Incomplete data, stochastic processes and finance

在金融、经济、抽样调查、生物统计学和其他统计学应用中，不完整或缺失的数据以及对缺失值进行推算的必要性是现代数据的共同特征。相关变量可能被忽略，因为它们难以获得或使分析复杂化。例如，如果股票A在时间t交易，另一只股票B极不可能在同一时间交易，在这种情况下，此时股票B的公允价值“缺失”。将交易时间四舍五入的通常做法是危险的，因为它会在估计中引入偏差，对高流动性股票和基准债券的影响可能有限，但对交易清淡或流动性差的资产的影响要大得多。为了数据分析的方便而天真地制造或“imputation”数据有引入偏见和破坏分析师初衷的风险；客观地从数据中提取所有相关信息。忽略那些缺少组件的情况或变量就像天真地填充数据一样危险；它也可能导致有偏见的结果。我的研究目标是为多元和单变量扩散模型（如金融中常用的模型）提供精确的归算技术，并利用它们来估计参数和其他问题，其中插值扩散值是有用的。在生物统计学中，通常存在响应变量或预测变量，这些变量可能有助于解释感兴趣的“响应变量”的行为，但它们并不仅仅是因为数据收集的费用而被收集。类似但不那么精确的信息可能很容易从卫生收集机构、医院数据库或简单廉价的测试中获得。在多大程度上，较便宜的变量可以用作较昂贵变量的代理？收集即使是一小部分情况下昂贵的变量，也经常有助于评估我们在忽略这些更昂贵的变量时牺牲效率和偏差的程度，并且通常允许以更低的成本提高效率。我的研究涉及数据的“适当”输入，即使用与我们观察到的准确分布相一致的数据输入，从而允许在这些缺失数据问题上使用更简单的完整数据技术。

Incomplete or missing data  and the necessity to impute missing values is a common feature of modern data in finance, economics, sample surveys, biostatistics, and other applications of statistics.  Relevant variables may be ignored either because they are difficult to obtain or  they complicate the analysis.  For example if  stock A is traded at time t, say,   another stock B is highly unlikely to be traded at exactly the same time in which case the fair value of stock B at this time is ``missing''. The usual practice of rounding the trading times up is dangerous in that it introduces biases in the estimates, possibly  of limited consequence for highly liquid equities and benchmark bonds, but much more significant  with thinly traded or illiquid assets.  Naive manufacture or ``imputation'' of data for the convenience of the data analysis runs the risk of introducing bias and undermining the original intent of the analyst; to objectively extract all pertinent information from the data. Ignoring those cases or variables which have missing components is as dangerous as naively filling in data; it too potentially leads to biased results. The objective of  my research is to make exact imputation techniques available to both multivariate and univariate diffusion models, such as those commonly used in finance, and exploit them for estimation of parameters and other problems in which interpolating the values of a diffusion is useful.    In biostatistics, there are often  response or predictor variables that  might help explain the behaviour of a ``response variable'' of interest, but they are not collected simply because of the expense of the data collection.  Similar but less precise information might be easily available from health collection agencies, hospital databases, or a simple cheap test. To what extent can the cheaper variables be used as proxies for the more expensive? Collection of the expensive  variables for even a fraction of the cases will often help assess the degree to which we sacrifice efficiency and bias when we omit these more expensive variables and usually allow for an improvement in efficiency with reduced cost.     My research concerns the "proper" imputation of data, i.e. the imputation of data using the exact distribution consistent with what we observe, so as permit use of simpler complete data techniques on these missing data problems.