{"id":"65ceccf1-b0f5-4f48-a723-d4ac1e4d6912","arxiv_id":"1908.07174","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims existence and construction of optimal multiple stopping times under nonlinear expectation, but the proof of the key left-continuity property is invalid.","lead":"This paper tries to solve optimal stopping with several exercise rights when uncertainty is modeled by nonlinear expectations, a framework that covers ambiguity. It claims to construct optimal stopping times by reducing the multi-stopping problem to repeated single-stopping problems, but a key proof step uses an inequality in the wrong direction.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.16's left-continuity proof uses E[ess sup f_n] ≤ sup E[f_n], which has the wrong direction and is false already for classical expectations; the existence of optimal multiple stopping times is therefore not established as written.","rationale":"The reader's verdict is REJECT, and the reader's weakest-assumption analysis correctly identifies the proof of Theorem 3.16 as the critical defect. I read the manuscript in good faith: the single-stopping theory in Sections 2 and the reduction of the multiple-stopping value function to a single-stopping value function in Theorem 3.5 and Theorem 4.3 are natural and largely sound. However, the central existence theorem depends on the continuity of the induced reward family. The specific inequality used to prove left-continuity is wrong in direction and is false even for linear expectations, so the paper currently does not provide a proof that Theorem 2.16 applies to u(S). Since the same invalid estimate is used in the d-time generalization and in the aggregation section, the flaw is not restricted to a single lemma. I do not see this as a matter of disagreement with a broader consensus; it is an internal correctness gap in the argument as written. The paper may be repairable with stronger uniformity or a different argument, but as submitted the central claim is not supported. Therefore I agree with the reader's REJECT verdict and see no change to that verdict.","tokens_in":29712,"tokens_out":5895,"duration_ms":63576,"concrete_test":"Check the asserted inequality in Theorem 3.16 with the classical expectation on a two-point probability space: take θ_n ↑ θ and two stopping times in S_{θ_n} giving ξ_1=a1_A and ξ_2=a1_{A^c}; then E[ess sup(ξ_1,ξ_2)]=a while sup(E[ξ_1],E[ξ_2])=a/2, so the displayed estimate fails in the simplest possible setting. Then attempt to re-derive the LCE of {~X(τ)} using only the valid ingredients of the paper (dominated convergence, UCE, and the dominance estimate (3.7)); if no such derivation exists under the stated hypotheses, Theorems 3.17, 4.13 and the Section 5 hitting-time results cannot be accepted as proved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the proof that the induced reward family ~X(τ)=max(u1(τ),u2(τ)) is left-continuous in E-expectation (LCE), because Theorem 2.16 requires full CE to produce the optimal θ* for the single-stopping problem u(S), and Theorem 3.17 (and by induction Theorem 4.13) builds the optimal multiple stopping times from that θ*. In the proof of Theorem 3.16, after defining X'(τ,θ)=X(τ,θ)1_{τ≥θ}−1_{τ<θ}, the paper bounds E[ess sup_{τ∈S_{θ_n}} |X'(τ,θ)−X(τ,θ_n)|] and then asserts E[ess sup_{τ∈S_{θ_n}} |X(τ,θ)−X(τ,θ_n)|] ≤ sup_{τ∈S_0} E[|X(τ,θ)−X(τ,θ_n)|]. This inequality has the wrong direction: monotonicity gives E[sup_i ξ_i] ≥ sup_i E[ξ_i], and the displayed bound is false even for the classical expectation. UCE only controls the supremum of expectations, not the expectation of the essential supremum. Without a valid estimate showing E[ess sup_{τ∈S_{θ_n}} |X(τ,θ)−X(τ,θ_n)|] → 0, the conclusion lim_n E[u1(θ_n)] = E[u1(θ)] is not proved, so ~X is not known to be LCE. Consequently Theorem 2.16 cannot be applied to the single-stopping problem for ~X, and the optimal stopping time constructed in Theorem 3.17 is unsupported. The same erroneous comparison reappears in the d-time induction in Proposition 4.12, and the aggregation results of Section 5 inherit the gap. The flaw is local in expression but central in logical position: without LCE of ~X, the reduction of multiple to single stopping does not yield an existence proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies optimal multiple stopping problems under filtration-consistent nonlinear expectations that are subadditive and positively homogeneous (upper expectations). The reward is modeled as a family of random variables indexed by stopping times rather than as an RCLL process, so no aggregation into a process is required. For the single stopping problem, the paper proves