Tauberian and Mercerian theorems and analysis of stochastic financial processes

我们引入了广义分数布朗运动，并利用Tauberian定理和一种新的预测理论方法证明了它的有限过去预测公式；通过对平稳时间序列应用一种新的预测理论方法，我们证明了偏自相关函数的一个令人惊讶地简单的表示定理，我们得到了它的一个带余项的精确渐近行为；我们引入了一类由无穷级连续时间AR方程描述的平稳增量过程。我们还考虑了具有平稳增量过程的SDE作为驱动力。通过这种方式，我们引入了具有长或短记忆的股票价格过程。在此过程中，我们讨论了金融市场的完备性和波动性的行为。我们使用一种新的预测理论方法，得到了与驱动力相关的创新过程的显式表示。这样就解决了金融市场中期望效用最大化的问题。在最简单的情况下，这是一个参数模型，与Black-Scholes模型相比，它只有两个额外的参数。我们观察到，该模型很好地描述了S&P500等真实的市场数据。这表明了该模型的实用性。

We introduced a generalized fractional Brownian motion and proved a finite-past prediction formula for it using Tauberian theorems and a new prediction-theoretic approach.By applying a new prediction-theoretic approach to stationary time series, we proved a surprisingly simple representation theorem for the partial autocorrelation function Using the result, we derived a precise asymptotic behavior with remainder for it.We introduced a class of stationary increments processes which are described by continuous-time AR-equations of infinite order. We also considered SDE with the stationary increments process as driving force. In this way, we introduced stock price processes with long or short memory. We discussed about completeness and behavior of volatility of the financial market with this stock price process. We obtained an explicit representation of the innovation process associated with the drinving force, using a new prediction-theoretic approach. In this way, we solved the problem of expected utility maximization problem in the financial market. In the simplest case, this is a paremetric model that has only two additional parameters compared with the Black-Scholes model. We observed that this model well describes the real market data such as S & P500. This shows the usefulness of the model.