Patterns for the waiting time in the context of discrete-time stochastic processes.

The aim here is to provide a deeper understanding on the concept of waiting time in application to multiple stochastic processes. This obliges us to work with the vector stochastic process which enables considering at least two stochastic process at simultaneous time instances. In the present study the plan is to master vector stochastic processes by developing the level crossing method. The reason that the previous level-crossing methods lack generality is based on their individual element studies, where the coupling between the components of the vector stochastic process had been simply neglected. In the present work by introducing the generalized level crossing method, consideration of coupling between the components has become possible. This enables analyzing and hence extracting information out of coupled processes usually faced when working in tensor environments. The results obtained by this technique state that in addition to the point distribution of the vector stochastic process, the coupling plays very effective, justifying its importance. To be most illustrative, when the components are Gaussian white noises, an analytic solution is obtained which would act as a measure for the effects of coupling. In a sense that when coupling is present in between stochastic processes, the waiting time would differ from the case of no coupling. The comparison of the two waiting times measures the efficiency of the coupling. By applying the two new concepts; instantaneous risk and instantaneous return to the market portfolio and risk management, the sense in which these two concepts can help is illustrated. This enables one to adjust its portfolio so how to minimize the expectation time (waiting time) for obtaining a specific profit connected to its risk.