existence of an optimal stopping time under a continuity condition on the reward family. For the multiple stopping problem, it defines an induced reward family ~X(τ)=max(u1(τ),u2(τ)) and proves that the multiple-stopping value function coincides with the single-stopping value function for ~X. The main technical step is the proof that ~X is continuous along stopping times in expectation, which then yields, by induction, existence of optimal multiple stopping times (Theorems 3.17 and 4.13). Section 5 gives aggregation and hitting-time representations under stronger regularity conditions.","tokens_in":30103,"tokens_out":4810,"duration_ms":48565,"significance":"If the main results were correct, they would provide a useful extension of multiple-stopping theory from classical expectations to sublinear expectations, with potential applications to swing options under Knightian uncertainty and to stopping problems when reward cannot be aggregated into a process. The paper builds on the established F-expectation framework of Bayraktar and Yao and uses the order relation from Kobylanski, Quenez, and Rouy-Mironescu; the proofs are original and there is no sign of circular dependence on the author's own prior work. The identification v=u in Theorem 3.5 and the construction in Propositions 3.6 and 4.4 are natural and clearly presented. However, a load-bearing inequality in the proof of the left-continuity of ~X is false, so the main existence theorems are not established as written.","major_comments":[{"comment":"The proof of the LCE property of ~X contains a false inequality. After defining X'(τ,θ)=X(τ,θ)1_{τ≥θ}-1_{τ<θ}, the paper bounds E[ess sup_{τ∈S_{θ_n}} |X(τ,θ)-X(τ,θ_n)|] by sup_{τ∈S_0} E[|X(τ,θ)-X(τ,θ_n)|], which then is sent to zero using the UCE assumption. This is the wrong direction: for any monotone (and in particular for any sublinear) expectation, E[ess sup_τ f_τ] ≥ sup_τ E[f_τ], and even for classical expectation one has E[sup f] ≥ sup E[f]. The displayed bound is therefore false already for the linear expectation. The UCE hypothesis only controls the supremum of expectations, not the expectation of the essential supremum. Consequently, the limit lim_n E[v'(θ_n)]=E[u1(θ)] is not proved, so the family ~X is not shown to be left-continuous in expectation. Since Theorem 2.16 requires full CE of the reward family to produce the optimal stopping time for u(S), the existence result in Theorem 3.17 is unsupported.","section":"Section 3, Theorem 3.16"},{"comment":"The same defective comparison reappears in the induction step for d stopping times. The proof states that |E[u^{(i),θ}(θ_n)] - E[u^{(i)}(θ_n)]| ≤ sup_{τ1,τ2∈S0} E[|X^{(i)}(τ1,τ2,θ)-X^{(i)}(τ1,τ2,θ_n)|] + E[ξ1_{θ_n<θ}], again replacing an expectation of an essential supremum by a supremum of expectations. As in Theorem 3.16, this inequality has the wrong direction and is false for classical expectations. Since Proposition 4.12 is the basis for the LCE of the induced reward family and hence for the induction in Theorem 4.13, the existence of optimal d-stopping times inherits the same gap.","section":"Section 4, Proposition 4.12"},{"comment":"Both theorems depend critically on the unproved LCE of ~X (or of the analogous induced family in the d-dimensional case). If the authors intend to repair the proof by strengthening the UCE condition to control E[ess sup |X(τ,θ)-X(τ,θ_n)|], that would be a substantive change in hypotheses and would require reworking the aggregation results in Section 5, which also rely on Theorem 3.16. As written, the central existence claim is not established.","section":"Theorem 3.17 and Theorem 4.13"}],"minor_comments":[{"comment":"The text contains several typos and misspellings: 'appropiate' in the Introduction, 'reftracting' in Section 1, 'deonte' in Section 2, 'mehtod' in the Introduction, 'postponsed' in Section 3, and 'defintion' in Section 5. These should be corrected.","section":"Throughout"},{"comment":"Definition 3.10 uses expressions E[|X(τ,σ)-X(τ,σ_n)|] before the paper explains in Remark 3.12 why these differences lie in Dom(E). It would improve readability to state explicitly at the definition that this membership follows from (D2) and (D3), or to move the remark earlier.","section":"Definition 3.10"},{"comment":"The proof of Theorem 5.11 refers to extending the definition of X^{(i)} to all stopping times via ~X^{(i)}(θ)=X^{(i)}(θ)1_{θ≥θ*}-1_{θ<θ*}; it would be helpful to check that this extension is indeed RC in the sense of Theorem 5.5 and that the subsequent hitting-time argument covers the boundary case θ=θ* explicitly.","section":"Theorem 5.11"}],"recommendation":"reject","confidential_remarks":"The paper addresses a meaningful