The main object of investigation in this paper is a very general regression model in optional setting – when an observed process is an optional semimartingale depending on an unknown parameter. It is well known that statistical data may present an information flow/filtration without ‘usual conditions’. The estimation problem is achieved by means of structural least squares (LS) estimates and their sequential versions. The main results of the paper are devoted to the strong consistency of such LS-estimates. For sequential LS-estimates, the property of fixed accuracy is proved. Finally, several illustrative examples from risk theory and mathematical finance are presented.

本文研究的主要对象是可选设置中的一个非常通用的回归模型——当观察到的过程是一个依赖于未知参数的可选半鞅时。众所周知，统计数据可能会在没有“通常情况”的情况下呈现信息流/过滤。估计问题是通过结构最小二乘 (LS) 估计及其顺序版本来实现的。本文的主要结果致力于此类 LS 估计的强一致性。对于序列 LS 估计，证明了固定精度的性质。最后，介绍了来自风险理论和数学金融的几个说明性示例。