] to explain the change of biodiversity also over time [20–23]. For example, Mao et al. [24] investigated a multidimensional stochastic Lotka–Volterra system driven by one-dimensional standard Brownian motion. They revealed that the environmental noise could suppress population explosion. Later, Mao [25] discussed a finite second moment of the stationary distribution under Brownian noise, which is very important in application. Bao et al. [26] and Bao and Yuan [27] considered a competitive Lotka–Volterra population model with Lévy jumps. Zhao et al. [28] studied the parameter estimation for stochastic Lotka–Volterra model by using the maximum likelihood method from continuous time observations. However, due to the limitation of instrument precision, it is impossible to observe the system from continuous time. Moreover, few literatures considered the consistency and asymptotic distribution of parameter estimators for stochastic Lotka–Volterra driven by α-stable noises. We consider the parameter estimation problem for discretely observed stochastic Lotka–Volterra model with small α-stable noises. The contrast function is given to obtain the least squares estimators. The consistency and asymptotic distribution of the estimators are discussed by Markov inequality, Cauchy–Schwarz inequality, and Gronwall’s inequality.The structure is that the stochastic Lotka–Volterra model driven by smallα-stable noises is introduced and the contrast function is given to obtain the least squares estimators in Section 2. In Section 3, the consistency of the estimators is proved and the asymptotic distribution of the estimators is studied. In Section 4, some simulations are made. The conclusion is given in Section 5.


## 2. Problem Formulation and Preliminaries

Ω,ℱ,ℙ is a basic probability space equipped with a right continuous and increasing family of σ-algebras ℱtt≥0 and Z=Zt,t≥0 is a strictly symmetric α-stable Lévy motion.A random variableη is said to have a stable distribution with index of stability α∈0,2, scale parameter σ∈0,∞, skewness parameter β∈−1,1, and location parameter μ∈−∞,∞ if it has the following characteristic function:(1)ϕηu=Eexpiuη=exp−σαuα1−iβsgnutanαπ2+iμu,ifα≠1,exp−σu1+iβ2πsgnulogu+iμu,ifα=1.We denoteη∼Sασ,β,μ. When μ=0, we say η is strictly α-stable; if in addition β=0, we call η symmetrical α-stable. Throughout this article, α-stable motion is strictly symmetrical and α∈1,2.We study the parametric estimation problem for stochastic Lotka–Volterra model driven by smallα-stable noises described by the following stochastic differential equation:(2)dXt=Xtθ−βXtdt+εXtdZt,t∈0,1,X0=x0,where θ and β are unknown parameters, ε∈0,1.Since the stochastic Lotka–Volterra model is driven by smallα-stable noises and due to the complexity of the α-stable noises, it is difficult to obtain the likelihood function. Thus, the maximum likelihood estimation cannot be used. Therefore, the contrast function is given to obtain least squares estimators.Consider the following contrast function:(3)ρn,εθ,β=∑i=1nXti−Xti−1−Xti−1α−βXti−1Δti−12ε2Xti−12Δti−1,where Δti−1=ti−ti−1=1/n.We obtain the estimators:(4)θ^n,ε=n∑i=1nXti−Xti−1∑i=1nXti−1+n2∑i=1nXti−12−n∑i=1nXti/Xti−1∑i=1nXti−12∑i=1nXti−12−n∑i=1nXti−12,β^n,ε=n2∑i=1nXti−Xti−1+n2∑i=1nXti−1−n∑i=1nXti/Xti−1∑i=1nXti−1∑i=1nXti−12−n∑i=1nXti−12.The work will be carried out under the following assumptions.Assumption 1.
θ and β are positive true values of the parameters.Assumption 2.
sup0≤t≤1EXt2<∞.


