{"id":"23789abe-b20c-413f-a427-1e2063bb2246","arxiv_id":"2607.18327","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The compensated split-step theta scheme for stochastic pantograph models with Poisson random measure is proven convergent, but its almost sure exponential stability theorem rests on an invalid inequality.","lead":"A numerical method for stochastic equations with jumps and a shrinking time delay is shown to converge, but the part that promises stable long-time behavior has a broken proof step. The convergence result may stand; the almost sure stability claim in the abstract should not be used as stated.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Stability proof rests on inequality (98), which is false: the preimage of [ηj]=l is not {l≤j<l+1}, and exponential weights make the claimed delayed-sum bound fail for η<1.","rationale":"I re-derived the preimage map and the weight comparison. The reader's weakest_assumption correctly identifies the fatal place: the paper's proof of Theorem 5.2 requires (98), and Theorem 5.3 says it follows by the same approach. The multiplicity bound (1+[1/η]) is not the issue; the exponential weights are. For a fixed l, the preimage j values are roughly l/η,...,(l+1)/η, so the weight ratio between the largest and smallest such j grows exponentially in l. Thus no constant can bound the delayed sum by the undelayed sum with the same weights, regardless of how Y* behaves. The explicit sequence above makes the failure transparent. I did not find any alternative argument in Section 5 that bypasses (98); the numerical examples are illustrative and cannot establish an inequality that is false. The convergence part of the paper appears substantially more plausible, so the rejection is best justified on the stability claim. Since this is exactly the reader's position, no verdict change is needed.","tokens_in":21351,"tokens_out":14698,"duration_ms":124883,"concrete_test":"Verify (98) directly. Fix η=1/2, ζ>0, Δt>0, and ε=ζΔt. For any large integer l, set n=2l+2, Y*_l=1, Y*_{2l+1}=1, and all other Y*_j=0. Then LHS(98)=e^{ζ(2l+2)Δt}, RHS(98)=(1+[1/η])e^{ζ(l+1)Δt}=3e^{ζ(l+1)Δt}. The inequality therefore reduces to e^{ζ(l+1)Δt} ≤ 3, which fails for l > (log 3)/(ζΔt)−1. Recomputing the passage from (98) to (99) with this sequence shows the coefficient of e^{ζ(j+1)Δt}|Y*_j|² is not controlled, so the Lyapunov argument and the discrete semi-martingale conclusion in (104)/(123) do not follow. This is a single finite computation that settles the validity of the key inequality.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorems 5.2 and 5.3 are not established because both depend on inequality (98). The paper states that [ηj]=l iff l≤j<l+1, and concludes that the delayed weighted sum is bounded by (1+[1/η]) times the undelayed sum with the same exponential weights (minus a tail). The preimage condition is wrong: [ηj]=l iff l/η ≤ j < (l+1)/η. For η<1, j can be much larger than l. Since e^{ζ(j+1)Δt} is increasing in j, the delayed term e^{ζ(j+1)Δt}|Y*_[ηj]|² can exceed the corresponding undelayed term with index [ηj] by a factor that grows like e^{ζ(1/η−1)lΔt}, which is unbounded as n→∞. No constant multiplicity factor can absorb this. Concretely, with η=1/2, ζΔt=ε, take l large, n=2l+2, and Y*_l=Y*_{2l+1}=1, all other Y*=0. The left side of (98) is e^{ζ(2l+2)Δt}, while the right side is 3e^{ζ(l+1)Δt}; for l large the inequality is e^{ζ(l+1)Δt} ≤ 3, which is false. Theorem 5.3 imports the same bound by 'proceeding by the same approach', so both almost-sure stability theorems are unsupported. Numerical examples cannot repair a false algebraic inequality; the convergence claim may still be salvageable, but the paper's stability contribution is not proved.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the stochastic pantograph equation with Poisson random measure, proposes a compensated split-step theta scheme, proves p-th moment convergence on finite intervals under local Lipschitz assumptions, and claims almost sure exponential stability of the numerical solution for theta in [0,1/2] and in (1/2,1] under Lyapunov-type conditions. The convergence part is a standard localization-plus-Gronwall argument. The stability proofs use a weighted Lyapunov inequality together with the discrete semimartingale convergence theorem. Numerical examples are provided for convergence rates and for almost sure exponential