An Extension of Massera’s Theorem for N -Dimensional Stochastic Differential Equations An Extension of Massera ’ s Theorem for N-Dimensional Stochastic Differential Equations

In this chapter, we consider a periodic SDE in the dimension n ≥ 2, and we study the existence of periodic solutions for this type of equations using the Massera principle. On the other hand, we prove an analogous result of the Massera ’ s theorem for the SDE considered.


Introduction
The theory of stochastic differential equations is given for the first time by Itô [7] in 1942. This theory is based on the concept of stochastic integrals, a new notion of integral generalizing the Lebesgue-Stieltjes one.
The stochastic differential equations (SDE) are applied for the first time in the problems of Kolmogorov of determining of Markov processes [8]. This type of equations was, from the first work of Itô, the subject of several investigations; the most recent include the generalization of known results for EDO, such as the existence of periodic and almost periodic solutions. It has, among others, the work of Bezandry and Diagana [1,2], Dorogovtsev [4], Vârsan [12], Da Prato [3], and Morozan and his collaborators [10,11].
The existence of periodic solutions for differential equations has received a particular interest. We quote the famous results of Massera [9]. In its approach, Massera was the first to establish a relation between the existence of bounded solutions and that of a periodic solution for a nonlinear ODE.
In this work, we will prove an extension of Massera's theorem for the following:

Preliminaries
Let Ω; F; F t f g t ≥ 0 ; P À Á be the complete probability space with a filtration F t f g t ≥ 0 satisfying the usual conditions is an increasing family of sub algebras containing negligible sets of F and is continuous at right.
The functions a t; x ð Þ : R þ Â R n ! R n and b t; x ð Þ : R þ Â R n ! R nÂm are measurable. We suppose that F t is the completion of σ W r ; t 0 ≤ r ≤ t f gfor all t ≥ t 0 , and the initial condition z is independent of W t , for t ≥ t 0 and E z j j p < ∞.
Suppose that the functions a t; x ð Þ and b t; x ð Þ satisfy the global Lipschitz and the linear growth conditions We know that if a and b satisfy these conditions, then the system (1) admits a single global solution.
We note by B the space of random F t Àmeasurable functions x t ð Þ for all t, satisfying the relation we consider in B the norm is the Banach space.

Markov property
The following result proves that the solution of the SDE (1) is a Markov process.
Then the process X t , solution of SDE (1), is a Markovian process with a transition function Let p s; x; t; A ð Þbe a transition function; we construct a Markov process with an initial arbitrary distribution. In a particular case, for t > s, we associate with the function p s; x; t; A ð Þ a family X s;z ð Þ t; ω ð Þof a Markov process such that the processes X s;z ð Þ t; ω ð Þexist with initial point z in s, i.e.,

Notions of periodicity and boundedness
Définition 1. A stochastic process X t; ω ð Þ is said to be periodic with period T T > 0 ð Þif its finite dimensional distributions are periodic with periodic T, i.e., for all m ≥ 0, and t 1 , t 2 , …t m ∈ R þ the joint distributions of the stochastic processes X t1þkT ω ð Þ, X t2þkT ω ð Þ, …X tmþkT ω ð Þ are independent of k k ∈ Z ð Þ: We denote X t0;z ð Þ ω ð Þ as the family of all Markov process for t 0 ∈ ℝ þ and z in L p : Remark 2. It is easy to see that all L p Àborné Markov processes X t , i:e∃M > 0; ∀t ≥ t 0 : X t k k if the transition function p s; X s ; t; A ð Þsatisfies the following not very restrictive assumption for some function β R ð Þ which tends to infinity as R ! ∞: In Eq. (4), we have R ∈ R * þ : The conditions of Lemma 1 are of little use for stochastic differential equations, since the properties of transition functions of such processes are usually not expressible in terms of the coefficients of the equation. So, in the following, we will give some new useful sufficient conditions in terms of uniform boundedness and point dissipativity of systems.
Lemme 2. If Markov families X t0;z ð Þ ω ð Þ with TÀperiodic transition functions are uniformly bounded uniformly stochastically continuous, then there is a TÀperiodic Markov process.

Main result
We assume that this SDE satisfies the conditions as in Section 2 after Eq. (1).
Suppose that H 1 ) the functions a t; x ð Þ and b t; x ð Þ are TÀperiodic in t.
H 2 ) the functions a t; x ð Þ and b t; x ð Þ satisfy the condition where ϕ is a concave non-decreasing function. We prove the Massera's theorem for the SDE in dimension n ≥ 2: An Extension of Massera's Theorem for N-Dimensional Stochastic Differential Equations http://dx.doi.org/10.5772/intechopen.73183