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.
- stochastic differential equations
- periodic solution
- Markov process
- Massera theorem
The theory of stochastic differential equations is given for the first time by Itô  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 . 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 , Vârsan , Da Prato , 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 . 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:
nonlinear SDE in dimension
Let be the complete probability space with a filtration satisfying the usual conditions
is an increasing family of sub algebras containing negligible sets of F and is continuous at right.
Let a Brownian motion , adapted to , i.e., is measurable. We consider the SDE
The functions and are measurable. We suppose that is the completion of for all and the initial condition is independent of , for and .
Suppose that the functions and satisfy the global Lipschitz and the linear growth conditions
We know that if and satisfy these conditions, then the system (1) admits a single global solution.
We note by the space of random measurable functions for all , satisfying the relation
we consider in the norm
is the Banach space.
2.1. Markov property
The following result proves that the solution of the SDE (1) is a Markov process.
Then the process solution of SDE (1), is a Markovian process with a transition function
Let be a transition function; we construct a Markov process with an initial arbitrary distribution. In a particular case, for , we associate with the function a family of a Markov process such that the processes exist with initial point in i.e.,
2.2. Notions of periodicity and boundedness
Définition 1. A stochastic process is said to be periodic with period if its finite dimensional distributions are periodic with periodic , i.e., for all and the joint distributions of the stochastic processes are independent of
Remark 1. If is periodic, then , are periodic, in this case, this process is said to be periodic in the wide sense.
Définition 2. The function for is said to be periodic if is periodic in
Définition 3. The Markov families are said to be uniformly bounded if
We denote as the family of all Markov process for and in Remark 2. It is easy to see that all borné Markov processes is uniformly bounded.
Remark 2. It is easy to see that all borné Markov processes is uniformly bounded.
Lemme 1. (, Theorem 3.2 and Remark 3.1, pp. 66–67) A necessary and sufficient condition for the existence of a Markov periodic with a given periodic transition function is that for some are uniformly stochastically continuous and
if the transition function satisfies the following not very restrictive assumption
for some function which tends to infinity as
In Eq. (4), we have
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 with periodic transition functions are uniformly bounded uniformly stochastically continuous, then there is a periodic Markov process.
Proof. By using a Markov inequality , we have
Then, for such that for all
that is, Eq. (4). From Lemma 1, we have a periodic Markov process.
3. Main result
Let the SDE
We assume that this SDE satisfies the conditions as in Section 2 after Eq. (1).
H1) the functions and are periodic in .
H2) the functions and satisfy the condition
where is a concave non-decreasing function.
Lemme 3. (, Lemme 3.4) Assume that and verify
then, the solutions of periodic SDE (6) are uniformly stochastically continuous.
We prove the Massera’s theorem for the SDE in dimension
Theorem 2. Under if the solutions of the SDE (6) are bounded, then there is a periodic Markov process.
Proof. We note by an -bounded solution of SDE (6), from Theorem 1, this solution is unique a Markov process that is measurable. Suppose that is a transition function of Markov process under and since depend of then this function is periodic in In the other hand, is concave non-decreasing function, we get
From the boundedness of , then under : such that
for By Lemma 3, we have is uniformly bounded and uniformly stochastically continuous, this gives, the conditions of Lemma 2 are verified, finally, we can conclude the existence of the periodic Markov process. □