Optimal Multiple Stopping Problem under Nonlinear Expectation

Hanwu Li

REVIEW 3 major objections 3 minor 17 references

Optimal Multiple Stopping Problem under Nonlinear Expectation

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Optimal multiple stopping problems under nonlinear expectation admit optimal times via induction.

desk verdict 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. read the letter →

arxiv 1908.07174 v1 pith:AV7C7AKW submitted 2019-08-20 math.PR

Hanwu Li This is my paper · ORCID

classification math.PR MSC 60G40

keywords nonlinearexpectationsoptimalstoppingmultipleKnightianuncertaintyambiguitysupermartingalesystemsrewardfamiliesaggregation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What 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.

The 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.

My 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.

Referee Report

3 major / 3 minor

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.

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 (3)

[Section 3, Theorem 3.16] 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 4, Proposition 4.12] 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.
[Theorem 3.17 and Theorem 4.13] 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.

minor comments (3)

[Throughout] 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.
[Definition 3.10] 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.
[Theorem 5.11] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the multiple-stopping reduction is proved from the nonlinear-expectation axioms rather than assumed.

full rationale

The derivation chain is self-contained relative to the imported F-expectation framework. Section 2 starts from the value function definition (2.1) and proves its supermartingale characterization (Proposition 2.7), the equivalence of optimality criteria (Proposition 2.9), and the existence of an optimal stopping time via the tau-lambda approximation (Lemmas 2.14-2.15 and Theorem 2.16). The multiple-stopping reduction in Theorem 3.5 is proved by showing that every pair reward is dominated by the induced reward ~X evaluated at the minimum of the two stopping times, and conversely that v dominates the smallest E-supermartingale over ~X; this is a derived equality, not a definitional restatement of the value function. The left-continuity result for the induced reward in Theorem 3.16 is also derived from UCE of the biadmissible family rather than assumed, and the induction in Theorem 4.13 uses the (d-1)-stopping result as an induction hypothesis together with the proved CE property of the induced reward. Citations to [1], [5], and [8] are external results, and no load-bearing premise is justified solely by a self-citation of the present author. The reader-flagged inequality E[ess sup |X(tau,theta)-X(tau,theta_n)|] <= sup E[|X(tau,theta)-X(tau,theta_n)|] in Theorem 3.16, if erroneous, is a correctness or soundness gap in proving LCE, not a circular reduction, because LCE is not among the assumptions used to define the value function or the optimal stopping time. No fitted parameters are used, and no prediction is renamed as an input. Therefore the paper merits a circularity score of 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper rests on the F-expectation axioms (H0)-(H7), the admissibility conditions on reward families, and additional uniform continuity and domination assumptions in the regularity and aggregation sections. None of these are fitted parameters; they are structural assumptions imported from prior literature. The main additional burden is the unproven LCE regularity, which fails due to the invalid inequality.

assumptions (5)

domain assumption F-expectation (E, Dom(E)) satisfies hypotheses (H0)-(H4), plus (H5) and, for existence results, (H6)-(H7).
The theory is built on this framework from [1]. Not proved here.
domain assumption Reward families are admissible, biadmissible, or d-admissible (Definitions 2.4, 3.1, 4.1).
The object of study is restricted to families satisfying positivity, measurability, and consistency along stopping times.
domain assumption For regularity and existence, the reward family is UCE (uniformly continuous along stopping times in E-expectation, Definitions 3.10, 4.10).
This stronger condition is assumed in Theorems 3.14, 3.16, 3.17, 4.11-4.13.
domain assumption For some RCE results, there exists an F-expectation ( ~E, Dom( ~E)) that dominates (E, Dom(E)) (Definition 3.11).
Used in Theorem 3.14 and Proposition 4.11. Not available for all F-expectations.
standard math Standard filtered probability space with usual conditions and finite horizon T.
Standard setting for stochastic processes.