stability.","tokens_in":21688,"tokens_out":12755,"duration_ms":113763,"significance":"If the stability results were valid, the paper would extend split-step theta methods to pantograph jump models in a useful way. The convergence part is a technically involved but largely routine adaptation of known techniques and appears essentially sound, apart from an unstated moment assumption. The main new contribution, however, is the almost sure exponential stability of the numerical scheme, and that contribution is not established: the key delayed-sum estimate (98) is false, and both Theorem 5.2 and Theorem 5.3 depend on it. I therefore cannot recommend acceptance.","major_comments":[{"comment":"The assertion before Eq. (98) that [ηj]=l ⇔ l≤j<l+1 is false. The correct preimage is l/η ≤ j < (l+1)/η. For η<1, j may be much larger than l, so the exponential weight e^{ζ(j+1)Δt} in the delayed sum can be larger than e^{ζ(l+1)Δt} by a factor growing with j−l. Concretely, take η=1/2, n=2l+2, and set Y*_l=Y*_{2l+1}=1, all other Y*=0. The left side of (98) is e^{ζ(2l+2)Δt}, while the right side is 3 e^{ζ(l+1)Δt}; for ζΔt>0 and l large the inequality fails. Since (98) is the key step converting the delayed sum into the undelayed sum in (99), Theorem 5.2 is not established.","section":"Section 5, before Eq. (98)"},{"comment":"The proof of Theorem 5.3 states that inequality (114) is obtained from (113) 'by proceeding by the same approach' used to get (99) from (97). That approach relies on the same incorrect delayed-sum bound (98), which is false for η<1. The coefficients in (114), in particular the term −2ξ2(1+[1/η])Δt, depend on that bound. Consequently Theorem 5.3 is also unsupported.","section":"Section 5, Theorem 5.3"},{"comment":"The theorem is stated under Assumption 2.1 only, but the proof uses Assumption 3.1 to bound E sup |x(t)|^ϱ and E sup |Y(t)|^ϱ in (62). Assumption 2.1, a local Lipschitz condition, does not by itself provide these moment bounds at the required order ϱ>p. Either state Assumption 3.1 (with uniform-in-p moment bounds) as a hypothesis of Theorem 4.1 or prove the needed bounds under the stated assumptions.","section":"Section 4, Theorem 4.1"}],"minor_comments":[{"comment":"The first term on the right-hand side should involve |Y_n^*|^2, not |Y_n|^2; this is what Assumption 5.1 gives and what is needed for the coefficient A(ζ,Δt) in (95). As printed, Eq. (91) is internally inconsistent with the subsequent algebra.","section":"Eq. (91)"},{"comment":"The expression '1−[1/η]' in the numerator should read '1+[1/η]' to be consistent with (101) and with the hypothesis ξ1>ξ2(1+[1/η]).","section":"Eq. (100)"},{"comment":"The sentence 'We have added that in the revised version' is a leftover from the submission process and should be removed.","section":"Section 2"},{"comment":"The proof is deferred to [17]. This dependence should be stated explicitly; as written, the theorem appears to be asserted without proof in the manuscript.","section":"Theorem 5.1"},{"comment":"The notation |\\tilde N(ds,du)|^2 is informal; clarify that this refers to the quadratic variation of the compensated Poisson measure.","section":"Eqs. (90) and (110)"}],"recommendation":"reject","confidential_remarks":"The convergence part may be salvageable, but the stability contribution is the paper's central advertised novelty and it rests on a demonstrably false inequality. I would not recommend a major-revision path within this submission. If the authors can prove the delayed-sum control under genuinely stronger hypotheses, a new submission could be considered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline: the convergence theorem is fine; the stability theorems are not supported because the key inequality (98) is false. Reject in current form, but the convergence half is solid enough to merit a referee.\n\nWhat's new: the combination of the compensated split-step theta scheme with a pantograph delay and Poisson random measure is not in the earlier literature (Yang–Jiang used stochastic theta for this model; Mo et al. did split-step theta for neutral SDDEs with jumps). The analytic machinery is otherwise standard: stopping times, BDG inequality, Kunita's inequality, discrete semimartingale convergence theorem.\n\nWhat's good: Theorem 4.1 and its supporting lemmas form a standard p-th moment convergence proof under local Lipschitz conditions. I checked the steps and they hold up; there's a missing θ in Lemma 4.2's bound and a typo in Eq. (91), but both are harmless. The numerical experiments are honest: they use a fine-step reference solution and report mean-square errors.