AI 中文总结
本文基于马利瓦因微积分和维纳混沌分析，开发新框架分析确定性等价物和动态风险溢价。指出经典阿罗-普拉特近似局限，动态构建确定性等价物，得出高阶动态风险溢价层次结构及系数表示，建立统一框架，为高阶风险度量开辟新视角。
AI 中文摘要
阿罗-普拉特近似是预期效用理论的基石之一，但其数学范围和与高阶风险偏好的关系仍未完全明晰。本文基于马利瓦因微积分和维纳混沌分析，开发了一个用于分析确定性等价物和动态风险溢价的新框架。首先指出经典阿罗-普拉特近似对任意消失风险序列并非渐近有效，进而通过布朗滤波考虑不确定性的逐步揭示来动态地构建确定性等价物。结合伊藤微积分、克拉克-奥康表示和维纳混沌分解，得出高阶动态风险溢价的完整层次结构，并以马利瓦因导数得到相应系数的显式表示。对于混合维纳混沌展开，包括审慎和节制在内的高阶偏好度量通过混沌分量之间的相互作用自然出现，并使用贝尔多项式表示进行刻画。二次高斯泛函和瓦西塞克利率模型的显式结果说明了该理论，并确定了一类广泛的正则维纳泛函，对于它们经典阿罗-普拉特近似作为主导项被恢复。这些结果建立了一个将预期效用理论、随机分析和维纳混沌展开联系起来的统一框架，为高阶确定性等价物和风险的动态度量开辟了新视角。
英文摘要
The Arrow-Pratt approximation is one of the cornerstones of expected utility theory, providing the classical local approximation of certainty equivalents and risk premia in terms of absolute risk aversion. Despite its widespread use, its mathematical scope and relationship to higher-order risk preferences remain only partially understood. This paper develops a new framework for the analysis of certainty equivalents and dynamic risk premia based on Malliavin calculus and Wiener chaos analysis. We first show that the classical Arrow-Pratt approximation is not asymptotically valid for arbitrary sequences of vanishing risks, thereby identifying precise limitations of the traditional theory. Motivated by this observation, we formulate certainty equivalents dynamically by considering the progressive revelation of uncertainty through a Brownian filtration. Combining Itô calculus, the Clark--Ocone representation and the Wiener chaos decomposition, we derive a complete hierarchy of higher-order dynamic risk premia and obtain explicit representations of the corresponding coefficients in terms of Malliavin derivatives. For mixed Wiener chaos expansions, higher-order preference measures, including prudence and temperance, emerge naturally through interactions between chaos components and are characterised using Bell polynomial representations. Explicit results for quadratic Gaussian functionals and the Vasicek interest-rate model illustrate the theory and identify a broad class of regular Wiener functionals for which the classical Arrow-Pratt approximation is recovered as the leading-order term. The results establish a unified framework linking expected utility theory, stochastic analysis and Wiener chaos expansions, opening a new perspective on higher-order certainty equivalents and the dynamic measurement of risk.
Comments47 pages
