Time Sensitive Functionals of Marked Random Measures in Real Time
Frisbee, Kizza M. Nandyose
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In this dissertation, we study marked random measures that model stochastic networks (under attacks), status of queueing systems during vacation modes, responses to cancer treatments (such as chemotherapy and radiation), hostile actions in economics and warfare. We extend the recently developed time sensitivity technique for investigating the processes’ behavior about a fixed threshold to a novel time sensitive technique in three important directions: (1) real-time monotone stochastic processes; (2) two-dimensional signed random measures; and (3) antagonistic stochastic games with two active players and one passive player. The need for the time sensitive feature in our study (i.e., an analytical association with real-time parameter ) allows stochastic control implementation in sharp contrast with time insensitive analysis very often occurring in the literature. To reach our objectives, we proceed with the classical approach of fluctuation analysis of a particle running through a random grid of a convex set that the particle is trying to escape using stand-alone techniques of stochastic expansion and Laplace transform. We investigate the status of the processes upon as well as the statuses at each time in a given observation time interval. For the monotone process, we target the first passage time, pre-first passage time, the status of the associated continuous time parameter process between these two epochs, and the status of the process upon these two epochs. We obtain analytically tractable formulas and demonstrate them on special cases of marked Poisson processes. Inspired by the monotone result, we embellish it to a two-dimensional signed random measure with position dependent marking. The real-valued component of the associated marked point process is non-monotone presenting an analytical challenge. We manage to investigate various characteristics of that component, including the nth drop or a sharp surge that find applications to finance (like option trading) and risk theory. Finally, we apply the technique to a class of antagonistic stochastic games of three players A,B, and T, of whom the first two are active and the third is a passive player. The active players exchange hostile attacks of random magnitudes with each other and also with player T exerted at random times. At some point (ruin time), one of the two active players is ruined, when the cumulative damages become unsustainable. We obtain the a closed form of the joint distribution functional representing the status of all players upon and also at each time prior to . We illustrate the game on a number of practical models, including stock option trading and queueing systems with vacations and (N,T)-policy.