AI 中文总结
本文针对CMIM类偏好，利用伯恩斯坦表示定理，推导了最优投资问题中价值函数和终端财富的高阶解析展开式，为投资决策敏感性分析提供了方法。
AI 中文摘要
[MSZ24]中提出的完全单调逆边际（CMIM）效用构成了一类易于处理的偏好，包含了数理金融中许多最重要的效用函数，如幂效用和指数效用。在随机占优市场中，其伯恩斯坦表示在对偶优化问题中诱导出隐藏的线性结构，这是本分析的基础。本文研究了CMIM类内投资者偏好扰动下最优投资的敏感性，利用伯恩斯坦表示定理，证明在随机占优条件下，伯恩斯坦测度的仿射扰动会诱导出对偶价值函数的仿射表示。因此，可通过标量预算方程分析优化问题对偏好的依赖，进而证明相关拉格朗日乘子关于扰动参数的解析性，并推导原价值函数和最优终端财富的任意阶收敛解析展开式，其显式递归公式通过贝尔多项式表示。
英文摘要
Completely monotonic inverse marginal (CMIM) utilities, introduced in [MSZ24], constitute a tractable class of preferences that includes many of the most important utility functions used in mathematical finance, such as power and exponential utilities. In stochastically dominant markets, their Bernstein representation induces a hidden linear structure in the dual optimization problem that serves as the foundation for the present analysis. In this paper, we investigate the sensitivity of optimal investment with respect to perturbations of investor preferences within the CMIM class. Exploiting Bernstein's representation theorem, we show that, under stochastic dominance, affine perturbations of Bernstein measures induce an affine representation of the dual value function. As a result, the dependence of the optimization problem on preferences can be analyzed through a scalar budget equation, allowing us to prove analyticity of the associated Lagrange multiplier with respect to the perturbation parameter and to derive convergent analytic expansions of arbitrary order for the primal value function and the optimal terminal wealth, with explicit recursive formulas expressed through Bell polynomials.
Comments33 pages, preliminary version
