  In this paper, we present the testing of four hypotheses on two streams of
observations that are driven by L\'evy 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 L\'evy processes with large or
small jumps. In either case, these give rise to four possibilities.
Infinitesimal generators are presented and analyzed. Bounds for infinitesimal
generators in terms of \emph{super-solutions} and \emph{sub-solutions} are
computed. An application of this procedure for the stochastic model is also
presented in relation to the financial market.
