Stochastic Differential Games with Multiple Modes and a Small Parameter |
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Authors: | K. M. Ramachandran |
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Affiliation: | 1. Department of Mathematics , University of South Florida , Tampa , Florida , USA ram@cas.usf.edu |
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Abstract: | Abstract Two persons stochastic differential games with multiple modes where the system is driven by a wideband noise is considered. The state of the system at time t is given by a pair (x ε(t), θε(t)), where θε(t) takes values in S = {1, 2,…, N} and ε is a small parameter. The discrete component θε(t) describes various modes of the system. The continuous component x ε(t) is governed by a “controlled process” with drift vector which depends on the discrete component θε (t). Thus, the state of the system, x ε(t), switches from one random path to another at random times as the mode θε(t) changes. The discrete component θε (t) is a “controlled Markov chain” with transition rate matrix depending on the continuous component. Both zero-sum and nonzero-sum games will be considered. In a zero-sum game, player I is trying to maximize certain expected payoff over his/her admissible strategies, where as player II is trying to minimize the same over his/her admissible strategies. This kind of game typically occurs in a pursuit-evation problem where an interceptor tries to destroy a specific target. Due to swift manuaring of the evador and the corresponding reaction by the interceptor, the trajectory keep switching rapidly at random times. We will show that the process (x ε(t), θε(t)) converges to a process whose evolution is given by a “controlled diffusion process” with switching random paths. We will also establish the existence of randomized δ -optimal strategies for both players. |
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Keywords: | Multiple modes Near optimal strategies Stochastic differential games Wideband noise Zero-sum game |
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