In this dissertation, we study the testing of hypotheses on streams of observations that are driven by Lévy processes. This is applicable for sequential decision making on the state of two-sensor systems. In one case, each sensor receives or does not receive a signal obstructed by noise. In another, each sensor receives data driven by Lévy processes with large or small jumps. In either case, these give rise to four possible outcomes for the hypotheses. Infinitesimal generators are presented and analyzed. Bounds for likelihood functions in terms of super-solutions} and sub-solutions are computed. As an application, we study a change point detection hypothesis test for the detection of the distribution of jump size in one-dimensional Lévy processes. This is shown to be implementable in relation to various classification problems for a crude oil price data set. Machine and deep learning algorithms are implemented to extract a specific deterministic component from the data set, and the deterministic component is implemented to improve the Barndorff-Nielsen & Shephard (BN-S) model, a commonly used stochastic model for derivative and commodity market analysis.