{"id":"7b4a6444-4bdc-401d-9cc4-97ec5945a1eb","arxiv_id":"2502.00788","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A positivity-preserving projection Euler scheme for α-stable driven linear volatility models converges with strong order 1/α in L^q for q < α.","lead":"This paper introduces an explicit Euler-Maruyama scheme with a positivity-preserving floor for a linear stochastic volatility model driven by α-stable noise, and proves a strong convergence rate of 1/α. The scheme gives a simple way to keep simulated volatility positive even when the driving process has heavy-tailed jumps.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.3, which controls the projection error in Theorem 4.4, is unproved: its tail-asymptotic step is invalid under the truncated Assumption 2.2 and no uniform-in-k rate is given, so the claimed 1/α order rests on a gap.","rationale":"The paper's central claim is the strong order 1/α for the positivity-preserving explicit EM scheme. The proof chain is Theorem 4.4 plus Lemma 4.3 controlling the floor. If Lemma 4.3 fails, the projection error term in (4.6) is uncontrolled, so the theorem is not established. The proof of Lemma 4.3 is genuinely incorrect as written: it invokes a power-law tail for the full symmetric stable process after Assumption 2.2 has removed exactly those negative jumps, and the convergence of the threshold to −1/κ is not uniform in X_k. This is not a mere cosmetic issue; the uniform control of P(X_k < Δ) is the only mechanism in the paper for bounding the difference between the raw update X_Δ and the floored value \\tilde X_Δ. The concern is repairable: a correct argument should use the truncated Lévy measure to show that a crossing below Δ requires a jump in a window of width O(Δ) near −1/κ, giving per-step probability O(Δ^2) and hence a union bound O(Δ). The paper does not supply this argument. I do not therefore move the verdict to REJECT: the claimed rate is consistent with the known EM theory for stable-driven SDEs and the scheme is plausible. The reader's verdict of CONDITIONAL remains appropriate, with the condition being a rigorous repair of Lemma 4.3. The reader's weakest-assumption pick, the truncation mismatch with simulations, is related but distinct; it concerns empirical support rather than the proof of the main theorem. Hence agreement is partial rather than full.","tokens_in":16621,"tokens_out":14428,"duration_ms":142620,"concrete_test":"Re-derive Lemma 4.3 under Assumption 2.2 by bounding the probability that a truncated stable increment falls in (−1/κ, −1/κ + cΔ) for the relevant c, then take a union bound over k ≤ T/Δ to obtain sup_k P(X_k < Δ) ≤ CΔ. If this bound fails, or if a simulation of the scheme with exactly truncated increments shows sup_k P(X_k < Δ) not tending to 0 as Δ → 0, then the projection term in (4.6) is not controlled and Theorem 4.5 does not follow. A direct numerical check is to estimate sup_k P(X_k < Δ) empirically for Δ = 2^−10, …, 2^−16 using the truncated jump generator and confirm the probability decays at least linearly in Δ.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.5 is obtained from Theorem 4.4 plus Lemmas 4.2 and 4.3. The only place the positivity projection is controlled is Lemma 4.3, used in (4.6) to bound E|X_Δ(η(s)) − \\tilde X_Δ(η(s))|^q by δ^q + 2^{q−1}Cε. Lemma 4.3's proof is not valid: for the case X_k > Δ it writes P(ΔL_k < ((1−μ)Δ − (1−λΔ)X_k)/(κX_k)) and uses the full-stable tail asymptotic P(ΔL_k < x) ∼ |x|^{−α}. Under Assumption 2.2 the Lévy measure has no mass below −1/κ, so this asymptotic is inapplicable near the threshold. The threshold is −1/κ + O(Δ) + (1−μ)Δ/(κX_k), and the last term is O(1) when X_k is of order Δ, so the claimed convergence of the threshold to −1/κ is not uniform. The proof also gives no rate and no union bound over the N = ⌊T/Δ⌋ steps, while Theorem 4.4 needs a uniform-in-s smallness of P(|X_Δ − \\tilde X_Δ| ≥ δ). Without a quantitative bound such as sup_k P(X_k < Δ) ≤ CΔ, or at least a uniform o(1) as Δ → 0, the ε–δ argument in Theorem 4.4 cannot be completed, and the 1/α strong order for the projected scheme is unproved as written. The reader's other concerns—the sign issue in J22 and the untruncated simulations—are real but secondary; this is the load-bearing gap for the central convergence claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies a scalar linear stochastic volatility model dx(t)=(µ-λx(t))dt+κx(t-)dL_α(t) driven by an α-stable process with α∈(1,2), under an assumption that negative jump sizes are bounded below by -1/κ. The authors prove existence and uniqueness of a global positive solution (Theorem 3.1), introduce an explicit Euler-Maruyama scheme with