{"id":"0fd575fe-1f50-4831-9a66-fc8026fd388b","arxiv_id":"2606.23648","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors reduce an optimal stopping problem with unobserved Bernoulli drift to a two-dimensional Markovian problem, characterize the continuation region as glued intervals whose boundaries satisfy monotonicity and ODEs, and obtain a threshold solution for the original one-dimensional problem unde","lead":"The paper solves an optimal stopping problem for a Wiener process with unobserved Bernoulli drift by filtering to a Markov process and then lifting the problem via an extra state variable y to characterize the stopping region in the plane before slicing back to the original line. A smart generalist might read it to see how partial-observation control problems can be reduced to explicit free-boundary problems in higher dimension.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the dependency on the structural assumptions that enable the interval property and semi-explicit form. Since the abstract presents the result as holding under those assumptions and the logical steps appear consistent, no additional load-bearing concern is identified beyond what the reader already flagged.","tokens_in":1878,"tokens_out":285,"duration_ms":20244,"concrete_test":"State the precise structural assumptions on the terminal cost (as used in the fixed-x analysis) and confirm they hold for a symmetric cost increasing in |z|; then numerically discretize the fixed-x optimal stopping problem for two representative x values and check whether the continuation set is a single bounded interval whose endpoints satisfy the claimed balancing condition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a conditional technical characterization of the continuation region and value function that follows from applying a foliation procedure under structural assumptions on the terminal cost. The abstract describes an internally consistent sequence: filtering to a Markov problem, lifting via the auxiliary parameter y, solving fixed-x sections as intervals via balancing conditions, gluing to obtain the 2D region with monotonic boundaries satisfying a coupled ODE system at regular points, and recovering a threshold solution on the y=0 slice. No internal inconsistency or gap in the logical flow is apparent from the given description.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript solves an optimal stopping problem for a Wiener process with unobserved Bernoulli drift, subject to a symmetric terminal cost that increases with distance from the origin and a positive running cost c. After filtering, the problem is Markovian in the centered state x. The authors introduce an auxiliary parameter y representing displacement from initial position to foliate the problem, solve the augmented problem in the (x,y)-plane by characterizing fixed-x continuation sections as bounded intervals via balancing conditions, glue them to obtain the 2D continuation region whose free boundaries satisfy monotonicity and a coupled ODE system at regular points, and recover a threshold-type solution for the original problem on the y=0 slice under the structural assumptions on the terminal cost.","tokens_in":2000,"tokens_out":442,"duration_ms":18676,"significance":"If the derivations hold, the paper offers a semi-explicit characterization of the value function and continuation region for this filtered optimal stopping problem, extending techniques like foliation and balancing conditions to this setting. This could be valuable for problems with partial observations in stochastic control, providing concrete structural results on the form of the solution. The direct derivation from filtered dynamics without fitted parameters is a strength.","major_comments":[],"minor_comments":[{"comment":"Abstract: the structural assumptions on the terminal cost are invoked to guarantee that fixed-x continuation sections are intervals with uniquely determined endpoints, but their precise form is not stated even at a high level; adding one sentence summarizing the key properties (e.g., convexity or growth conditions) would help readers evaluate the scope.","section":"Abstract"},{"comment":"The phrase 'semi-explicit form' for the value function is used repeatedly; specifying whether this means an integral representation, an explicit formula in terms of the boundaries, or a numerical ODE solution would clarify the degree of explicitness achieved.","section":null},{"comment":"The description of the coupled ODE system for the free boundaries at regular points would benefit from a brief indication of the variables involved (e.g., which derivatives appear) to aid readability before the full derivation.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful summary of our manuscript and the positive assessment of its contributions. The referee's description accurately reflects the filtering approach, the foliation by the auxiliary parameter y, the characterization of the continuation region via balancing conditions and gluing, and the recovery of the threshold solution on the y=0 slice. The recommendation for minor revision is noted; however, the major comments section contains no specific points.","responses":[],"tokens_in":1362,"tokens_out":103,"duration_ms":16246,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that this paper gives a semi-explicit description of the continuation region for optimal stopping of a Wiener process with hidden Bernoulli drift, under a symmetric terminal cost that grows with distance from zero plus a constant running cost. After filtering they introduce the extra state variable y for displacement from the initial position x. This lifts the problem to the plane. For each fixed x they solve a one-dimensional problem whose continuation set is either empty or a single bounded interval, with the endpoints fixed by a balancing condition. Gluing these intervals across x produces the two-dimensional continuation region; its free boundaries are monotone and obey a coupled ODE system at regular points. When the y=0 slice sits inside that region the original problem reduces to a threshold rule.\n\nThe foliation step and the reduction to explicit intervals per fixed x are the genuinely new pieces. They turn what would otherwise be a hard two-dimensional free-boundary problem into a collection of one-dimensional ones that can be characterized directly from the filtered dynamics. The construction is direct, avoids self-referential definitions, and produces a concrete geometric picture of the solution set.