Wavelet-Based Stochastic Analysis of Probability Distributions in Financial Markets: A Measure Theory Approach

Abstract

The purpose of this research was to investigate the applicability of wavelet-based stochastic analysis to determine the probability distribution of financial markets employing measure theory. The research aimed at giving more insights about the markets and the volatility behaviour using sophisticated mathematical models. Historical data of the financial market was analyzed using wavelet transform to obtain the time series in terms of its frequency content. Stochastic analysis methods were then used to describe the distributional characteristics of market changes. Measure theory was incorporated to improve the accuracy of the probability estimates, which gave a strong mathematical base for the evaluation of market behaviour. The wavelet-based approach was able to detect relevant frequency bands associated with the fluctuations of the market, and the results pointed out that the data were non-Gaussian distributed. The use of the measure-theoretic approach was an improvement to the previous probability theory since it facilitated better modelling of the risks and uncertainties in the financial markets especially in cases of extreme events. The study proved that the application of stochastic analysis based on wavelet and measure theory can be used as a strong tool in analyzing the probability distribution of financial markets. It can enhance the accuracy of forecast of market volatility, thus making it as a useful resource in research for scholars, and in practice for practitioners in financial risk management.