problem and the overall structure is sensible, but the flaw in the proof of Theorem 3.16 is central and invalidates the main existence theorem. Since the error is a false inequality rather than a local gap, and the proposed repair would require changing the main hypotheses, I cannot recommend publication in the present form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper takes on a real problem—multiple stopping under Knightian uncertainty without aggregating the reward into a process—and the reduction of the d-stopping problem to a single stopping problem via an induced reward family is clean and genuinely new. But the proof that the induced family is left-continuous in E-expectation (Theorem 3.16) uses an inequality with the wrong direction, and that breaks the existence theorem.\n\nWhat is actually new: the combination of F-expectations (Bayraktar–Yao) with the non-aggregated reward framework (Kobylanski–Quenez–Rouy-Mironescu). Section 2 gives a careful treatment of single stopping for admissible families; Propositions 2.7 and 2.9 and Theorem 2.16 are solid. The value-function identity v(S)=u(S) in Theorem 3.5 and the construction of optimal multiple stopping times from optimal single stopping times in Proposition 3.6 are the real contributions. The aggregation section is a natural extension and looks mostly fine as far as I can tell.\n\nThe soft spot is exactly where the stress-test note points. In Theorem 3.16, to show LCE of ~X, the paper needs E[ess sup |X(τ,θ)-X(τ,θ_n)|] → 0. It bounds this by sup_τ E[|X(τ,θ)-X(τ,θ_n)|]. That inequality is backwards: monotonicity gives E[sup f] ≥ sup E[f]. It is false already for classical expectation. So the LCE conclusion is unproved. Since Theorem 3.17 and the induction in Proposition 4.12/Theorem 4.13 rely on applying Theorem 2.16 to ~X, the central existence claim is unsupported as written. The flaw is local in expression but load-bearing; it is not a typo, though it might be repairable with a stronger uniform-continuity assumption or a different argument.\n\nMy overall take: worth engaging. The reduction theorem and the single-stopping machinery are valuable, and the flaw is specific enough that a serious author could fix it. I would not publish as is, but I would send it to a referee and ask them to focus on Theorem 3.16.","headline":"Genuinely useful reduction of multiple to single stopping under nonlinear expectations, but a false inequality in the left-continuity proof leaves the main existence theorem unsupported.","tokens_in":30637,"tokens_out":2758,"would_cite":false,"duration_ms":27930,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Optimal multiple stopping problems under nonlinear expectation admit optimal times via induction.","keywords":["nonlinear expectations","optimal stopping","multiple stopping","Knightian uncertainty","ambiguity","supermartingale systems","reward families","aggregation"],"falsifier":"Take the classical expectation, which satisfies all the paper's hypotheses, and choose a bounded, nonnegative biadmissible reward with $\\theta_n\\uparrow\\theta$; compute both sides of the inequality in the proof of Theorem 3.16. If an explicit example shows $\\lim_n E[u_1(\\theta_n)]\\neq E[u_1(\\theta)]$ for the induced reward, then Theorem 3.17 is false as stated; if the left limit always agrees despite the bad inequality, the theorem survives and only the proof needs repair.","tokens_in":29468,"feed_emoji":"🎯","tokens_out":8933,"duration_ms":81307,"temperature":0.7,"pith_summary":"This paper claims that the optimal multiple stopping problem can be solved when uncertainty is modeled by a nonlinear expectation, meaning the agent does not know the probability law precisely. The reward is not required to be a stochastic process: it is a family of random variables indexed by stopping times, and the value function is an essential supremum over allowed exercise tuples. The central device is a reduction: the value of a $d$-exercise problem equals the value of a single-stopping problem for an induced reward family formed by the maxima of value functions of the $(d-1)$-exercise problem. By induction, optimal $d$-tuples of stopping times are constructed, and under stronger uniformity assumptions the rewards and values aggregate into processes so that the optimal times are first hitting times. A sympathetic reader would care because this extends the standard machinery for swing options and multiple exercise rights from classical expectation to ambiguity, at the cost of assuming the nonlinear expectation is subadditive and positively homogeneous.","feed_headline":"Optimal multiple stopping times exist under nonlinear expectation","feed_subtitle":"A