- Open Access
On the oscillation of solutions for a class of second-order nonlinear stochastic difference equations
Advances in Difference Equations volume 2014, Article number: 91 (2014)
In this paper, we investigate the asymptotical behavior for a partial sum sequence of independent random variables, and we derive a law of the iterated logarithm type. It is worth to point out that the partial sum sequence needs not to be an independent increment process. As an application of the theory established, we also give a sufficient criterion on the almost sure oscillation of solutions for a class of second-order stochastic difference equation of neutral type.
To date, the asymptotic behavior of the solutions to deterministic difference equations has been discussed in many papers. Among them there are many papers about the oscillation of the solutions to deterministic difference equations. In a related field, the asymptotic behavior of the solutions to stochastic difference equation was discussed in many papers, and there have been also very fruitful achievements. However, there is little known about the oscillation of the solutions of stochastic difference equations. Recently Appleby and Rodkina  and Appleby et al.  first investigated the oscillation of the solutions of first-order nonlinear stochastic difference equations. In , the authors considered the following equation:
The solution of (1.1) can be expressed as
where is a sequence of independent and identically distributed random variables. Note that the sequence () has the independent increment property, and as a result the authors can analyze the limit behavior of system (1.1) by the law of the iterated logarithm and they obtain a beautiful result, i.e., the solution of (1.1) is an almost sure oscillation under some sufficient conditions. Motivated by , in this paper we investigate the oscillation of the solution for the following second-order nonlinear stochastic difference equation:
Here is the forward difference operator. This equation can be viewed as a stochastic analog of the following classical deterministic difference equations:
The solution of (1.3) can be expressed as
where is determined by and . The proof of (1.6) is to be given in Section 4. Denoting for any , it implies . It is obvious that is not independent of even though is a sequence of independent random variables. That is to say that does not have the independent increment property, which means that we do not directly adopt law of the iterated logarithm to to obtain their limit behavior. However, under some restrictions we can use a roundabout way to analyze the limit behavior on by the law of the iterated logarithm, then we give some sufficient conditions on the almost sure oscillation for (1.3). These results and proofs are deferred to the following sections.
2 Definitions and assumptions
Throughout this paper, the following notation, definitions, and assumptions are needed. ℕ and ℝ denote, respectively, the positive integer numbers and real numbers. Let for every . denotes a complete probability space. is a random variable sequence defined on . We suppose that filtration is naturally generated, namely that . We use the standard abbreviations ‘a.s.’ and ‘i.i.d.’ instead of ‘almost surely’ and ‘independent identically distribution’, respectively. For simplicity, we denote throughout this paper.
For (1.3), the following elementary assumptions are needed.
(A2.1) for every ,
(A2.2) for every ,
(A2.3) F is assumed to be Borel measurable and to obey for , and ,
(A2.4) is assumed to be independent identically distributed random variable sequence defined on , and, moreover, , .
Definition 2.1 is called a solution of (1.3) with initial values , . is constituted by and , , where is obtained by steps of iteration of (1.3) with initial values , .
Definition 2.2 The solution of (1.3) is said to be a.s. oscillatory if
where ‘i.o.’ stands for infinitely often.
Definition 2.3 Equation (1.3) is said to be a.s. oscillatory if its any solution is a.s. oscillation.
3 Law of the iterated logarithm
The classical Kolmogorov law of the iterated logarithm is an effective tool in studying the limit behavior of partial sum of independent random variable sequence (see ). In 1973, Chow and Teicher  generalized the classical results and obtained the following law of the iterated logarithm for weighted averages.
Theorem 3.1 (Iterated logarithm laws of weighted averages)
If are i.i.d. random variables with , and are real constants satisfying
for some C in , then
On the above results, notice that is independent of .
Now we establish a new result about law of iterated logarithm type. Suppose that satisfies for every , and is an i.i.d. random variable sequence defined on with , . For , set
Here we appoint if . It is obvious that is not independent of , but we have
Note that is a monotony increasing sequence, hence we give the following hypothesis:
(C.1) There exist constants and such that .
Lemma 3.2 If (C.1) holds, then
Proof Set , for every , we have
In view of (C.1), for any fixed positive integer number m, there exists such that for every . As n is sufficiently large, we have
It is clear that
In view of (3.6) and (3.7), as n is sufficiently large, we get
Letting in the above formula (3.9), and combining (3.8) and (C.1), we have
Setting in the above inequalities, we obtain (3.3).
Let , , . According to (3.3), we have () and for every . Therefore
Taking the limit on both sides of the above equation, result (3.4) follows.
Equation (3.5) can be proved similarly. □
Lemma 3.3 (Law of the iterated logarithm on defined by (3.1))
If (C.1) holds, then
Here α is described as (C.1).