\n\nThe obvious limitation is that everything rests on structural assumptions on the terminal cost that force each fixed-x slice to be a single interval with unique balancing points. The abstract does not quantify how restrictive those assumptions are, so the method applies only inside that class; outside it the interval property and the semi-explicit form collapse. Full verification that the constructed function satisfies the variational inequality and that the ODE system is correctly derived will have to be checked in the proofs, which are not visible here.\n\nThis is written for people who work on filtering plus optimal stopping. A reader who needs concrete tools for similar partially observed problems will find the foliation device useful. It is worth sending to a serious referee because the technical device is original, the claims are specific, and the logical flow from filtering to foliation to glued boundaries is internally consistent.","headline":"The paper uses a foliation by an auxiliary displacement y to reduce the filtered optimal stopping problem to a family of one-dimensional interval problems whose boundaries are set by balancing conditions, then glues them into a two-dimensional region with coupled ODE boundaries.","tokens_in":2505,"tokens_out":485,"would_cite":false,"duration_ms":23381,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Optimal stopping for a Wiener process with hidden Bernoulli drift is solved by lifting the problem to two dimensions via foliation, then slicing back.","keywords":["optimal stopping","Wiener process","unobserved Bernoulli drift","filtering","continuation region","free boundary","foliation"],"falsifier":"A concrete terminal cost obeying the stated structural assumptions for which the optimal continuation set at some fixed x consists of two disjoint intervals rather than one or none.","tokens_in":2771,"feed_emoji":"","tokens_out":748,"duration_ms":11816,"temperature":0.7,"pith_summary":"The paper reduces the original one-dimensional optimal stopping problem, after filtering the unobserved drift, to a Markovian problem whose solution requires an auxiliary parameter y that tracks displacement from the starting point. By solving the lifted two-dimensional problem for each fixed starting position x and then restricting to the slice y=0, the authors obtain an explicit description of the continuation region under structural assumptions on the terminal cost. The value function takes a semi-explicit form, the two-dimensional free boundaries obey monotonicity and a coupled ODE system, and the original problem admits a threshold solution whenever the y=0 slice intersects the continuation region. A sympathetic reader cares because the reduction turns an intractable filtering-plus-stopping problem into a concrete geometric construction whose boundaries can be computed or approximated directly.","feed_headline":"Foliation by displacement solves hidden-drift optimal stopping","feed_subtitle":"Lifting the filtered problem to the plane yields semi-explicit intervals and ODEs for the boundaries before slicing back to the original lin","key_machinery":"Foliation by the auxiliary displacement parameter y, which lifts the filtered one-dimensional problem to the plane so that fixed-x sections become intervals whose endpoints are fixed by a balancing condition before gluing produces the two-dimensional continuation region.","core_discovery":"Under suitable structural assumptions on the terminal cost, each fixed-x continuation section is either empty or a single bounded interval whose endpoints are determined uniquely by a balancing condition; the value function is given in semi-explicit form, the two-dimensional continuation region is obtained by gluing, its free boundaries satisfy natural monotonicity and at regular points a coupled system of ODEs, and the original problem admits a threshold-type solution whenever the horizontal slice y=0 enters the two-dimensional continuation region.","pith_inferences":["The same foliation technique may apply to other optimal stopping problems with partial observations whose filtered state lives on the line but whose value depends on an auxiliary displacement variable.","Numerical solution of the coupled ODE system for the free boundaries would yield computable approximations to the optimal stopping set for concrete terminal costs.","If the structural assumptions on the terminal cost are dropped, the continuation sections may fragment into multiple intervals and the semi-explicit characterization would no longer hold."],"forward_implications":["The two free boundaries of the two-dimensional continuation region are monotone and, at regular points, satisfy a coupled system of ordinary differential equations.","Whenever the slice y=0 lies inside the two-dimensional continuation region, the original one-dimensional problem has a threshold-type solution.","The value function of the lifted problem is given in semi-explicit form once the interval endpoints are known.","The original problem is recovered by restricting the two-dimensional solution to the plane y=0."],"fun_headline_variants":["Foliation by y solves hidden-drift optimal stopping","Plane lifting gives bounded intervals for drift stopping","Displacement parameter yields semi-explicit stopping regions","Glued continuation intervals solve unobserved Bernoulli drift"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The terminal cost must satisfy structural assumptions that force each fixed-x continuation section to be either empty or a single bounded interval with uniquely determined endpoints.","fun_headline_variants_meta":{"raw":{"variants":["Foliation by y solves hidden-drift optimal stopping","Plane lifting gives bounded intervals for drift stopping","Displacement parameter yields semi-explicit stopping regions","Glued continuation intervals solve unobserved Bernoulli drift"]},"model":"grok-4.3","cost_usd":0.003802,"raw_usage":{"total_tokens":2010,"prompt_tokens":763,"num_sources_used":0,"completion_tokens":56,"cost_in_usd_ticks":38024500,"prompt_tokens_details":{"text_tokens":763,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1191,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":763,"tokens_out":56,"duration_ms":7324,"temperature":1.0,"reasoning_tokens":1191,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T06:49:59.013196+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete terminal cost obeying the stated structural assumptions for which the optimal continuation set at some fixed x consists of two disjoint intervals rather than one or none.","supporting_citations":[],"review_version":1}