d-exercise reward reduces to nested single stopping problems; extra regularity gives hitting times.","key_machinery":"The machinery has three parts. First, a filtration-consistent nonlinear expectation $(E,\\mathrm{Dom}(E))$ satisfying hypotheses (H0)-(H7); in particular subadditivity and positive homogeneity make the stopping problem a robust 'max over strategies, max over priors' problem. Second, a $d$-admissible reward family $\\{X(\\tau)\\}_{\\tau\\in S^d_0}$, a collection of random variables indexed by stopping-time tuples that agree when two tuples agree almost surely; this replaces the usual RCLL reward process. Third, the induced reward $\\hat X(\\theta)=\\max_i u^{(i)}(\\theta)$ and the approximation scheme $\\tau_\\lambda(S)=\\mathrm{ess\\,inf}\\{\\tau\\in S_S:\\lambda v(\\tau)\\le X(\\tau)\\}$, whose limit as $\\lambda\\uparrow1$ is the minimal optimal single-stopping time. The same scheme, applied inductively to the families $X^{(i)}(\\cdot,\\theta)$, yields the optimal multiple stopping tuple and, in the aggregation section, the hitting-time representation.","core_discovery":"The paper's central claim is the reduction theorem $v(S)=u(S)$, where $v(S)=\\mathrm{ess\\,sup}_{\\tau\\in S^d_S} E_S[X(\\tau)]$ is the value of the $d$-stopping problem and $u(S)=\\mathrm{ess\\,sup}_{\\tau\\in S_S} E_S[\\hat X(\\tau)]$ is the value of a single-stopping problem with the induced reward $\\hat X(\\theta)=\\max_i u^{(i)}(\\theta)$; each $u^{(i)}$ is the value of the $(d-1)$-stopping problem in which one exercise is fixed at $\\theta$. Once the induced family $\\hat X$ is known to be continuous along stopping times in $E$-expectation, the single-stopping theorem produces a minimal optimal time for $u$, and that time is the minimum of the optimal $d$-tuple. The paper proves right-continuity of $\\hat X$ through a dominance condition and left-continuity through a modification of the reward that restores adaptedness. The conclusion is an inductive existence theorem for optimal multiple stopping times and, under stronger regularity, a representation of those times as hitting times of aggregated progressive processes.","pith_inferences":["Editorial inference: the reduction $v(S)=u(S)$ itself needs no regularity of the reward family, so the only real obstacle to a fully general existence theorem is left-continuity of $\\hat X$; a correct dominated-convergence argument would remove the proof gap without changing the statement.","Editorial inference: for $g$-expectations with convex generators, the dominance condition is satisfied, so the right-continuity half of the theorem should hold; the left-continuity half could be tested on a finite-horizon binomial analogue, where the questionable inequality can be checked exactly.","Editorial inference: since the single-stopping value is the smallest $E$-supermartingale system dominating the reward family, the multiple-stopping value should satisfy a dynamic programming principle over stopping-time-indexed families; that would give a nonlinear Snell-envelope theory without process aggregation.","Editorial inference: if left-continuity is dropped, the $\\lambda$-approximation still gives $(1-\\lambda)$-optimal stopping times, so the practical effect of the gap may be a quantitative worst-case error rather than a total failure of existence."],"forward_implications":["If the paper is right, a $d$-exercise American or swing option under ambiguity has a well-defined value $v(S)=\\mathrm{ess\\,sup}_{\\tau\\in S^d_S}E_S[X(\\tau)]$, and an optimal exercise strategy exists.","The $d$-stopping problem can be solved by solving $d$ nested single-stopping problems; each step only needs the induced reward $\\hat X$, so the original reward family need not be aggregated into a process.","When the expectation is subadditive and positively homogeneous, the value is conservative under ambiguity: it is a supremum over stopping strategies of an upper expectation.","Under uniform continuity of the reward in $E$-expectation and a dominance condition, the optimal stopping times coincide with first hitting times of aggregated right-continuous value processes, recovering the classical picture.","The minimal optimal stopping tuple has a recursive characterization: its minimum is the minimal optimal time for the induced single-stopping problem, and the other coordinates are minimal optimal times for the reduced $(d-1)$-stopping problems."],"supporting_citations":[{"why":"Establishes the filtration-consistent nonlinear expectation framework, the domain hypotheses (H0)-(H4), and the extension of conditional expectations to arbitrary stopping times used throughout the paper.","marker":"[1]"},{"why":"Provides the