The problem of portfolio allocation in the context of stocks evolving in random environments, that is with volatility and returns depending on random factors, has attracted a lot of attention. The problem of maximizing a power utility at a terminal time with only one random factor can be linearized thanks to a classical distortion transformation. In the present paper, we address the situation with several factors using a perturbation technique around the case where these factors are perfectly correlated reducing the problem to the case with a single factor. Our proposed approximation requires to solve numerically two linear equations in lower dimension instead of a fully nonlinear HJB equation. A rigorous accuracy result is derived by constructing sub- and super-solutions so that their difference is at the desired order of accuracy. We illustrate our result with a particular model for which we have explicit formulas for the approximation. In order to keep the notations as explicit as possible, we treat the case with one stock and two factors and we describe an extension to the case with two stocks and two factors.

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具有相关随机波动率因子的最优投资

股票在随机环境中演变的背景下的投资组合分配问题，即波动性和回报取决于随机因素，已经引起了很多关注。由于经典失真变换，可以线性化仅使用一个随机因素在终端时间最大化电力效用的问题。在本文中，我们围绕这些因素完全相关的情况使用扰动技术解决了具有多个因素的情况，将问题减少到具有单个因素的情况。我们提出的近似要求在数值上求解两个较低维度的线性方程，而不是完全非线性的 HJB 方程。严格的精度结果是通过构建子解和超解得到的，这样它们的差异就在期望的精度等级上。我们用一个特定的模型来说明我们的结果，我们有明确的近似公式。为了使符号尽可能明确，我们用一只股票和两个因子来处理案例，并描述了对两只股票和两个因子的案例的扩展。