Incomplete data, stochastic processes and finance

在金融、经济学、抽样调查、生物统计学和其他统计学应用中，不完整或缺少数据的必要性是现代数据的一个共同特征。相关变量可能会被忽略，因为它们很难获得，或者它们会使分析复杂化。例如，如果一只股票A在时间t交易，比方说，另一只股票B不太可能在同一时间交易，在这种情况下，股票B的公允价值在此时是“缺失的”。将交易时间向上舍入的常见做法是危险的，因为它会在估计中引入偏差，可能对高流动性的股票和基准债券影响有限，但对成交稀少或非流动性资产的影响要大得多。为了方便数据分析而天真地制造数据或对数据进行“推定”，可能会引入偏差，破坏分析师的初衷；客观地从数据中提取所有相关信息。忽视那些有缺失成分的案例或变量就像幼稚地填写数据一样危险；它也可能导致有偏见的结果。我研究的目的是使精确的补偿技术可用于多变量和单变量扩散模型，例如那些在金融中常用的模型，并将它们用于参数估计和其他有用的问题，其中对扩散的值进行内插是有用的。在生物统计学中，通常有一些反应变量或预测变量可能有助于解释感兴趣的“反应变量”的行为，但收集这些变量并不是因为数据收集的费用。类似但不那么精确的信息可能很容易从卫生收集机构、医院数据库或简单的廉价测试中获得。在多大程度上，较便宜的变量可以被用来代替较昂贵的变量？收集即使是一小部分情况下昂贵的数据变量，往往有助于评估当我们省略这些更昂贵的变量时，我们牺牲效率和偏差的程度，并通常允许在降低成本的情况下提高效率。我的研究关注数据的“适当”归因，即使用与我们观察到的一致的准确分布来归算数据，以便允许使用更简单的完整数据技术来解决这些丢失的数据问题。

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.