## 3. Main Results and Proofs

X0=Xt0,t≥0 is the solution to the underlying ordinary differential equation under the true value of the parameters:(5)dXt0=Xt0θ−βXt0dt,X00=x0.Discretizing equation (2), we obtain(6)Xti−Xti−1=θ∫ti−1tiXsds−β∫ti−1tiXs2ds+ε∫ti−1tiXsdZs.Then, a more explicit decomposition forθ^n,ε and β^n,ε can be given(7)θ^n,ε=θ∑i=1n∫ti−1tiXsds1/n∑i=1nXti−1−θ∑i=1n∫ti−1tiXs/Xti−1ds1/n∑i=1nXti−121/n∑i=1nXti−12−1/n∑i=1nXti−12+β∑i=1n∫ti−1tiXs2/Xti−1ds1/n∑i=1nXti−12−β∑i=1n∫ti−1tiXs2ds1/n∑i=1nXti−11/n∑i=1nXti−12−1/n∑i=1nXti−12+ε∑i=1n∫ti−1tiXsdZs1/n∑i=1nXti−1−ε∑i=1n∫ti−1tiXs/Xti−1dZs1/n∑i=1nXti−121/n∑i=1nXti−12−1/n∑i=1nXti−12,β^n,ε=θ∑i=1n∫ti−1tiXsds−θ∑i=1n∫ti−1tiXs/Xti−1ds1/n∑i=1nXti−11/n∑i=1nXti−12−1/n∑i=1nXti−12+β∑i=1n∫ti−1tiXs2/Xti−1ds1/n∑i=1nXti−1−β∑i=1n∫ti−1tiXs2ds1/n∑i=1nXti−12−1/n∑i=1nXti−12+ε∑i=1n∫ti−1tiXsdZs−ε∑i=1n∫ti−1tiXs/Xti−1dZs1/n∑i=1nXti−11/n∑i=1nXti−12−1/n∑i=1nXti−12.Before giving the theorems, we need to establish some preliminary results.Lemma 1 (see [29]).
Z is a strictly α-stable Lévy process and ϕ∈La.s.α. Then,(8)∫0tϕsdZs=Z′∘∫0tϕ+αsds−Z′′∘∫0tϕ−αsds,a.s.
IfZ is symmetric, that is, β=0, then, there exists some α-stable Lévy process Z′=dZ, such that(9)∫0tϕsdZs=Z′°∫0tϕsαds,a.s.Lemma 2.
Whenε⟶0 and n⟶∞, we have(10)sup0≤t≤1Xt−Xt0⟶P0.Proof.
Integrating on both sides of equation (2) and (8), we have(11)Xt−Xt0=θ∫0tXs−Xs0ds−β∫0tXs2−Xs02ds+ε∫0tXsdZs.
By using the Cauchy–Schwarz inequality, we obtain(12)Xt−Xt02≤3θ2∫0tXs−Xs0ds2+3β2∫0tXs2−Xs02ds2+3ε2∫0tXsdZs2≤3tθ2∫0tXs−Xs02ds+12tβ2K2∫0tXs−Xs02ds+3ε2∫0tXsdZs2=3tθ2+4β2K2∫0tXs−Xs02ds+3ε2∫0tXsdZs2≤3tθ2+4β2K2∫0tXs−Xs02ds+3ε2sup0≤t≤1∫0tXsdZs2,where K is the upper bound of Xt.
According to Gronwall’s inequality, we have(13)Xt−Xt02≤3ε2e3t2θ2+4β2K2sup0≤t≤1∫0tXsdZs2.
Squaring on both sides of equation (17),(14)sup0≤t≤1Xt−Xt0≤3εe3/2θ2+4β2K2sup0≤t≤1∫0tXsdZs.
By the Markov inequality, for any givenδ>0, when ε⟶0, we have(15)ℙ3εe3/2θ2+4β2K2sup0≤t≤1∫0tXsdZs>δ≤δ−13εe3/2θ2+4β2K2Esup0≤t≤1∫0tXsdZs≤Cδ−13εe3/2θ2+4β2K2E∫01Xsαds1/α≤Cδ−13εe3/2θ2+4β2K2K⟶0,namely,(16)sup0≤t≤1Xt−Xt0⟶P0.
The proof is complete.Lemma 3.
Whenε⟶0 and n⟶∞, we have(17)1n∑i=1nXti−12⟶P∫01Xt02dt.Proof.
As(18)1n∑i=1nXti−12=1n∑i=1nXti−102+1n∑i=1nXti−12−Xti−102,we obtain(19)1n∑i=1nXti−102⟶P∫01Xt02dt.
According to Lemma2, when ε⟶0 and n⟶∞, we have(20)1n∑i=1nXti−12−Xti−102≤1n∑i=1nXti−1+Xti−10Xti−1−Xti−10≤2Ksup0≤t≤1Xt−Xt0⟶P0.
