On the controller-stopper problems with controlled jumps. (English) Zbl 1422.91081
Summary: We analyze the continuous time zero-sum and cooperative controller-stopper games of
I. Karatzas and W. D. Sudderth [Ann. Probab. 29, No. 3, 1111–1127 (2001; Zbl 1039.60043)],
I. Karatzas and I.-M. Zamfirescu [Ann. Probab. 36, No. 4, 1495–1527 (2008; Zbl 1142.93040)]
I. Karatzas and I.-M. Zamfirescu [Appl. Math. Optim. 53, No. 2, 163–184 (2006; Zbl 1136.93047)]
when the volatility of the state process is controlled as in
[E. Bayraktar and Y.-J. Huang, SIAM J. Control Optim. 51, No. 2, 1263–1297 (2013; Zbl 1268.49045)]
but additionally when the state process has controlled jumps. We perform this analysis by first resolving the stochastic target problems of
H. M. Soner and N. Touzi [SIAM J. Control Optim. 41, No. 2, 404–424 (2002; Zbl 1011.49019); J. Eur. Math. Soc. (JEMS) 4, No. 3, 201–236 (2002; Zbl 1003.49003)]
with a cooperative or a noncooperative stopper and then embedding the original problem into the latter set-up. Unlike in
[E. Bayraktar and Y.-J. Huang, SIAM J. Control Optim. 51, No. 2, 1263–1297 (2013; Zbl 1268.49045)]
our analysis relies crucially on the stochastic Perron method of
E. Bayraktar and M. Sîrbu [SIAM J. Control Optim. 51, No. 6, 4274–4294 (2013; Zbl 1285.49019)]
but not the dynamic programming principle, which is difficult to prove directly for games.
MSC:
| 91A15 | Stochastic games, stochastic differential games |
| 91A55 | Games of timing |
| 91A12 | Cooperative games |
| 49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
| 60G40 | Stopping times; optimal stopping problems; gambling theory |
Keywords:
controller-stopper problems; stochastic target problems; stochastic Perron’s method; viscosity solutionCitations:
Zbl 1039.60043; Zbl 1142.93040; Zbl 1136.93047; Zbl 1268.49045; Zbl 1011.49019; Zbl 1003.49003; Zbl 1285.49019References:
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