Detecting the jumps from the data: Elastic anomaly detection algorithms and parameter estimation of uncertain differential equations with jumps

Abstract

In financial and actuarial modeling, alongside diverse other applied domains, stochastic differential equation models incorporating jump components, used to characterize the dynamics of financial variables, have witnessed a marked rise in prominence in recent years. The jump component serves to capture event-driven uncertainties, including corporate defaults, operational failures, or insured events.

However, to detect the jump component is a very vital but challenging issue. Uncertain differential equations are used widely to model many complicated phenomena in financial market, physics, engineering, and so on. One of key research issues in this area is to estimate the parameters of the corresponding equations based on observations from their solutions.

One parameter estimation framework for uncertain differential equations with jumps, combining numerical algorithms and moment methods, is proposed in this paper. To be more precise, one anomaly detection algorithm is designed to preprocess the data first; then the process of parameter estimation is implemented by the method of moments. To illustrate our method, some numerical examples are given. Additionally, empirical studies of quarterly government consumption expenditure data for Australia as well as Microsoft's stock prices are also presented. With a throughly comparative study with other jump detection methods as well as a study with its stochastic counterparts for real data, our model outperforms all others. We conclude this paper with some possible directions and remarks.