Optimal Multiple Stopping For Non-linear Expectations

There are already many results about optimal stopping problem under lin-ear expectation. In recent years, as the developments of non-linear expectation theory, optimal stopping for non-linear expectations begin to attract ours at-tention. In paper [1] and paper [2], Bayraktar and Yao defined a collection of non-linear expectations in which non-linear expectation εg is a particular case. And they studied the optimal single stopping problem under this collection of expectations. Inspired by this two articles, we consider to extend the result about optimal multiple stopping under linear expectations to the non-linear expecta-tion case. Our aim is to solve the optimal multiple stopping times problem under non-linear expectations. We find that there are similar theory in both the linear expectation and the non-linear expectations, including that the value function v(s)=esssupθ∈Ss,Tdε[ψ(θ)|Fs](here ε is non-linear expectation)is still a super-martingale, the existence and computation of stopping time θ*which make v(s) obtain its supremum.