专家意见与多元正态漂移股票收益的对数效用最大化
Expert Opinions and Logarithmic Utility Maximization for Multivariate Stock Returns with Gaussian Drift
摘要：
本文研究了具有多维股票收益的金融市场中的最优交易策略，其中漂移是一个不可观测的多变量Ornstein-Uhlenbeck过程。关于漂移的信息是通过观察股票收益和专家意见获得的。后者提供离散时间点漂移当前状态的无偏估计。投资者最大化终端财富预期对数效用的最佳交易策略取决于过滤器，过滤器是给定可用信息的漂移的条件期望。我们状态过滤方程来描述其动态不同的信息设置。在专家意见之间，这是卡尔曼滤波器。滤波器的条件协方差矩阵遵循Riccati型常微分方程。我们依靠矩阵Riccati方程的基本理论来研究它们的性质。首先，我们考虑有限时间内越来越多的专家意见的协方差矩阵的渐近行为。其次，我们状态的协方差矩阵的收敛条件在一个无限的时间范围内定期到达的专家意见。最后，我们推导出投资者的最优交易策略。终端财富的最优期望对数效用，即价值函数，是条件协方差矩阵的泛函。因此，我们对协方差矩阵的分析使我们能够推导出价值函数的性质。
英文摘要：
This paper investigates optimal trading strategies in a financial market with multidimensional stock returns where the drift is an unobservable multivariate Ornstein-Uhlenbeck process. Information about the drift is obtained by observing stock returns and expert opinions. The latter provide unbiased estimates on the current state of the drift at discrete points in time. The optimal trading strategy of investors maximizing expected logarithmic utility of terminal wealth depends on the filter which is the conditional expectation of the drift given the available information. We state filtering equations to describe its dynamics for different information settings. Between expert opinions this is the Kalman filter. The conditional covariance matrices of the filter follow ordinary differential equations of Riccati type. We rely on basic theory about matrix Riccati equations to investigate their properties. Firstly, we consider the asymptotic behaviour of the covariance matrices for an increasing number of expert opinions on a finite time horizon. Secondly, we state conditions for the convergence of the covariance matrices on an infinite time horizon with regularly arriving expert opinions. Finally, we derive the optimal trading strategy of an investor. The optimal expected logarithmic utility of terminal wealth, the value function, is a functional of the conditional covariance matrices. Hence, our analysis of the covariance matrices allows us to deduce properties of the value function.
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