a floor projection X_{k+1}↦X_{k+1}∨Δ, establish uniform moment bounds, and claim strong convergence of order 1/α in L^q for q∈[1,α) (Theorem 4.5). Numerical experiments with several parameter sets are reported as supporting the predicted rate.","tokens_in":16955,"tokens_out":14575,"duration_ms":120583,"significance":"If the main theorem were fully proved, the paper would make a useful contribution: an explicit, positivity-preserving EM scheme for a linear α-stable-driven SDE with the optimal strong order 1/α, in line with known rates for EM for stable noise. The argument is constructive, contains no fitted parameters, and the claimed order is empirically falsifiable; the simulations are consistent with the slope 1/α. However, the proof of the key projection lemma is currently invalid, and the model is not consistently defined (truncated Lévy measure vs full stable measure used in lemmas and simulations). These issues are local in nature but must be repaired before the result can be accepted.","major_comments":[{"comment":"The paper never defines the truncated Lévy measure that Assumption 2.2 requires. Eq. (2.2) states ν(dz)=C_α|z|^{-α-1}dz on R\\{0}, the full symmetric stable measure, and Lemma 2.4 and Lemma 2.5 are stated for the full α-stable process. If negative jumps are restricted to z>-1/κ, then the driver in (2.1) is no longer the α-stable process used in the lemmas and in the Section 5 simulations; the proofs and the numerics describe different processes. Please state the truncated measure explicitly (e.g., ν(dz)=C_α|z|^{-α-1}1_{z>-1/κ}dz), give the corresponding Itô formula, and adapt all tail estimates and simulation increments accordingly.","section":"§2, Eq. (2.2) / Assumption 2.2"},{"comment":"The proof applies the full-stable tail asymptotic P(ΔL_k-1/κ is implemented. Since the theoretical results rely on the truncation, please state the simulation procedure for the truncated process.","section":"§5"},{"comment":"Some entries are formatted inconsistently (e.g., Table 2, α=1.1 row uses 0.0551×10^{-3}); please unify the notation.","section":"§5, Tables 1-3"},{"comment":"The displayed nonlinear Gronwall bound contains e(t) and e1(t) with different definitions; please verify the application against [27], in particular the denominator 1-[1-e1(t)]^{q/α}.","section":"Lemma 4.1 / Theorem 4.4"},{"comment":"Reference [3] is an unpublished preprint; if a peer-reviewed version exists, cite it.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main issue is the unproved projection lemma. I believe the result is likely true and the proof can be repaired with a quantitative tail estimate under the truncated measure, but the present text does not contain it. The model inconsistency between Assumption 2.2 and the full stable Lévy measure is also substantive and should be addressed head-on. I would not recommend rejection on novelty grounds; the contribution is of appropriate scope for the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The contribution is real: a fully explicit EM scheme with a positivity floor for a linear volatility SDE driven by an α-stable process, and they claim strong L^q order 1/α. That's a natural extension of the implicit schemes by Li-Taguchi and Li-Liu, and I don't know of an explicit version in the literature. The existence and moment-bound sections are standard but mostly correct, and the numerical experiments show nice slopes.\n\nThe soft spots are not cosmetic. First, in Theorem 3.1's J22 estimate, they drop the positive term −0.5 log(1+κz) for negative large jumps and then bound the remaining integral. Dropping a positive term gives a lower bound, not an upper bound, so the displayed inequality is reversed. It's probably fixable, but as written it's wrong.\n\nThe bigger issue is Lemma 4.3 and its use in Theorem 4.4. The proof uses the stable tail asymptotic P(ΔL_k < x) ∼ |x|^{-α} at a threshold that approaches −1/κ, exactly where the truncated Lévy measure has no mass and the power-law asymptotic stops being valid. The argument also needs a uniform-in-k bound on the probability of falling below Δ; the proof gives only a pointwise limit, no rate, and no union bound over the N steps. Without something like sup_k P(XΔ(t_k)<Δ) ≤ CΔ, the ε-δ step in Theorem 4.4 can't be completed. This is the load-bearing gap for the 1/α claim.\n\nThere's also a mismatch between theory and simulation. Assumption 2.2 truncates negative jumps, but the numerical section appears to use standard symmetric α-stable increments. The scheme itself is positive by construction, but the experiment no longer tests the theorem's assumptions. They need to simulate the truncated process.