Open access peer-reviewed chapter

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

Written By

Boudref Mohamed Ahmed, Berboucha Ahmed and Osmanov Hamid Ibrahim Ouglu

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

DOI: 10.5772/intechopen.73183

From the Edited Volume

## Differential Equations

Edited by Terry E. Moschandreou

Chapter metrics overview

View Full Metrics

## Abstract

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.

### Keywords

• 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

dx=atxdt+btxdWt

## 2. Preliminaries

Let ΩFFtt0P be the complete probability space with a filtration Ftt0 satisfying the usual conditions

• Ftt0 is an increasing family of sub algebras containing negligible sets of F and is continuous at right.

F=σt0Ft.

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

dx=atxdt+btxdWtxt0=z.E1

in ΩFFtt0P.

The functions atx:R+×RnRn and btx:R+×RnRn×m are measurable. We suppose that Ft is the completion of σWrt0rt for all tt0, and the initial condition z is independent of Wt, for tt0 and Ezp<.

Suppose that the functions atx and btx satisfy the global Lipschitz and the linear growth conditions

k>0,tR+,x,yRn:atxa(ty)+btxb(ty)kxy

and

atxp+btxpkp1+xp

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 Ft measurable functions xt for all t, satisfying the relation

supt0Ext2,

we consider in B the norm

xB=supt0Ex212

B.B is 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 that atx and btx satisfy the hypothesis of the theorem ([5], Th. 1, p. 461) and that Xtxs is a process such that for st,) for all t>t0 is a solution of SDE

Xtxs=x+tsauXtxudu+tsbuXtxudWuE2

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

ptxsA=PXtxsA.

Let psxtA be 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 psxtA a family Xsztω of a Markov process such that the processes Xsztω exist with initial point z in s, i.e.,

PXsztω=z=1E3

### 2.2. Notions of periodicity and boundedness

Définition 1. A stochastic process Xtω is said to be periodic with period T T>0 if its finite dimensional distributions are periodic with periodic T, i.e., for all m0, and t1,t2,tmR+ the joint distributions of the stochastic processes Xt1+kTω,Xt2+kTω,Xtm+kTω are independent of k kZ.

Remark 1. If Xtω is Tperiodic, then mt=EXt, vt=VarXt are Tperiodic, in this case, this process is said to be Tperiodic in the wide sense.

Définition 2. The function psxtA=PXtA/Xs for 0st, is said to be periodic if psxt+sA is periodic in s.

Définition 3. The Markov families Xt0zω are said to be puniformly bounded p>2, if α>0,θα>0,tt0:

zB,pαXt0zωB,pθα

We denote Xt0zω as the family of all Markov process for t0+ and z in Lp.

Remark 2. It is easy to see that all Lpborné Markov processes Xt, i.eM>0tt0:XtB,ppM is puniformly bounded.

Lemme 1. ([6], Theorem 3.2 and Remark 3.1, pp. 66–67) A necessary and sufficient condition for the existence of a Markov Tperiodic Xt0zω with a given Tperiodic transition function psxtA, is that for some t0,z,Xt0zω are uniformly stochastically continuous and

limRlimLinf1Lt0t0+Lpt0ztU¯R,pdt=0E4

if the transition function psXstA satisfies the following not very restrictive assumption

αR=supzUβR,p0<t0,tt0<Tpt0ztU¯R,pR0E5

for some function βR which tends to infinity as R.

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

UR,p=xRn:xp<R
U¯R,p=xRn:xpR

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 Xt0zω with Tperiodic transition functions are uniformly bounded uniformly stochastically continuous, then there is a Tperiodic Markov process.

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

pt0ztU¯R,p=1RPXt0=zEXt0zωp1RPzXt0zωB,pp

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

zB,pαXt0zωB,pθα

we get

pt0ztU¯R,p1RPzθpα

Then

0limRlimLinf1Lt0t0+Lpt0ztU¯R,pdtlimR1RPzθpαlimLinf1Lt0t0+Ldt=limRθpαRPz=0,

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

## 3. Main result

Let the SDE

dx=atxdt+btxdWtxt0=z,Ezp<E6

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

Suppose that

H1) the functions atx and btx are T periodic in t.

H2) the functions atx and btx satisfy the condition

atxp+btxpϕxp,p>2E7

where ϕ is a concave non-decreasing function.

Lemme 3. ([13], Lemme 3.4) Assume that atxand btx verify

Eatxp+Ebtxpη,p>2

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. Under H1,H2, if the solutions of the SDE (6) are Lpbounded, then there is a Tperiodic Markov process.

Proof. We note by Xt0ztω an Lp-bounded solution of SDE (6), from Theorem 1, this solution is unique a Markov process that is Ft measurable. Suppose that pt0ztA is a transition function of Markov process Xt0ztω, under H1 and since pt0ztA depend of atx,btx then this function is T periodic in t. In the other hand, ϕ is concave non-decreasing function, we get

xpϕExp

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

Ea(tXt0ztω)p+Eb(tXt0ztω)p<η

for p>2. By Lemma 3, we have Xt0ztω is p uniformly bounded and p uniformly stochastically continuous, this gives, the conditions of Lemma 2 are verified, finally, we can conclude the existence of the T periodic Markov process. □

## References

1. 1. Bezandry PH, Diagana T. Existence of almost periodic solutions to some stochastic differential equations. Applicable Analysis. 2007;86(7):819-827. MR 2355540 (2008i: 60089)
2. 2. Bezandry PH, Diagana T. Square-mean almost periodic solutions nonautonomous stochastic differential equations. Electronic Journal of Differential Equations. 2007;117:10. (electronic) MRMR2349945 (2009e: 34171)
3. 3. Da Prato G. Periodic and almost periodic solutions for semilinear stochastic equations. Stochastic Analysis and Applications. 1995;13(1):13-33
4. 4. Dorogovtsev A. Existence of periodic solutions or abstract stochastic equations. Asymptotic periodicity of the Cauchy problem (in Russsian). Teorija na Verojatnost i Matematika Statistika. 1988;39:47-52
5. 5. Guikhman I, Skorokhod A. Introduction à la Théorie des Processus Aléatoires. Moscou: Mir; 1980
6. 6. Has’minskii RZ. Stochastic Stability of Differential Equations. Second ed. Berlin Heidelberg: Springer-Verlag; 2012
7. 7. Itô K. On stochastic differential equations. Memoirs of the American Mathematical Society. 1951;4. (Russian translation: Mathematika. 1957;1(1):78-116. MR 12 #724
8. 8. Mao XR. Stochastic Differential Equations and Applications. Chichester: Horwood; 1997
9. 9. Massera JL. The existence of periodic solutions of systems of differential equations. Duke Mathematical Journal. 1950;17:457-475
10. 10. Morozan T, Tudor C. Almost periodic solutions of affine Itô equations. Stochastic Analysis and Applications. 1989;7(4):451-474. MR 1040479 (91k: 60064)
11. 11. Tudor C. Almost periodic solutions of affine stochastic evolution equations. Stochastics and Stochastics Reports. 1992;38(4):251-266. MR1274905 (95e: 60058)
12. 12. Vârsan C. Asymptotic almost periodic solutions for stochastic differential equations. Tohoku Mathematical Journal. 1986;41:609-618
13. 13. Xu DY, Huang YM, Yang ZG. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete and Continuous Dynamical Systems. 2009;24(3):1005-1023

Written By

Boudref Mohamed Ahmed, Berboucha Ahmed and Osmanov Hamid Ibrahim Ouglu

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