Optimal stopping for non-linear expectations—Part I

Optimal stopping for non-linear expectations—Part II

“…(1) For any i ∈ I, (S2) and the right-continuity of H i imply that except on a null set N (i) H i s,t ≥ C H , for any 0 ≤ s < t ≤ T, thus H i ν,ρ ≥ C H , ∀ν, ρ ∈ S 0,T with ν ≤ ρ, a.s. (2.4) (2) If (2.3) is assumed for some j ∈ I, the right-continuity of H j and (2.4) imply that except on a null set N C H ≤ H j s,t ≤ ζ j , for any 0 ≤ s < t ≤ T, thus C H ≤ H j ν,ρ ≤ ζ j , ∀ν, ρ ∈ S 0,T with ν ≤ ρ, a.s. Then Lemma 3.2 of [1] implies that (2.2) holds for j. Hence we see that (2.3) is a stronger condition than (2.2).…”

“…(S1) For any i ∈ I, H i 0 = 0, a.s. and (S2) There exists a C H < 0 such that for any i ∈ I, essinf s,t∈D T ;s<t H i s,t ≥ C H , a.s. (S3) For any ν ∈ S 0,T and i, j ∈ I, it holds for any 0 ≤ s < t ≤ T that H k s,t = H i ν∧s,ν∧t + H j ν∨s,ν∨t , a.s., where k = k  i, j, ν  ∈ I is the index defined in Definition 3.2(2) of [1].…”

“…We build this paper on the results of [1] and analyze the optimal stopping problems for nonlinear expectations. The background, literature review and the motivation for these problems are provided in the introduction section of [1].…”

“…The background, literature review and the motivation for these problems are provided in the introduction section of [1]. The notation used in this paper is outlined in Section 1.1 of [1].…”

On the Robust Optimal Stopping Problem

“…[26], [14], Appendix D of [20]) and has been applied to various problems stemming from mathematical finance, the most important example of which is the computation of the super hedging price of the American contingent claims [6,17,18,22]. Optimal stopping under Knightian uncertainty/nonlinear expectations/risk measures or the closely related controller-stopper-games have attracted a lot of attention in the recent years: [23,24,16,8,9,32,2,3,4,5,7,25]. In this literature, the set of probabilities is assumed to be dominated by a single probability or the controller is only allowed to influence the drift.…”