\n\nAll that said, I think the underlying claim is likely true — the rate is what you'd expect for EM with α-stable noise and a small perturbation from the floor. The paper deserves a serious referee, because the idea is useful and the problems are repairable. Send it out; expect major revision but don't desk reject.\n\nWho this is for: people working on numerical methods for jump SDEs in mathematical finance. It's a subfield-level contribution, not a breakthrough.","headline":"A useful explicit positivity-preserving scheme for α-stable-driven SDEs, with a plausible 1/α rate, but the key projection lemma is genuinely under-proved and the numerics don't match the assumptions.","tokens_in":17520,"tokens_out":3794,"would_cite":false,"duration_ms":35879,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G51","60H35","65C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that an explicit, positivity-preserving Euler-Maruyama scheme for linear α-stable stochastic volatility models converges in L^q with strong order 1/α.","keywords":["stochastic volatility models","α-stable process","positivity preserving Euler-Maruyama scheme","strong convergence","Lévy process","positive solution","explicit scheme"],"falsifier":"Run the same scheme on a spectrally negative $\\alpha$-stable process whose Lévy measure includes jumps below $-1/\\kappa$. If the true SDE still has a positive solution and the scheme still converges at order $1/\\alpha$, then Assumption 2.2 is not needed and the theorem's stated hypothesis is too strong; if positivity fails or the error slope on a log-log plot departs from $1/\\alpha$, the assumption is doing real work.","tokens_in":16389,"feed_emoji":"📈","tokens_out":9247,"duration_ms":81179,"temperature":0.7,"pith_summary":"This paper introduces a linear stochastic volatility model driven by an α-stable Lévy process, $dx(t) = (\\mu - \\lambda x(t))dt + \\kappa x(t-)dL_\\alpha(t)$, and proves that under a lower bound on negative jumps the model has a unique positive global solution. It then constructs an explicit Euler-Maruyama scheme that preserves positivity by replacing any value below the step size $\\Delta$ with $\\Delta$ itself. The main result is that this scheme converges in $L^q$, for every $q \\in [1, \\alpha)$, with strong order $1/\\alpha$: $\\sup_{0 \\le t \\le T} E|x(t) - \\tilde{X}_\\Delta(\\eta(t))|^q \\le C\\Delta^{q/\\alpha}$. Because the scheme is explicit and positivity is built in, it offers a practical way to simulate fat-tailed volatility paths without solving implicit equations at each step.","feed_headline":"Floored explicit EM keeps α-stable volatility positive at order 1/α","feed_subtitle":"A simple floor at the step size lets explicit Euler-Maruyama converge at strong order 1/α for heavy-tailed volatility.","key_machinery":"The load-bearing object is the projected explicit Euler-Maruyama scheme (4.1): at each step the raw update $X_{k+1}$ is floored at the step size, $\\tilde{X}_{k+1} = X_{k+1} \\vee \\Delta$, and the floored value is the one used in the drift and diffusion of the next step. This floor keeps the numerical process strictly positive. The convergence argument works through a continuous-time interpolation $\\tilde{x}_\\Delta(t)$ that agrees with $X_\\Delta$ at grid points, and measures three discrepancies: the extrapolation error between $\\tilde{x}_\\Delta$ and the true solution, the within-step error between $\\tilde{x}_\\Delta$ and $X_\\Delta$, and the projection error between $X_\\Delta$ and its floor. The rate $1/\\alpha$ emerges from the maximal inequality for $\\alpha$-stable integrals and the $q$-th moment growth of stable increments, which together convert a stochastic integral with respect to $L_\\alpha$ into a term of size $\\Delta^{q/\\alpha}$.","core_discovery":"The paper's central claim is that positivity and strong convergence of order $1/\\alpha$ can be achieved simultaneously by a fully explicit numerical method for SDEs driven by symmetric $\\alpha$-stable noise with $\\alpha \\in (1,2)$. The proposed method updates $X_{k+1} = X_k + (\\mu - \\lambda \\tilde{X}_k)\\Delta + \\kappa \\tilde{X}_k \\Delta L_k$ and then sets $\\tilde{X}_{k+1} = X_{k+1} \\vee \\Delta$, so the numerical solution is never below the step size. Theorem 4.5 states that for step sizes $\\Delta \\in (0, (\\mu-1)/\\lambda \\wedge 1/\\lambda)$ and $q \\in [1, \\alpha)$, the $L^q$ error between the true solution $x(t)$ and the projected numerical solution $\\tilde{X}_\\Delta(\\eta(t))$ is bounded by $C\\Delta^{q/\\alpha}$. The proof combines Ito's formula with a truncated Lévy measure, a maximal inequality for $\\alpha$-stable stochastic integrals, and a nonlinear Gronwall inequality; the error decomposes into the continuous-time