Proof According to (3.1), it is clear that is an i.i.d. random variable sequence and, moreover, , . Setting , for any , it is obvious that , which satisfies the conditions of Theorem 3.1. In , letting , , it is clear that () by (C.1). By (3.3), it is found that there is such that . It is also known that which satisfies the conditions of Theorem 3.1. Hence
So there is with such that all equalities of of the left side of (3.13)-(3.16) hold on . Therefore by (3.13), for there exists such that
By (3.15) and (3.16), it is clear that
is a bounded sequence. Therefore there is a and such that
By (3.2)-(3.5) and (3.17)-(3.18), we get
Hence (3.11) holds.
Equation (3.12) can be proved similarly. □
To proceed the study, we give another assumption:
(C.2) There exist such that .
Note that in (C.2). It is obvious that the conclusions of Lemma 3.2 and Lemma 3.3 are also right when (C.2) replaces (C.1).
4 The main results
In this section, we give the main results on the oscillation of the solution of (1.3).
Let be an any solution of (1.3) with arbitrary initial values . Set
By (1.3), one obtains
Let in the above equation, and one has
So for any , one has
Theorem 4.1 Suppose that (1.3) satisfies, respectively, (A2.1)-(A2.4), then (1.3) is an almost sure oscillation under condition (C.2).
Proof Suppose that the result is not right, then (1.3) must have a solution, denoted as , and it is not an almost sure oscillation. That is to say at least one is not true between and .
1. Firstly, we assume that holds. For this case, it implies that there is with such that the following equation holds:
as . Here . By virtue of (3.3) of Lemma 3.2, we have
By Lemma 3.3, we obtain
Therefore there exists with such that for any
Setting , it is obvious that and (4.2) and (4.3) are also true for any . For any , we have
So for any , we have
as . Hence
Therefore the left-hand side of (4.4) is nonnegative.
On the right-hand side of (4.4), we have
due to (). Therefore we find that it is an oscillation by (4.3). This is a contradiction.
2. Secondly, we assume that holds. We may get a contradiction for case 2 similar to case 1. Thus we finish the proof of Theorem 4.1. □
Remark 1 If the condition (C.1) replaces the condition (C.2) of Theorem 4.1 and the other conditions are not changed, then the conclusions of Theorem 4.1 cannot be guaranteed to be right as .
The example is as follows.
Example 1 Take , , ,
in (1.3), then (1.3) becomes the following special equation:
here is assumed to satisfy (A2.4) and be locally bounded, i.e., there is and with such that , , .
It is clear that r, f satisfy, respectively, (A2.1) and (A2.2), and F satisfies (A2.3), and (), i.e., satisfies (C.1) but it does not satisfy (C.2).
Now we illustrate that (4.6) is not an a.s. oscillation. Let be a solution of (4.4) with initial values , , then we have
for every . Here is determined by , and .
About the terms of (4.7), the following assertions are right.
, a.s. ().
There is finite value measurable function defined on such that , a.s. ().
Proof of the assertions (i) Setting , , , because is an i.i.d. random variable sequence and, moreover, , , then the have the same properties. It is obvious that
By Theorem 3.1, we have
So we have
By (4.5) and , we have
It is obvious that , . So we get
By (4.7) and the above assertions (i)-(iii), we obtain
for every . Here and are mutually independent.
We choose , satisfying . Due to for any , and
as , one obtains
Thus on by (4.9). Therefore is not an almost sure oscillation, and consequently, (4.6) is not almost surely oscillatory by Definition 2.3. □
Appleby J, Rodkina A: On the oscillation of solutions of stochastic difference equations with state-independent perturbations. Int. J. Differ. Equ. 2007, 2(2):139–164.
Appleby J, Rodkina A, Schurz H: On the oscillations of stochastic difference equations. Mat, Enseñ. Univ. 2009, 17(2):1–10.
Loève M: Probability Theory. 4th edition. Springer, New York; 1978.
Chow Y, Teicher H: Iterated logarithm laws for weighted averages. Z. Wahrscheinlichkeitstheor. Verw. Geb. 1973, 26(2):87–94. 10.1007/BF00533478
Yan S, Wang J, Liu X: Fundamentals of Probability. Science Press, Beijing; 1982.
This work was supported in part by the National Natural Science Foundation of China under Grants no. 11101054, Hunan Provincial Natural Science Foundation of China under Grant no. 12JJ4005 and the Scientific Research Funds of Hunan Provincial Science and Technology Department of China under Grants no. 2010FJ6036.
The authors declare that they have no competing interests.
The authors contributed equally in this paper. They read and approved the final manuscript.
About this article
Cite this article
Yu, Z., Zhu, E. & Zeng, J. On the oscillation of solutions for a class of second-order nonlinear stochastic difference equations. Adv Differ Equ 2014, 91 (2014). https://doi.org/10.1186/1687-1847-2014-91
- second-order nonlinear stochastic difference equations
- almost sure oscillation