Open access peer-reviewed chapter

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

By Boudref Mohamed Ahmed, Berboucha Ahmed and Osmanov Hamid Ibrahim Ouglu

Submitted: October 24th 2017Reviewed: December 18th 2017Published: May 23rd 2018

DOI: 10.5772/intechopen.73183

Downloaded: 784


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

1. 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:

nonlinear SDE in dimension n2


2. Preliminaries

Let ΩFFtt0Pbe the complete probability space with a filtration Ftt0satisfying the usual conditions

  • Ftt0is an increasing family of sub algebras containing negligible sets of Fand is continuous at right.


Let a Brownian motion Wt, adapted to Ftt0, i.e., W0=0,t0,Wtis Ftmeasurable. We consider the SDE


in ΩFFtt0P.

The functions atx:R+×RnRnand btx:R+×RnRn×mare measurable. We suppose that Ftis the completion of σWrt0rtfor all tt0,and the initial condition zis independent of Wt, for tt0and Ezp<.

Suppose that the functions atxand btxsatisfy the global Lipschitz and the linear growth conditions




We know that if aand bsatisfy these conditions, then the system (1) admits a single global solution.

We note by Bthe space of random Ftmeasurable functions xtfor all t, satisfying the relation


we consider in Bthe norm


B.Bis the Banach space.

2.1. Markov property

The following result proves that the solution of the SDE (1) is a Markov process.

Theorem 1. ([5], Th. 2, p. 466) Assume thatatxandbtxsatisfy the hypothesis of the theorem ([5], Th. 1, p. 461) and thatXtxsis a process such that forst,)for allt>t0is a solution of SDE


Then the processXt,solution of SDE(1), is a Markovian process with a transition function


Let psxtAbe 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 psxtAa family Xsztωof a Markov process such that the processes Xsztωexist with initial point zin s,i.e.,


2.2. Notions of periodicity and boundedness

Définition 1.A stochastic processXtωis said to be periodic with periodTT>0if its finite dimensional distributions are periodic with periodicT, i.e., for allm0,andt1,t2,tmR+the joint distributions of the stochastic processesXt1+kTω,Xt2+kTω,Xtm+kTωare independent ofkkZ.

Remark 1.IfXtωisTperiodic, thenmt=EXt, vt=VarXtareTperiodic, in this case, this process is said to beTperiodic in the wide sense.

Définition 2.The functionpsxtA=PXtA/Xsfor0st,is said to be periodic ifpsxt+sAis periodic ins.

Définition 3.The Markov familiesXt0zωare said to bepuniformly boundedp>2,ifα>0,θα>0,tt0:


We denote Xt0zωas the family of all Markov process for t0+and zin Lp.

Remark 2.It is easy to see that allLpborné Markov processesXt,i.eM>0tt0:XtB,ppMispuniformly bounded.

Lemme 1.([6], Theorem 3.2 and Remark 3.1, pp. 66–67) A necessary and sufficient condition for the existence of a MarkovTperiodicXt0zωwith a givenTperiodic transition functionpsxtA,is that for somet0,z,Xt0zωare uniformly stochastically continuous and


if the transition functionpsXstAsatisfies the following not very restrictive assumption


for some functionβRwhich tends to infinity asR.

In Eq. (4), we have RR+:


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 familiesXt0zωwithTperiodic transition functions are uniformly bounded uniformly stochastically continuous, then there is aTperiodic Markov process.

Proof.By using a Markov inequality [13], we have


Then, for α>0,θα>0,such that for all tt0


we get




that is, Eq. (4). From Lemma 1, we have a Tperiodic Markov process.


3. Main result

Let the SDE


We assume that this SDE satisfies the conditions as in Section 2 after Eq. (1).

Suppose that

H1) the functions atxand btxare Tperiodic in t.

H2) the functions atxand btxsatisfy the condition


where ϕis a concave non-decreasing function.

Lemme 3.([13], Lemme 3.4) Assume thatatxandbtxverify


then, the solutions of periodic SDE (6) are uniformly stochastically continuous.

We prove the Massera’s theorem for the SDE in dimension n2.

Theorem 2.UnderH1,H2,if the solutions of the SDE(6) areLpbounded, then there is aTperiodic Markov process.

Proof.We note by Xt0ztωan Lp-bounded solution of SDE (6), from Theorem 1, this solution is unique a Markov process that is Ftmeasurable. Suppose that pt0ztAis a transition function of Markov process Xt0ztω,under H1and since pt0ztAdepend of atx,btxthen this function is Tperiodic in t.In the other hand, ϕis concave non-decreasing function, we get


From the Lpboundedness of Xt0ztω, then under H2: η>0such that


for p>2.By Lemma 3, we have Xt0ztωis puniformly bounded and puniformly stochastically continuous, this gives, the conditions of Lemma 2 are verified, finally, we can conclude the existence of the Tperiodic Markov process. □

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Boudref Mohamed Ahmed, Berboucha Ahmed and Osmanov Hamid Ibrahim Ouglu (May 23rd 2018). An Extension of Massera’s Theorem for N-Dimensional Stochastic Differential Equations, Differential Equations - Theory and Current Research, Terry E. Moschandreou, IntechOpen, DOI: 10.5772/intechopen.73183. Available from:

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