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: 254

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

  • Ftt0is 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,Wtis Ftmeasurable. We consider the SDE

dx=atxdt+btxdWtxt0=z.E1

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

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

and

atxp+btxpkp1+xp

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

supt0Ext2,

we consider in Bthe norm

xB=supt0Ex212

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 that atxand btxsatisfy the hypothesis of the theorem ([5], Th. 1, p. 461) and that Xtxsis a process such that for st,)for all t>t0is 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 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.,

PXsztω=z=1E3

2.2. Notions of periodicity and boundedness

Définition 1. A stochastic process Xtωis said to be periodic with period TT>0if 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+kTωare independent of kkZ.

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

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

Définition 3. The Markov families Xt0zωare said to be puniformly bounded p>2,if α>0,θα>0,tt0:

zB,pαXt0zωB,pθα

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

Remark 2. It is easy to see that all Lpborné Markov processes Xt,i.eM>0tt0:XtB,ppMis 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 Xt0zωwith a given Tperiodic transition function psxtA,is that for some t0,z,Xt0zωare uniformly stochastically continuous and

limRlimLinf1Lt0t0+Lpt0ztU¯R,pdt=0E4

if the transition function psXstAsatisfies the following not very restrictive assumption

αR=supzUβR,p0<t0,tt0<Tpt0ztU¯R,pR0E5

for some function βRwhich 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 Xt0zωwith 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 Tperiodic 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 atxand btxare Tperiodic in t.

H2) the functions atxand btxsatisfy the condition

atxp+btxpϕxp,p>2E7

where ϕis a concave non-decreasing function.

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

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 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

xpϕExp

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

Ea(tXt0ztω)p+Eb(tXt0ztω)p<η

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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