Time Sensitive Analysis of Antagonistic Stochastic Processes and Applications to Finance and Queueing

Abstract

This paper deals with a class of antagonistic stochastic games of three players A, B, and C, of whom the first two are active players and the third is a passive player. The active players exchange hostile attacks at random times T = {t 1 , t 2 , . . .} of random magnitudes with each other and also with player C. Player C does not respond to any attacks (that are regarded as a collateral damage). There are two sustainability thresholds M and T are set so that when the total damages to players A and B cross M and T , respectively, the underlying player is ruined. At some point t ν (ruin time), one of the two active players will be ruined. Player C's damages are sustainable and some rebuilt. Of interest are the ruin time t ν and the status of all three players upon t ν as well as at any time t prior to t ν . We obtain an analytic formula for the joint distribution of the named processes and demonstrate its closed form in various analytic and computational examples. In some situations pertaining to stock option trading, stock prices (player C) can fluctuate. So in this case, it is of interest to predict the first time when an underlying stock price drops or significantly drops so that the trader can exercise the call option prior to the drop and before maturity T . Player A monitors the prices upon times T assigning 0 damage to itself if the stock price appreciates or does not change and assumes a positive integer if the price drops. The times T are themselves damages to player B with threshold T . The "ruin" time is when threshold M is crossed (i.e., there is a big price drop or a series of drops) or when the maturity T expires whichever comes first. T hus a p rior a ction i s n eeded a nd i ts t ime is predicted. We illustrate the applicability of the game on a number of other practical models, including queueing systems with vacations and (N,T)-policy.