Sub-maxingale versus strong sub-maxingale

Determine whether every sub-maxingale M=(M_t)_{t∈[0,T]} on a filtered probability space (Ω,(F_t)_{t∈[0,T]},P), defined via the conditional essential supremum operator by the property esssup_{F_u} M_t ≥ M_u for all 0 ≤ u ≤ t ≤ T, is necessarily a strong sub-maxingale; that is, ascertain whether for every stopping time τ the stopped process M^τ=(M_{t∧τ})_{t∈[0,T]} remains a sub-maxingale under the same conditional essential supremum operator.

Background

The paper introduces a maxingale framework based on the conditional essential supremum operator, defining sub-maxingales as processes M satisfying esssup_{F_u} M_t ≥ M_u for all u ≤ t, and strong sub-maxingales as processes whose stopped versions M^τ remain sub-maxingales for all stopping times τ.

In classical martingale theory under conditional expectation, Doob’s optional stopping theorem implies that submartingales enjoy certain stability under stopping (with suitable boundedness/integrability conditions). However, whether the analogous stability holds in this new maxingale framework based on conditional suprema is explicitly left unresolved.

The authors provide a characterization (Proposition S-SM) showing that M is strong sub-maxingale if and only if for all stopping times S and τ one has esssup_{F_S}(M_τ) ≥ M_{S∧τ}, but they do not establish that every sub-maxingale automatically satisfies this condition. Hence, the general implication remains open.