Pathwise estimation of the stochastic functional Kolmogorov-type system

Abstract

In this paper we stochastically perturb the functional Kolmogorov-type system

$$\dot{x}(t)=\mathrm{diag}(x_1(t),\dots ,x_n(t))f(x_t)$$

into the stochastic functional differential equation

$$\mathrm{d}x(t)=\mathrm{diag}(x_1(t),\dots ,x_n(t))[f(x_t)\mathrm{d}t+g(x_t)\mathrm{d}w(t)]\text{.}$$

This paper studies pathwise estimation of the solution to this equation. As the applications, this paper also discusses the pathwise estimation of the solutions of various stochastic Lotka–Volterra-type systems.

Introduction

The functional Kolmogorov-type system for n interacting species is described by the n-dimensional functional differential equation

$$\dot{x}(t)=\mathrm{diag}(x_1(t),\dots ,x_n(t))f(x_t)\text{,}$$

where $x=(x_1,\dots ,x_n{)}^{\mathrm{T}},\mathrm{diag}(x_1,\dots ,x_n)$ represents the $n\times n$ matrix with all elements zero except those on the diagonal which are $x_1,\dots ,x_n,f=(f_1,\dots ,f_n{)}^{\mathrm{T}}$ and $x_t\in C([-\tau ,0];{\mathbb{R}}^n)$ is defined by $x_t(\theta )=x(t+\theta ),\theta \in [-\tau ,0]$. There is an extensive literature concerned with the dynamics of this system and the Lotka–Volterra-type system as its special case and we here only mention [1], [2], [3], [4], [5].

On the other hand, population systems are often subject to environmental noise. Recently, stochastic Lotka–Volterra systems receive the increasing attention. Bahar and Mao [6] and Mao [7] reveal that the noise plays an important role to suppress the growth of the solution, and [8], [9] show the stochastic system behaves similarly with the deterministic system under different stochastic perturbations, respectively. These indicate clearly that different structures of environmental noise may have different effects on Lotka–Volterra systems.

However, little is as yet known about properties of stochastic functional Kolmogorov-type systems although they may be seen as the generalized stochastic functional Lotka–Volterra systems. This paper will examine the pathwise estimation of stochastic functional Kolmogorov-type systems under the general noise structures. Consider the n-dimensional stochastic functional Kolmogorov-type system

$$\mathrm{d}x(t)=\mathrm{diag}(x_1(t),\dots ,x_n(t))[f(x_t)\mathrm{d}t+g(x_t)\mathrm{d}w(t)]$$

on $t⩾0$, where $w(t)$ is a scalar Brownian motion, and

$$f=(f_1,\dots ,f_n{)}^{\mathrm{T}}:C([-\tau ,0];{\mathbb{R}}^n)\to {\mathbb{R}}^n,g=(g_1,\dots ,g_n{)}^{\mathrm{T}}:C([-\tau ,0];{\mathbb{R}}^n)\to {\mathbb{R}}^n\text{.}$$

Here both f and g are locally Lipschitz continuous. Clearly, Eq. (2) includes the following two equations:

$$\mathrm{d}x(t)=\mathrm{diag}(x_1(t),\dots ,x_n(t))[f(x_t)\mathrm{d}t+g(x(t))\mathrm{d}w(t)]\text{,}$$

$$\mathrm{d}x(t)=\mathrm{diag}(x_1(t),\dots ,x_n(t))[f(x(t))\mathrm{d}t+g(x_t)\mathrm{d}w(t)]\text{.}$$

When $\tau =0$, all Eqs. (2)–(4) return to

$$\mathrm{d}x(t)=\mathrm{diag}(x_1(t),\dots ,x_n(t))[f(x(t))\mathrm{d}t+g(x(t))\mathrm{d}w(t)]\text{.}$$

Since this paper mainly examines the pathwise estimation of the solution to stochastic functional Kolmogorov-type system, we assume that there exists a unique global positive solution (strong solution) for all discussed equations.

In Section 2, we give some necessary notations and lemmas. To show our idea clearly, Section 3 studies the pathwise estimation for general stochastic functional differential equations. Applying the result of Section 3, we give various conditions under which stochastic functional Kolmogorov systems show the nice pathwise properties in Section 4. As applications of Sections 3 and 4, Section 5 discusses several special equations, including various stochastic Lotka–Volterra systems.