According to equations (23) and (24), we obtain(21)1n∑i=1nXti−12⟶P∫01Xt02dt.
The proof is complete.
Applying the same methods used in Lemma3, we have(22)1n∑i=1nXti−1⟶P∫01Xt0dt.
In the following theorem, the consistency of the least squares estimators is proved.Theorem 1.
Whenε⟶0, n⟶∞, and εn1−1/α⟶0, the least squares estimators θ^n,ε and β^n,ε are consistent, namely,(23)θ^n,ε⟶Pθ,β^n,ε⟶Pβ.Proof.
According to Lemma3, we have(24)1n∑i=1nXti−12−1n∑i=1nXti−12⟶P∫01Xt0dt2−∫01Xt02dt.
Whenε⟶0 and n⟶∞, we obtain(25)θ∑i=1n∫ti−1tiXsds1n∑i=1nXti−1⟶Pθ∫01Xtdt∫01Xt0dt,θ∑i=1n∫ti−1tiXsXti−1ds1n∑i=1nXti−12⟶Pθ∫01XtXt0dt∫01Xt02dt.
According to Lemma2, we have(26)θ∑i=1n∫ti−1tiXsds1n∑i=1nXti−1−θ∑i=1n∫ti−1tiXsXti−1ds1n∑i=1nXti−12⟶Pθ∫01Xt0dt2−∫01Xt02dt.
From equations (28) and (29), we obtain(27)β∑i=1n∫ti−1tiXs2Xti−1ds1n∑i=1nXti−12−β∑i=1n∫ti−1tiXs2ds1n∑i=1nXti−1⟶P0.
By the Markov inequality, we have(28)Pε∑i=1n∫ti−1tiXsdZs>δ≤δ−1ε∑i=1nE∫ti−1tiXsdZs≤Cδ−1ε∑i=1nE∫ti−1tiXsαds1/α≤CKδ−1εn1−1/α⟶0,where C is constant.
Asε⟶0, n⟶∞, and εn1−1/α⟶0, we obtain(29)ε∑i=1n∫ti−1tiXsdZs⟶P0.
According to Lemma2, we obtain(30)ε∑i=1n∫ti−1tiXsXti−1dZs≤ε∑i=1n1Xti−1∫ti−1tiXsdZs≤ε∑i=1n1Xti−10+1Xti−1−1Xti−10∫ti−1tiXsdZs≤ε∑i=1n1Xti−10∫ti−1tiXsdZs+εsup0≤t≤11Xt−1Xt0∫ti−1tiXsdZs.
By the Markov inequality, we have(31)P|ε∑i=1n|1Xti−10∫ti−1tiXsdZs>δ≤δ−1ε∑i=1n1Xti−10E∫ti−1tiXsdZs≤Cδ−1ε∑i=1n1Xti−10E∫ti−1tiXsαds1/α≤CKδ−1εn1−1/α1n∑i=1n1Xti−10⟶P0.
Asε⟶0, n⟶∞, and εn1−1/α⟶0, we obtain(32)ε∑i=1n1Xti−10∫ti−1tiXsdZs⟶P0.
According to Lemma2, when ε⟶0 and n⟶∞,(33)εsup0≤t≤11Xt−1Xt0∫ti−1tiXsdZs⟶P0.
From equations (37) and (38), we have(34)ε∑i=1n∫ti−1tiXsXti−1dZs⟶P0.
According to equations (27), (30), (31), (33), and (39), when ε⟶0, n⟶∞, and εn1−1/α⟶0,(35)θ^n,ε⟶Pθ.
Using the same methods in Theorem1, we obtain(36)θ∑i=1n∫ti−1tiXsds−θ∑i=1n∫ti−1tiXsXti−1ds1n∑i=1nXti−1⟶P0,(37)β∑i=1n∫ti−1tiXs2Xti−1ds1n∑i=1nXti−1−β∑i=1n∫ti−1tiXs2ds⟶Pβ∫01Xt0dt2−∫01Xt02dt.
Together with the results that(38)ε∑i=1n∫ti−1tiXsdZs⟶P0,(39)ε∑i=1n∫ti−1tiXsXti−1dZs1n∑i=1nXti−1⟶P0,when ε⟶0, n⟶∞, and εn1−1/α⟶0, we have(40)β^n,ε⟶Pβ.
The proof is complete.Theorem 2.
Whenε⟶0, n⟶∞, and nε⟶∞,(41)ε−1θ^n,ε−θ⟶d∫01Xt0αdt1/α∫01Xt0dt−∫01Xt02dt∫01Xt0dt2−∫01Xt02dtSα1,0,0,ε−1β^n,ε−β⟶d∫01Xt0αdt1/α−∫01Xt0dt∫01Xt0dt2−∫01Xt02dtSα1,0,0.Proof.