\n\nThe problem: Section 5. Inequality (98) is load-bearing. The paper claims that for any l, [ηj]=l iff l≤j<l+1. The correct condition is l/η ≤ j < (l+1)/η. The stated condition is wrong. Worse, even if you correct the preimage count, the exponential weights make the inequality fail. For η<1, j can be much larger than l, so e^{ζ(j+1)Δt} is exponentially larger than e^{ζ(l+1)Δt}. Take η=1/2, ζΔt=ε, set Y*_l=1 and all other Y*=0, and let n be large enough so that j=2l+1 is included. Then the left side of (98) contains e^{ζ(2l+2)Δt}, while the right side is at most 3e^{ζ(l+1)Δt}. For large l the inequality fails. Since both Theorem 5.2 and Theorem 5.3 import this bound, the a.s. exponential stability claims are not proven. Numerical simulations can't fix a false algebraic inequality.\n\nVerdict: reject in current form. The authors should either produce a correct argument for (98) or state the stability results as conjectures. The convergence part could be resubmitted as a standalone result, though the novelty is modest.\n\nSerious referee? Yes, I'd send this out rather than desk-reject, because the convergence part is correct and the stability error is subtle enough that a referee might catch something I missed.","headline":"Convergence is solid; the almost-sure stability theorems rest on a false inequality and are not established.","tokens_in":22218,"tokens_out":4289,"would_cite":false,"duration_ms":36391,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H10","60H35","65C30","65L20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that the compensated split-step theta method converges in the p-th moment to the exact solution of stochastic pantograph models with Poisson random measure, and that its numerical paths are almost surely exponentially stab","keywords":["stochastic pantograph equations","Poisson random measure","compensated split-step theta method","strong convergence","p-th moment convergence","almost sure exponential stability","jump-diffusion models"],"falsifier":"Take η=0.2, Y*_j=1 for all j, and ζΔt=1; the left side of inequality (98) then contains terms e^{j+1} with j ranging up to n-1, while the right side is only about 6 times the partial sum up to [η(n-1)] ≈ 0.2n. For n large, the right side is smaller than the left by a factor of order e^{0.8n}, so a direct numerical evaluation of (98) for such n settles whether the key stability estimate holds.","tokens_in":21184,"feed_emoji":"📉","tokens_out":8472,"duration_ms":84625,"temperature":0.7,"pith_summary":"The paper aims to establish that a single numerical scheme, the compensated split-step theta method, can handle stochastic pantograph models with Poisson jumps both accurately and stably. On finite time intervals, the claim is p-th moment convergence: as the step size goes to zero, the numerical paths approach the exact solution in a strong averaged sense, even though the model combines an unbounded memory delay, Brownian noise, and jump discontinuities. On infinite horizons, the claim is almost sure exponential stability: the numerical solution eventually decays at an exponential rate at least half that of the exact solution, under explicit inequalities linking the drift coefficients to the delay ratio η. The practical value, if the claims hold, is that one adjustable scheme covers the full range from explicit to implicit discretizations while preserving the qualitative long-time behavior of the underlying jump-diffusion model.","feed_headline":"Split-step theta method converges and is almost surely stable","feed_subtitle":"One theta scheme gives finite-time accuracy and long-time decay for jump pantograph models","key_machinery":"The central object is the compensated split-step theta scheme (equations (6)–(7)), a one-parameter numerical method that performs an implicit correction of the drift and then an explicit update with Brownian and compensated Poisson increments. The delayed argument x(ηt) is handled by piecewise-constant interpolation to the nearest grid point from the left, producing the index [ηn]. The finite-time argument uses stopping times and Gronwall-type estimates over drift, diffusion, and jump terms. The stability argument is carried by a Lyapunov-type weighted-sum inequality (98) — which bounds the