具有分數Brown運動的分數階中立型隨機微分方程解的存在唯一性

摘 要：研究了一類在Hilbert空間中具有分數Brown運動的分數階中立型隨機微分方程，利用Picard逐步逼近法得到了其非Lipschitz條件和弱化的線性增長條件下粘性解的新的存在唯一性的充分條件.所提的研究方法使得先前一些研究結果得到了拓展.最后通過具有分數Brown運動的隨機非線性波動方程驗證了理論的有效性.

關鍵詞：分數階隨機微分方程;分數Brown運動;粘性解;存在唯一性

中圖分類號：O175.14 文獻標志碼：A文章編號：1000-2367（2025）03-0104-08

1 預備知識

2 解的存在唯一性

3 例 子

4 結 論

本文討論了一類具有分數Brown運動的分數階中立型隨機微分方程.運用逐步逼近法給出了非Lipschitz條件和弱化的線性增長條件下粘性解的存在性和唯一性的充分條件，其中Lipschitz條件視為其特殊情形.并通過具有分數Brown運動的隨機非線性波動方程的應用驗證了理論的有效性.本文的方法避免了使用較強的Lipschitz條件，對于研究隨機非線性系統的解的存在唯一性較為有效.該方法亦可用于討論時滯中立型隨機微分方程解的存在唯一性和穩定性，為研究其他中立型隨機微分方程提供了理論依據.

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Existence and uniqueness of solutions to fractional neutral stochastic

differential equations with fractional brownian motionLi Guoping¹， Han Ting^1，2

（1. Xinhua College， Ningxia University， Yinchuan 750021， China;

2. College of Mathematics and Statistics， Ningxia University， Yinchuan 750021， China）

Abstract： In this paper， we consider a class of fractional-order neutral stochastic" differential" equations with fractional Brownian motion by using picard step by step in a Hilbert space. A novel sufficient condition for the existence and uniqueness of mild solutions is obtained in conditions of the non-Lipschitz condition and the weakened linear growth. The result generalizes a few previous known results. An application to the stochastic nonlinear wave equation with fractional Brownian motion is given to illustrate the validity of the obtained theory.

Keywords： fractional" stochastic" differential" equations; fractional Brownian motion; mild solution; existence and uniqueness

［責任編校 陳留院 楊浦］