According to the explicit decomposition forθ^n,ε, we have(42)ε−1θ^n,ε−θ=ε−1θ∑i=1n∫ti−1tiXsds−1/n∑i=1nXti−11/n∑i=1nXti−11/n∑i=1nXti−12−1/n∑i=1nXti−12+ε−1θ1−∑i=1n∫ti−1tiXs/Xti−1ds1/n∑i=1nXti−121/n∑i=1nXti−12−1/n∑i=1nXti−12+ε−1β∑i=1n∫ti−1tiXs2/Xti−1ds1/n∑i=1nXti−12−ε−1β∑i=1n∫ti−1tiXs2ds1/n∑i=1nXti−11/n∑i=1nXti−12−1/n∑i=1nXti−12+∑i=1n∫ti−1tiXsdZs1/n∑i=1nXti−1−∑i=1n∫ti−1tiXs/Xti−1dZs1/n∑i=1nXti−121/n∑i=1nXti−12−1/n∑i=1nXti−12.
From Lemma2, when ε⟶0, n⟶∞, and nε⟶∞,(43)∑i=1n∫ti−1tiXsds−1n∑i=1nXti−1⟶P0,ε−1n−1∑i=1nXti−1≤ε−1n−1∑i=1nXti−1=ε−1n−1∑i=1nXti−1−Xti−10+Xti−10≤ε−1n−1∑i=1nXti−1−Xti−10+Xti−10≤ε−1n−1sup0≤t≤1Xt−Xt0+Xt0⟶P0.
From equations (45) and (46), we have(44)ε−1θ∑i=1n∫ti−1tiXsds−1/n∑i=1nXti−11/n∑i=1nXti−11/n∑i=1nXti−12−1/n∑i=1nXti−12⟶P0,ε−1θ1−∑i=1n∫ti−1tiXs/Xti−1ds1/n∑i=1nXti−121/n∑i=1nXti−12−1/n∑i=1nXti−12⟶P0,(45)ε−1β∑i=1n∫ti−1tiXs2/Xti−1ds1/n∑i=1nXti−12−ε−1β∑i=1n∫ti−1tiXs2ds1/n∑i=1nXti−11/n∑i=1nXti−12−1/n∑i=1nXti−12⟶P0.
According to Lemma2, we obtain(46)∑i=1n∫ti−1tiXsdZs=∑i=1n∫ti−1tiXs0dZs+∑i=1n∫ti−1tiXs−Xs0dZs.
Using the Markov inequality and Holder’s inequality, for any givenδ>0, we have(47)P∑i=1n∫ti−1tiXs−Xs0dZs>δ≤δ−1∑i=1nE∫ti−1tiXs−Xs0dZs≤Cδ−1∑i=1nE∫ti−1tiXs−Xs0αds1/α≤Cδ−1∑i=1nEsup0≤t≤1Xt−Xt0n−1/α1/2≤Cδ−1n1−1/αEsup0≤t≤1Xt−Xt0⟶0,as ε⟶0, n⟶∞, and n1−1/α⟶0.
Moreover,(48)∑i=1n∫ti−1tiXs0dZs=∫01∑i=1nXs01ti−1,tisdZs=Z′°∫01∑i=1nXs01ti−1,tisαds,where Z′=dZ.
According to Lemma1, we have(49)∫01∑i=1nXs01ti−1,tisαds⟶∫01Xs0αds,(50)Z′°∫01∑i=1nXs01ti−1,tisαds⟶a.s.Z′°∫01Xs0αds.
From equations (53) and (54),(51)∑i=1n∫ti−1tiXs0dZs⟶d∫01Xt0α1/αSα1,0,0.
According to equations (45)–(51) and (55), we have(52)ε−1θ^n,ε−θ⟶d∫01Xt0αdt1/α∫01Xt0dt−∫01Xt02dt∫01Xt0dt2−∫01Xt02dtSα1,0,0.
From equation (8), we obtain(53)ε−1β^n,ε−β=ε−1θ∑i=1n∫ti−1tiXsds−ε−1θ∑i=1n∫ti−1tiXs/Xti−1ds1/n∑i=1nXti−11/n∑i=1nXti−12−1/n∑i=1nXti−12+ε−1β∑i=1n∫ti−1tiXs2/Xti−1ds1/n∑i=1nXti−1−ε−1β∑i=1n∫ti−1tiXs2ds1/n∑i=1nXti−12−1/n∑i=1nXti−12+∑i=1n∫ti−1tiXsdZs−∑i=1n∫ti−1tiXs/Xti−1dZs1/n∑i=1nXti−11/n∑i=1nXti−12−1/n∑i=1nXti−12−ε−1β.
According to equations (25)–(27) and (55), we have(54)ε−1θ∑i=1n∫ti−1tiXsds−ε−1θ∑i=1n∫ti−1tiXs/Xti−1ds1/n∑i=1nXti−11/n∑i=1nXti−12−1/n∑i=1nXti−12⟶P0,(55)ε−1β∑i=1n∫ti−1tiXs2/Xti−1ds1/n∑i=1nXti−1−ε−1β∑i=1n∫ti−1tiXs2ds1/n∑i=1nXti−12−1/n∑i=1nXti−12−ε−1β⟶P0,(56)∑i=1n∫ti−1tiXsdZs−∑i=1n∫ti−1tiXs/Xti−1dZs1/n∑i=1nXti−11/n∑i=1nXti−12−1/n∑i=1nXti−12⟶d∫01Xt0αdt1/α−∫01Xt0dt∫01Xt0dt2−∫01Xt02dtSα1,0,0.
From equations (54)–(56), we have(57)ε−1β^n,ε−β⟶d∫01Xt0αdt1/α−∫01Xt0dt∫01Xt0dt2−∫01Xt02dtSα1,0,0.
The proof is complete.