exponentially weighted delayed sum by a multiple of the undelayed weighted sum — together with the dis","core_discovery":"On the paper's own terms, the central discovery is that the compensated split-step theta method—a one-parameter family that includes compensated Euler–Maruyama at θ=0 and compensated split-step backward Euler at θ=1—preserves both strong convergence and long-time almost-sure exponential decay for the stochastic pantograph model with Poisson jumps. Theorem 4.1 states that lim_{Δt→0} E[sup_{0≤t≤T}|x(t)-Y(t)|^p] = 0 under local Lipschitz conditions. Theorems 5.2 and 5.3 state that limsup_{n→∞} log|Y_n|/(nΔt) ≤ -γ2/2 almost surely, where the decay rate γ2 is determined by drift-dominance conditions involving ξ1, ξ2, and the delay ratio η.","pith_inferences":["Because the stability condition involves the delay ratio η through the factor (1+[1/η]), the result suggests a general pattern: for proportional-delay equations, larger delay ratios demand stronger drift domination; quantifying the exact dependence for other delay mappings is a natural next step.","The reported numerical rate γ2/2 is half the exact solution's guaranteed rate γ; a natural extension is to test numerically whether this factor 1/2 is sharp or an artifact of the proof technique.","The same weighted Lyapunov argument should extend to stochastic pantograph models with Markov switching or multiple delays, provided the delay map has bounded preimage multiplicity and monotone exponential weights."],"forward_implications":["For finite-horizon simulation, shrinking Δt drives the p-th moment error to zero, so the scheme can serve as a reliable path simulator for pantograph models with jumps under local Lipschitz coefficients.","For long-time simulation, the numerical paths inherit almost sure exponential decay, meaning trajectories do not blow up and the scheme can be used to study asymptotic behavior.","Setting θ=0 or θ=1 recovers the compensated Euler–Maruyama and compensated split-step backward Euler methods, so the convergence and stability results apply to both common variants.","The sufficient stability condition is expressed directly in terms of the drift, diffusion, and jump coefficients (ξ1, ξ2, η), giving users an explicit check before running the scheme."],"fun_headline_variants":["Theta scheme for jump pantographs: convergence and a.s. stability","Compensated split-step theta: converges and decays a.s. for jump pantographs","Jump pantograph model: split-step theta is convergent and stable a.s.","Convergence and a.s. stability for jump pantographs via theta scheme","Split-step theta converges and decays a.s. for pantographs with Poisson jumps"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The stability theorems rest on inequality (98), which asserts that a weighted sum over delayed indices is controlled by a multiple of the same weighted sum without delay; that absorption requires both a bounded number of preimages under j↦[ηj] and no larger exponential weight on the delayed terms, and if either fails for η<1, the Lyapunov argument and Theorems 5.2–5.3 collapse.","fun_headline_variants_meta":{"raw":{"variants":["Theta scheme for jump pantographs: convergence and a.s. stability","Compensated split-step theta: converges and decays a.s. for jump pantographs","Jump pantograph model: split-step theta is convergent and stable a.s.","Convergence and a.s. stability for jump pantographs via theta scheme","Split-step theta converges and decays a.s. for pantographs with Poisson jumps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000927,"raw_usage":{"total_tokens":3769,"prompt_tokens":668,"completion_tokens":3101,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":412,"completion_tokens_details":{"reasoning_tokens":3001}},"tokens_in":412,"tokens_out":3101,"duration_ms":18551,"temperature":1.0,"reasoning_tokens":3001,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T19:43:05.025042+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take η=0.2, Y*_j=1 for all j, and ζΔt=1; the left side of inequality (98) then contains terms e^{j+1} with j ranging up to n-1, while the right side is only about 6 times the partial sum up to [η(n-1)] ≈ 0.2n. For n large, the right side is smaller than the left by a factor of order e^{0.8n}, so a direct numerical evaluation of (98) for such n settles whether the key stability estimate holds.","supporting_citations":[],"review_version":1}