On the Controller-Stopper Problems with Controlled Jumps

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Abstract

We analyze the continuous time zero-sum and cooperative controller-stopper games of Karatzas and Sudderth (Ann Probab 29(3):1111–1127, 2001), Karatzas and Zamfirescu (Ann Probab 36(4):1495–1527, 2008) and Karatzas and Zamfirescu (Appl Math Optim 53(2):163–184, 2006) when the volatility of the state process is controlled as in Bayraktar and Huang (SIAM J Control Optim 51(2):1263–1297, 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 J Control Optim 41(2):404–424, 2002; J Eur Math Soc 4(3):201–236, 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 J Control Optim 51(2):1263–1297, 2013) our analysis relies crucially on the Stochastic Perron method of Bayraktar and Sîrbu (SIAM J Control Optim 51(6):4274–4294, 2013) but not the dynamic programming principle, which is difficult to prove directly for games.

Keywords

Controller-stopper problems Stochastic target problems Stochastic Perron’s method Viscosity solution 

Mathematics Subject Classification

Primary 93E20 Secondary 49L25 60J75 60G40 

Notes

Acknowledgements

E. Bayraktar is supported in part by the National Science Foundation under Grant DMS-1613170 and the Susan M. Smith Professorship.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.University of MichiganAnn ArborUSA
  2. 2.Goldman SachsNew YorkUSA

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