Hilfer分数阶随机时滞微分方程的大偏差原理

Large Deviation Principle of Hilfer Fractional Stochastic Delay Differential Equation

摘要

文章研究一类具有乘性噪声的Hilfer分数阶随机时滞微分方程的大偏差原理（Large Deviation Principle, LDP）。基于大偏差原理与Laplace原理等价性，利用弱收敛方法，通过验证骨架方程的紧性及扰动系统的收敛性，得到该类方程的LDP. 文章以金融市场模型为例，通过数值模拟展示了如何利用LDP分析小噪声扰动下系统的渐近行为，验证了理论结果。

Abstract

This paper investigates the Large Deviation Principle (LDP) for a class of Hilfer fractional stochastic delay differential equations with multiplicative noise. Based on the equivalence between the LDP and the Laplace principle, the weak convergence method is employed to establish the LDP for such equations by verifying the compactness of the skeleton equation and the convergence of the perturbed system. A financial market model is presented as an example, and numerical simulations are conducted to illustrate how the LDP can be used to analyze the asymptotic behavior of the system under small noise perturbations, thereby validating the theoretical results.

关键词

分数阶微积分 / 随机微分方程 / 大偏差原理 / 弱收敛

Key words

Fractional calculus / Stochastic differential equation / Large Deviation Principle / Weak convergence

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基金资助

国家自然科学基金项目(11961069)

新疆维吾尔自治区自然科学基金项目(2025D01E16)

新疆优秀青年科技人才培训计划项目(2019Q022)

新疆师范大学青年拔尖人才项目(XJNUQB2022-14)