Abstract
In this chapter, we will explain some numerical methods to deal with two person stochastic differential games that are developed by Kushner ([107, 108] and [109]). Almost all of the materials of this chapter are extracted from these three excellent works of Kushner. The Markov chain approximation method is a powerful and more popularly used class of methods introduced in early 1990s by Kushner for the numerical solution of almost all of the standard forms of stochastic strategy problems [106, 109], Kushner and Ramachandran [113], and Kushner and Chamberlain [110]. The idea of the Markov chain approximation method is to first approximate the controlled diffusion dynamics by a suitable Markov chain on a finite state space with a discretization parameter h > 0, then approximate the payoff functions. One solves the game problem for the simpler chain model, and then proves that the value functions associated with equilibrium or δ -equilibrium strategies for the chain converge to the value functions associated with equilibrium or δ -equilibrium strategies for the diffusion model, as δ → 0. This is method is intuitive and it uses approximations which are “physically” close to the original problem. Extensions to approximations for two-person differential games with discounted, finite time, stopping time, and pursuit-evasion games were given in Kushner [108] for reflected diffusion models where the strategies for the two players are separated in the dynamics and payoff rate functions. An extension to two-player stochastic dynamic games with the same systems model, but where the payoff function is ergodic is given in Kushner [107].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Atlantis Press
About this chapter
Cite this chapter
Ramachandran, K.M., Tsokos, C.P. (2012). Some Numerical Methods. In: Stochastic Differential Games. Theory and Applications. Atlantis Studies in Probability and Statistics, vol 2. Atlantis Press. https://doi.org/10.2991/978-94-91216-47-3_8
Download citation
DOI: https://doi.org/10.2991/978-94-91216-47-3_8
Published:
Publisher Name: Atlantis Press
Print ISBN: 978-94-91216-46-6
Online ISBN: 978-94-91216-47-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)