On the controller-stopper problems with controlled jumps
Abstract
We analyze the continuous time zero-sum and cooperative controller-stopper games of Karatzas and Sudderth [Annals of Probability, 2001], Karatzas and Zamfirescu [Annals of Probability, 2008] and Karatzas and Zamfirescu [Applied Mathematics and Optimization, 2005] when the volatility of the state process is controlled as in Bayraktar and Huang [SIAM Journal on Control and Optimization, 2013] but additionally when the state process has controlled jumps. We perform this analysis by first resolving the stochastic target problems (of Soner and Touzi [SIAM Journal on Control and Optimization, 2002; Journal of European Mathematical Society, 2002]) with a cooperative or a non-cooperative stopper and then embedding the original problem into the latter set-up. Unlike in Bayraktar and Huang [SIAM Journal on Control and Optimization, 2013] our analysis relies crucially on the Stochastic Perron method of Bayraktar and Sirbu [SIAM Journal on Control and Optimization, 2013] but not the dynamic programming principle, which is difficult to prove directly for games.
Cite
@article{arxiv.1609.03954,
title = {On the controller-stopper problems with controlled jumps},
author = {Erhan Bayraktar and Jiaqi Li},
journal= {arXiv preprint arXiv:1609.03954},
year = {2017}
}
Comments
Final version. To appear in Applied Mathematics and Optimization. Keywords: Controller-stopper problems, controlled jumps, stochastic target problems, stochastic Perron's method, viscosity solutions