不可观测的牛熊机制转换模型下具有交易费的最优投资消费问题的研究

All the time, the research on the optimal investment and consumption strategies with transaction costs and bull-bear regime switching has been the common hotspot of financial industry and academic circles. Although the experts at home and abroad have achieved admirable results，few references have considered the case with transaction costs and unobservable bull-bear regime switching at the same time. This project considers the impact of these two issues and builds a more realistic investment and consumption model. We will study the proportional transaction costs case as well as the fixed transaction costs case popular on the market. Firstly, we will transform the original problem into an observable one through the introduction of the probability of being in a bull market. Secondly, we will apply the stochastic optimal control theory to derive the partial differential equation for the value function of the optimal problem. Under the proportion transaction costs, it is a variational inequality with gradient constraint; while under the fixed transaction costs, it is an obstacle problem with the obstacle depending on the solution of the problem. Thirdly, through numerical computation and PDE theoretical analysis methods, we will provide the appearance and the analytic properties of the optimal strategies. Eventually, through analysis combined with numerical simulation, we will show the relationship between the optimal trading strategy and the trend following as well as the contra-trend strategy popular on the market.

关于交易费用以及牛熊机制转换影响下的最优投资消费策略的研究一直以来都是理论金融学界和实证金融学界共同关注的热点问题。虽然目前国内外专家已经取得令人赞赏的成果，但是很少有文献考虑了交易费用和不可观测的牛熊机制转换共存的情况。本项目拟考虑两者共存的情况，建立更加贴近现实的投资消费模型。我们将同时考虑市场上流行的比例交易费用与固定交易费用。首先，我们将通过引进市场处于牛市的概率把问题转化为可观测的最优问题。接着，运用随机最优控制理论，我们将推导出最优问题的价值函数满足的偏微分方程。在比例交易费用下，它是一个带梯度约束的变分不等式；在固定交易费用下，它是一个障碍问题，障碍函数依赖于问题的解。进一步，通过数值计算以及偏微分方程的理论分析方法，我们将给出最优策略的直观刻画以及分析性质。最后，我们将通过分析结合数值模拟等手段揭示交易策略与目前市场上流行的趋势交易与反趋势交易之间的关系。

具有交易费用的最优投资消费问题因为同时具有重要的实际应用价值和数学理论价值一直是金融学界研究的一个热点问题。虽然目前国内外专家已经取得令人赞赏的成果，但是很少有文献在不可观测的牛熊机制转换模型下考虑交易费用对投资策略的影响。本项目对不可观测的牛熊机制转换模型中具有比例和固定交易费用的最优投资消费策略进行了数值描述和理论分析。此外，我们还研究了交易费用对股票流动性溢价的影响。关于交易费用对股票流动性溢价的影响理论金融学界和实证金融学界一直存在分歧。理论金融学界的研究表明，在单一机制模型，甚至是可观测的机制转换模型中，交易费用对股票流动性的影响远远小于市场上观察到的实际数据。而我们首次发现，在不可观测的牛熊机制转换模型中交易费用对股票流动性溢价的影响有了大幅度的提高以至于接近市场上观察到的实际数据。