Pathwise estimation of the stochastic functional Kolmogorov-type system
Introduction
The functional Kolmogorov-type system for n interacting species is described by the n-dimensional functional differential equationwhere represents the matrix with all elements zero except those on the diagonal which are and is defined by . 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 systemon , where is a scalar Brownian motion, and Here both f and g are locally Lipschitz continuous. Clearly, Eq. (2) includes the following two equations:When , all Eqs. (2)–(4) return toSince 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.
Section snippets
Preliminaries
Throughout this paper, unless otherwise specified, we use the following notations. Let be the Euclidean norm in . If A is a vector or matrix, its transpose is denoted by . If A is a matrix, its trace norm is denoted by . Let , and let . Denoted by the family of continuous functions from to with the norm , which forms a Banach space. Let and . For any , let
A general result
To show our idea about the pathwise estimation of solutions clearly, we first consider the following general n-dimensional stochastic functional differential equation:on . Here are locally Lipschitz continuous and is an m-dimensional Brownian motion. Assume that Eq. (10) almost surely admits a unique global positive strong solution. Then we have the following pathwise estimation. Theorem 3.1 Let . For the global positive
Stochastic functional Kolmogorov-type systems
This section mainly applies the result of the previous section to Eq. (2) as well as Eqs. (3) and (4). For any and , we firstly list the following conditions for both f and g that we will need.
- (H1)
There exist and the probability measure on such that
- (H2)
There exist and the probability measure on such that
- (H3)
There exist and the
Some special equations
In this section, we apply the results in the previous two sections to some special equations to obtain some pathwise estimation, which includes some existing results for some stochastic Lotka–Volterra equations. Firstly, applying Corollaries 4.3 and 4.5 to Eq. (5), conditions (H3) and (H4) should be replaced by conditions (H) and (H), which implies . We have the following result. Theorem 5.1 For the global positive solution of Eq. (5), if one of the following conditions hold
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