single-stopping theory for nonlinear expectations with process rewards, the optimality criteria, and the g-expectation examples and counterexamples for hypothesis (H5).","marker":"[2]"},{"why":"Supplies the continuous-time optimal stopping result under ambiguity for RCLL reward processes and the first hitting time representation that the aggregation section extends to multiple stopping.","marker":"[5]"},{"why":"Defines filtration-consistent nonlinear expectations and their relation to g-expectations, which serve as the main model class and as source of the F-expectation axioms.","marker":"[6]"},{"why":"Provides the classical theory of optimal stopping for a family of random variables indexed by stopping times, including supermartingale systems and essential supremum arguments adapted here.","marker":"[7]"},{"why":"Gives the linear-expectation multiple stopping problem with reward families, the partial order on d-tuples, the minimal optimal stopping characterization, and the hitting-time aggregation results that this paper generalizes.","marker":"[8]"}],"fun_headline_variants":["Nonlinear expectation: nested singles solve the d-stop problem","Multiple stopping under nonlinear expectation via nested singles","Nested single stops solve multiple stopping under nonlinearity","With regularity, multiple stops become hitting times","Optimal multiple stops under nonlinear expectation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction depends on the induced reward after one exercise remaining continuous from the left as stopping times increase; the proof of that fact uses a comparison between the expectation of a maximum and the maximum of expectations that can fail for nonlinear expectations.","fun_headline_variants_meta":{"raw":{"variants":["Nonlinear expectation: nested singles solve the d-stop problem","Multiple stopping under nonlinear expectation via nested singles","Nested single stops solve multiple stopping under nonlinearity","With regularity, multiple stops become hitting times","Optimal multiple stops under nonlinear expectation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001128,"raw_usage":{"total_tokens":4668,"prompt_tokens":902,"completion_tokens":3766,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":518,"completion_tokens_details":{"reasoning_tokens":3696}},"tokens_in":518,"tokens_out":3766,"duration_ms":27861,"temperature":1.0,"reasoning_tokens":3696,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:25:19.627960+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the classical expectation, which satisfies all the paper's hypotheses, and choose a bounded, nonnegative biadmissible reward with $\\theta_n\\uparrow\\theta$; compute both sides of the inequality in the proof of Theorem 3.16. If an explicit example shows $\\lim_n E[u_1(\\theta_n)]\\neq E[u_1(\\theta)]$ for the induced reward, then Theorem 3.17 is false as stated; if the left limit always agrees despite the bad inequality, the theorem survives and only the proof needs repair.","supporting_citations":[{"cited_title":"and Yao, S","cited_arxiv_id":null,"evidence_quote":"Establishes the filtration-consistent nonlinear expectation framework, the domain hypotheses (H0)-(H4), and the extension of conditional expectations to arbitrary stopping times used throughout the paper."},{"cited_title":"and Yao, S","cited_arxiv_id":null,"evidence_quote":"Provides the single-stopping theory for nonlinear expectations with process rewards, the optimality criteria, and the g-expectation examples and counterexamples for hypothesis (H5)."},{"cited_title":"and Riedel, F","cited_arxiv_id":null,"evidence_quote":"Supplies the continuous-time optimal stopping result under ambiguity for RCLL reward processes and the first hitting time representation that the aggregation section extends to multiple stopping."},{"cited_title":"and Peng, S","cited_arxiv_id":null,"evidence_quote":"Defines filtration-consistent nonlinear expectations and their relation to g-expectations, which serve as the main model class and as source of the F-expectation axioms."},{"cited_title":"and Zhang, J","cited_arxiv_id":null,"evidence_quote":"Provides the classical theory of optimal stopping for a family of random variables indexed by stopping times, including supermartingale systems and essential supremum arguments adapted here."},{"cited_title":"(1981) Les aspects probabilistes du controle stochastique","cited_arxiv_id":null,"evidence_quote":"Gives the linear-expectation multiple stopping problem with reward families, the partial order on d-tuples, the minimal optimal stopping characterization, and the hitting-time aggregation results that this paper generalizes."}],"review_version":1}