## 4. Simulation

In this experiment, we generate a discrete sampleXti−1i=1,…,n and compute θ^n,ε and β^n,ε from the sample. Let x0=0.2. For every given true value of the parameters-θ,β, the size of the sample is represented as“Size n” and given in the first column of the table. In Table 1, ε=0.1, the size is increasing from 1000 to 5000. In Table 2, ε=0.01, the size is increasing from 10,000 to 50,000. The tables list the value of “θ^n,ε”, “β^n,ε,” and the absolute errors of least squares estimators.Table 1
Least squares estimator simulation results ofθ and β.
True valueAverage valueAbsolute errorθ,βSizenθ^n,εβ^n,εθ^n,ε−θβ^n,ε−β(1, 2)10001.15262.13680.15260.136830001.03792.01510.03790.015150001.00102.00070.00100.0007(3, 4)10003.16434.15380.16430.153830003.02364.03200.02360.032050003.00124.00140.00120.0014Table 2
Least squares estimator simulation results ofθ and β.
True valueAverage valueAbsolute errorθ,βSizenθ^n,εβ^n,εθ^n,ε−θβ^n,ε−β(1, 2)10,0001.10412.09250.10410.092530,0001.00532.00870.00530.008750,0001.00052.00070.00050.0007(3, 4)10,0003.11544.10250.11540.102530,0003.01274.00890.01270.008950,0003.00094.00030.00090.0003Two tables illustrate that whenn is large enough and ε is small enough, the obtained estimators are very close to the true parameter value. Therefore, the methods used in this paper are effective and the obtained estimators are good.In Figure1, θ=1; when ε=0.1, the size is increasing from 1000 to 5000; when ε=0.01, the size is increasing from 10,000 to 50,000. In Figure 2, β=2; when ε=0.1, the size is increasing from 1000 to 5000; when ε=0.01, the size is increasing from 10,000 to 50,000. Two figures illustrate that when n is large enough and ε is small enough, the obtained estimators are very close to the true parameter value.Figure 1
The simulation of the estimatorθ^n,ε with θ=1.
(a)(b)Figure 2
The simulation of the estimatorβ^n,ε with β=2.
(a)(b)


## 5. Conclusion

The parameter estimation problem for discretely observed stochastic Lotka–Volterra model driven by smallα-stable noises has been studied. The contrast function has been given to obtain the least squares estimators. The consistency and asymptotic distribution of the least squares estimators have been discussed by using the Markov inequality, Cauchy–Schwarz inequality, and Gronwall’s inequality. Further research topics will include parameter and state estimation for partially observed stochastic system driven by α-stable noises.



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