3. Main result

Let the SDE

$$\begin{cases} dx = a(t, \mathbf{x})dt + b(t, \mathbf{x})dW\_t \\ \mathbf{x}\_0 = \mathbf{z}, \ E|\mathbf{z}|^p < \infty \end{cases} \tag{6}$$

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

Suppose that


$$\|\|a(t, \mathbf{x})\|\|^p + \|\|b(t, \mathbf{x})\|\|^p \le \phi(\|\|\mathbf{x}\|\|^p) \,\_t p > 2 \tag{7}$$

where ϕ is a concave non-decreasing function.

Lemme 3. ([13], Lemme 3.4) Assume that a tð Þ ; x and b tð Þ ; x verify

$$E(||a(t, \mathfrak{x})||^p) + E(||b(t, \mathfrak{x})||^p) \le \eta\_\prime p > 2$$

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

We prove the Massera's theorem for the SDE in dimension n ≥ 2:

Theorem 2. Under Hð Þ<sup>1</sup> , Hð Þ<sup>2</sup> , if the solutions of the SDE (6) are Lp�bounded, then there is a T�periodic Markov process.

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

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Proof. We note by <sup>X</sup>ð Þ <sup>t</sup>0;<sup>z</sup> ð Þ <sup>t</sup>; <sup>ω</sup> an Lp-bounded solution of SDE (6), from Theorem 1, this solution is unique a Markov process that is Ft�measurable. Suppose that p tð Þ <sup>0</sup>; z; t; A is a transition function of Markov process <sup>X</sup>ð Þ <sup>t</sup>0;<sup>z</sup> ð Þ <sup>t</sup>; <sup>ω</sup> , under ð Þ <sup>H</sup><sup>1</sup> and since p tð Þ <sup>0</sup>; <sup>z</sup>; <sup>t</sup>; <sup>A</sup> depend of a tð Þ ; x ,b tð Þ ; x then this function is T�periodic in t: In the other hand, ϕ is concave nondecreasing function, we get

$$E\phi(|\mathbf{x}|^p) \le \phi(E|\mathbf{x}|^p)$$

From the Lp�boundedness of <sup>X</sup>ð Þ <sup>t</sup>0;<sup>z</sup> ð Þ <sup>t</sup>; <sup>ω</sup> , then under ð Þ <sup>H</sup><sup>2</sup> : <sup>∃</sup><sup>η</sup> <sup>&</sup>gt; 0 such that

$$E\left\|a\left(t,X^{(t\_0,z)}(t,\omega)\right)\right\|^p + E\left\|b\left(t,X^{(t\_0,z)}(t,\omega)\right)\right\|^p < \eta^\perp$$

for <sup>p</sup> <sup>&</sup>gt; <sup>2</sup>: By Lemma 3, we have <sup>X</sup>ð Þ <sup>t</sup>0;<sup>z</sup> ð Þ <sup>t</sup>; <sup>ω</sup> is <sup>p</sup>�uniformly bounded and <sup>p</sup>�uniformly stochastically continuous, this gives, the conditions of Lemma 2 are verified, finally, we can conclude the existence of the <sup>T</sup>�periodic Markov process. □
