Asymptotic Analysis for Functional Stochastic Differential Equations by Jianhai Bao George Yin & Chenggui Yuan

Author:Jianhai Bao, George Yin & Chenggui Yuan
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

As a result, the desired assertion follows from the Gronwall inequality.

Lemma 3.12

Under (A3) and (H1)–(H2),

(3.35)

for any , and

Proof

For notation simplicity, set By (3.31), one has

which further implies

(3.36)

For any , by Hölder’s inequality [99, Theorem 7.1, p. 39] and Lemma 3.5, we obtain from (3.30) and (3.34) that

(3.37)

where in the last step we have used Lemma 3.11. In what follows, we divide the following five cases to show (3.35).

Case 1: and . (3.35) holds by carrying out the argument to derive (3.37).

Case 2: and . It is easily seen that

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