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Cite this review

Pith. "Pith review of Optimal Multiple Stopping Problem under Nonlinear Expectation." pith.science (2026). https://pith.science/paper/AV7C7AKW

@misc{pith2026190807174,
  author       = {Pith},
  title        = {Pith review of: Optimal Multiple Stopping Problem under Nonlinear Expectation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AV7C7AKW}},
  note         = {Machine review of arXiv:1908.07174}
}

read the original abstract

In this paper, we study the optimal multiple stopping problem under the filtration consistent nonlinear expectations. The reward is given by a set of random variables satisfying some appropriate assumptions rather than an RCLL process. We first construct the optimal stopping time for the single stopping problem, which is no longer given by the first hitting time of processes. We then prove by induction that the value function of the multiple stopping problem can be interpreted as the one for the single stopping problem associated with a new reward family, which allows us to construct the optimal multiple stopping times. If the reward family satisfies some strong regularity conditions, we show that the reward family and the value functions can be aggregated by some progressive processes. Hence, the optimal stopping times can be represented as hitting times.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

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Bayraktar, E. and Yao, S. (2011) Optimal stopping for nonlinear expectations-part I. Stochastic Processes and their Applications, 121, 185-211

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Bayraktar, E. and Yao, S. (2011) Optimal stopping for nonlinear expectations-part II. Stochastic Processes and their Applications, 121, 212-264

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Bender, C. and Schoenmakers, J. (2006) An iterative method for multiple stopping: convergence and stability. Adv. Appl. Probab. 38, 729-749

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Carmona, R. and Touzi, N. (2008) Optimal multiple stopping and valuation of swing options. Math. Finan., 18, 239-268

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Cheng, X. and Riedel, F. (2013) Optimal stopping under ambiguity in continuous time. Math. Finan. Econ., 7, 29-68

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Coquet, F., Hu, Y., M\' e min, J. and Peng, S. (2002) Filtration-consistent nonlinear expectations and related g -expectations. Probab. Theory Relat. Fields, 123, 1-27

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Kobylanski, M., Quenez, M.C. and Rouy-Mironescu, E. (2011) Optimal multiple stopping time problem. Ann. Appl. Probab., 21(4), 1364-1399

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

and Yao, S

Bayraktar, E. and Yao, S. (2011) Optimal stopping for nonlinear expectations-part I. Stochastic Processes and their Applications, 121, 185-211

work page 2011

[2] [2]

and Yao, S

Bayraktar, E. and Yao, S. (2011) Optimal stopping for nonlinear expectations-part II. Stochastic Processes and their Applications, 121, 212-264

work page 2011

[3] [3]

and Schoenmakers, J

Bender, C. and Schoenmakers, J. (2006) An iterative method for multiple stopping: convergence and stability. Adv. Appl. Probab. 38, 729-749

work page 2006

[4] [4]

and Touzi, N

Carmona, R. and Touzi, N. (2008) Optimal multiple stopping and valuation of swing options. Math. Finan., 18, 239-268

work page 2008

[5] [5]

and Riedel, F

Cheng, X. and Riedel, F. (2013) Optimal stopping under ambiguity in continuous time. Math. Finan. Econ., 7, 29-68

work page 2013

[6] [6]

and Peng, S

Coquet, F., Hu, Y., M\' e min, J. and Peng, S. (2002) Filtration-consistent nonlinear expectations and related g -expectations. Probab. Theory Relat. Fields, 123, 1-27

work page 2002

[7] [7]

and Zhang, J

Ekren, I., Touzi, N. and Zhang, J. (2014) Optimal stopping under nonlinear expectation. Stochastic Processes and their Applications, 124, 3277-3311

work page 2014

[8] [8]

(1981) Les aspects probabilistes du controle stochastique

El Karoui, N. (1981) Les aspects probabilistes du controle stochastique. Ecole d'Et\' e de Probabilit\' e s de Saint-Flour IX. Lecture Notes in Math 876, 73-238. Springer Verlag, Berlin

work page 1981