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Sensitivity analysis for expected utility maximization in incomplete Brownian market models

机译：不完全布朗市场模型中预期效用最大化的敏感性分析

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摘要

We examine the issue of sensitivity with respect to model parameters for the problem of utility maximization from final wealth in an incomplete Samuelson model and mainly, but not exclusively, for utility functions of positive-power type. The method consists in moving the parameters through change of measure, which we call a weak perturbation , decoupling the usual wealth equation from the varying parameters. By rewriting the maximization problem in terms of a convex-analytical support function of a weakly-compact set, crucially leveraging on the work (Backhoff and Fontbona in SIAM J Financ Math 7(1):70–103, 2016), the previous formulation let us prove the Hadamard directional differentiability of the value function with respect to the market price of risk, the drift and interest rate parameters, as well as for volatility matrices under a stability condition on their Kernel, and derive explicit expressions for the directional derivatives. We contrast our proposed weak perturbations against what we call strong perturbations , where the wealth equation is directly influenced by the changing parameters. Contrary to conventional wisdom, we find that both points of view generally yield different sensitivities unless e.g. if initial parameters and their perturbations are deterministic.
机译：对于不完整的Samuelson模型中最终财富带来的效用最大化问题，我们主要针对模型参数检验敏感性问题，主要但并非唯一地针对正幂类型的效用函数。该方法包括通过量度的变化来移动参数，我们称其为微扰，将通常的财富方程式与变化的参数解耦。通过重写弱紧集的凸分析支持函数来最大化最大化问题，从而至关重要地利用了该工作（SIAM J Financ Math 7（1）：70–103，2016中的Backhoff和Fontbona），以前的公式让我们证明价值函数相对于风险的市场价格，漂移和利率参数以及在其内核处于稳定状态下的波动性矩阵的Hadamard方向可微性，并导出方向性导数的显式表达式。我们将拟议的弱摄动与所谓的强摄动进行对比，强扰动的财富方程直接受到参数变化的影响。与传统观点相反，我们发现，两种观点通常会产生不同的敏感性，除非，例如。如果初始参数及其扰动是确定性的。

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