Mathematics and Statistics 9(4): 481-500, 2021 DOI: 10.13189/ms.2021.090408 http://www.hrpub.org Time Sensitive Analysis of Antagonistic Stochastic Processes and Applications to Finance and Queueing Jewgeni H. Dshalalow 1,* , Kizza Nandyose 2 , Ryan T. White 1 1 Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA 2 Intel Corporation, USA Received March 3, 2021; Revised June 2, 2021; Accepted June 15, 2021 Cite This Paper in the following Citation Styles (a): [1] Jewgeni H. Dshalalow, Kizza Nandyose, Ryan T. White, ”Time Sensitive Analysis Of Antagonistic Stochastic Processes And Applications To Finance And Queueing,” Mathematics and Statistics, Vol.9, No.4, pp. 481-500, 2021. DOI: 10.13189/ms.2021.090408 (b): Jewgeni H. Dshalalow, Kizza Nandyose, Ryan T. White, (2021). Time Sensitive Analysis Of Antagonistic Stochastic Processes And Applications To Finance And Queueing. Mathematics and Statistics, 9(4), 481-500. DOI: 10.13189/ms.2021.090408 Copyright ©2021 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License 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 sus- tainability 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 is 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. Keywords Random Walk, Independent and Stationary Increments Processes, Fluctuations of Stochastic Processes, Marked Point Processes, First Passage Time, Signed Marked Random Measures, Time Sensitive Analysis 1 Introduction In this article we study the behavior of bivariate signed marked random measures around specified thresholds in which underlying marks are competing against each other. There are various applications of such settings found in stochastic mod- eling, including finance and queueing. It is very convenient to formalize the problem using some terminology of antagonistic stochastic games of several players, even though our interpreta- tion does not include traditional game-theoretical components such as payoff functions and iterative optimization. To serve this objective, we consider a generic game of three players, A, B, and C of whom A and B are “active players” who are involved in a periodic exchange of random hostile ac- tions aimed at inflicting damages upon each other, while player C is passive being a collateral damage of the active part of the game. Player C periodically recovers from the blows and he does not respond to the actions from the other players. In a nutshell, such a game can be described as follows. At ran- dom times t 0 ,t 1 ,t 2 ..., players A and B exchange with simul- taneous hostile actions inflicting casualties of random magni- tudes, x 0 ,x 1 ,x 2 ,..., and Δ 0 , Δ 1 , Δ 2 ,..., respectively, until one of the two players becomes ruined. The latter is stipu-