标题： 基于随机偏微分方程的市场限价订单模型，参数估计与投资优化 显示英文标题

标题： A Model of Market Limit Orders By Stochastic PDE's, Parameter Estimation, and Investment Optimization

摘要： 在本文中，我们引入了一个完全连续且时间变化的市场限价单演化模型，该模型基于Zheng和Sowers（2012）获得的一类随机偏微分方程解的存在性、唯一性和正则性。 与文献中提出的几种模型相反，该模型从随机PDE中继承了时间和价格上的完全连续性，因此特别适用于交易速度极快的情况，例如由高频交易者（HFT's）提供的交易。 我们首先详细阐述了该模型及其相关参数的精确定义，并在给定固定参数集的情况下，从相关的数学结果中证明了其存在性和唯一性。 然后，我们使用最大似然和最小均方误差估计方法，在某些标准（如AIC）下统计推导出该模型的参数估计方案，以适应不同数量的参数。 最后，作为该模型的一个典型经济学和金融学应用案例，我们通过分析投资者或交易者在其模型和参数作为外生变量时效用函数的随机（Itô）演化，在静态和动态意义上解决了投资优化问题。 证明了两个定理，它们提供了确定最佳（限价）价格和时间点进行交易的标准。 显示英文摘要

摘要： In this paper we introduce a completely continuous and time-variate model of the evolution of market limit orders based on the existence, uniqueness, and regularity of the solutions to a type of stochastic partial differential equations obtained in Zheng and Sowers (2012). In contrary to several models proposed and researched in literature, this model provides complete continuity in both time and price inherited from the stochastic PDE, and thus is particularly suitable for the cases where transactions happen in an extremely fast pace, such as those delivered by high frequency traders (HFT's). We first elaborate the precise definition of the model with its associated parameters, and show its existence and uniqueness from the related mathematical results given a fixed set of parameters. Then we statistically derive parameter estimation schemes of the model using maximum likelihood and least mean-square-errors estimation methods under certain criteria such as AIC to accommodate to variant number of parameters . Finally as a typical economics and finance use case of the model we settle the investment optimization problem in both static and dynamic sense by analysing the stochastic (It\^{o}) evolution of the utility function of an investor or trader who takes the model and its parameters as exogenous. Two theorems are proved which provide criteria for determining the best (limit) price and time point to make the transaction.