具有高斯漂移的金融市场中专家意见的扩散近似
Diffusion Approximations for Expert Opinions in a Financial Market with Gaussian Drift
摘要：
本文研究了一个金融市场，其中收益依赖于一个不可观测的高斯漂移过程。虽然收益率的观察产生有关潜在漂移的信息，但我们也将离散时间专家意见作为外部信息来源。为了估计隐藏漂移，考虑给定可用观测值的漂移的条件分布是至关重要的，即所谓的滤波器。对于投资者观察的回报过程和离散时间的专家意见，我们详细研究了过滤器的渐近行为的专家意见的到来的频率趋于无穷大。在我们的设置中，更高频率的专家意见是以准确性为代价的，这意味着随着专家意见频率的增加，专家意见的方差变得更大。我们考虑一个模型，其中信息日期是确定的和等距的，另一个模型，其中的信息日期到达随机的跳跃时间的泊松过程。在这两种情况下，我们推导出极限定理，说明从观察离散时间的专家意见所获得的信息是渐近相同的，从观察一定的扩散过程，可以解释为一个连续时间的专家。我们使用我们的极限定理推导出所谓的扩散近似的过滤器的高频离散时间的专家意见。这些扩散近似对于导出效用最大化问题的简化近似解是非常有帮助的。
英文摘要：
This paper investigates a financial market where returns depend on an unobservable Gaussian drift process. While the observation of returns yields information about the underlying drift, we also incorporate discrete-time expert opinions as an external source of information. For estimating the hidden drift it is crucial to consider the conditional distribution of the drift given the available observations, the so-called filter. For an investor observing both the return process and the discrete-time expert opinions, we investigate in detail the asymptotic behavior of the filter as the frequency of the arrival of expert opinions tends to infinity. In our setting, a higher frequency of expert opinions comes at the cost of accuracy, meaning that as the frequency of expert opinions increases, the variance of expert opinions becomes larger. We consider a model where information dates are deterministic and equidistant and another model where the information dates arrive randomly as the jump times of a Poisson process. In both cases we derive limit theorems stating that the information obtained from observing the discrete-time expert opinions is asymptotically the same as that from observing a certain diffusion process which can be interpreted as a continuous-time expert. We use our limit theorems to derive so-called diffusion approximations of the filter for high-frequency discrete-time expert opinions. These diffusion approximations are extremely helpful for deriving simplified approximate solutions of utility maximization problems.
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