extrapolation error, the jump-induced error, and the projection bias, each of which is shown to be $O(\\Delta^{q/\\alpha})$.","pith_inferences":["If the $1/\\alpha$ rate is the minimax rate for any method driven by $\\alpha$-stable increments, then this explicit scheme is rate-optimal; the paper proves only the upper bound.","The mismatch between the theory, which truncates negative jumps, and the simulations, which use the full symmetric stable measure, suggests the truncation may be a proof device rather than a computational requirement; testing on a process with genuinely unbounded negative jumps would settle this.","The same floor construction should extend to super-linear mean-reverting drifts, a future direction the paper names, but the Ito estimates would need modification because the drift is no longer linear.","For option pricing, positive heavy-tailed volatility paths can now be simulated explicitly, opening Monte Carlo pricing for models such as $\\alpha$-CIR without implicit solves."],"forward_implications":["By construction, $\\tilde{X}_\\Delta(\\eta(t)) \\ge \\Delta > 0$, so the simulated volatility path never crosses zero.","The $L^q$ error bound $\\sup_{0\\le t\\le T} E|x(t)-\\tilde{X}_\\Delta(\\eta(t))|^q \\le C\\Delta^{q/\\alpha}$ holds uniformly over the whole interval $[0,T]$, not just at grid points.","The explicit update requires only one stable increment per step, so the method is as cheap as the classical Euler-Maruyama scheme while preserving positivity.","The same rate $1/\\alpha$ is obtained for every $q \\in [1,\\alpha)$, with a constant independent of the step size."],"supporting_citations":[{"why":"Defines stable distributions and supplies the q-th moment bound $E|L_\\alpha(t)|^q \\le Ct^{q/\\alpha}$ used in Lemma 2.4.","marker":"[22]"},{"why":"Provides the Ito formula for Lévy-type SDEs used to prove positivity and moment estimates.","marker":"[1]"},{"why":"Supplies the maximal inequality for $\\alpha$-stable stochastic integrals used to bound the jump term in the convergence proof.","marker":"[28]"},{"why":"Gives the nonlinear Gronwall inequality used to close the error bounds in Lemmas 4.1 and Theorem 4.4.","marker":"[27]"},{"why":"Inspires the positivity-preserving projection device and provides an implicit positivity-preserving scheme for the jump-extended CIR process.","marker":"[15]"},{"why":"Establishes strong convergence orders for Euler approximations of multiplicative stable-noise SDEs, the baseline this paper extends to an explicit positivity-preserving construction.","marker":"[17]"}],"fun_headline_variants":["Floored explicit EM: positivity and order 1/α for stable volatility","Explicit EM with floor: strong order 1/α for α-stable volatility","No negativity: explicit Euler projects onto Δ, still order 1/α","Floor at Δ: explicit method for α-stable SDEs hits order 1/α","Explicit, positivity, 1/α: new EM for stable volatility"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on Assumption 2.2, which says the negative jumps of the driving stable process are never smaller than $-1/\\kappa$; if arbitrarily large negative jumps are allowed, the true solution may be pushed to or through zero, and the positivity theorem plus the convergence target would no longer hold.","fun_headline_variants_meta":{"raw":{"variants":["Floored explicit EM: positivity and order 1/α for stable volatility","Explicit EM with floor: strong order 1/α for α-stable volatility","No negativity: explicit Euler projects onto Δ, still order 1/α","Floor at Δ: explicit method for α-stable SDEs hits order 1/α","Explicit, positivity, 1/α: new EM for stable volatility"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000682,"raw_usage":{"total_tokens":3060,"prompt_tokens":874,"completion_tokens":2186,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":490,"completion_tokens_details":{"reasoning_tokens":2078}},"tokens_in":490,"tokens_out":2186,"duration_ms":17708,"temperature":1.0,"reasoning_tokens":2078,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T17:44:05.102441+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same scheme on a spectrally negative $\\alpha$-stable process whose Lévy measure includes jumps below $-1/\\kappa$. If the true SDE still has a positive solution and the scheme still converges at order $1/\\alpha$, then Assumption 2.2 is not needed and the theorem's stated hypothesis is too strong; if positivity fails or the error slope on a log-log plot departs from $1/\\alpha$, the assumption is doing real work.","supporting_citations":[{"cited_title":"Samorodnitsky and M","cited_arxiv_id":null,"evidence_quote":"Defines stable distributions and supplies the q-th moment bound $E|L_\\alpha(t)|^q \\le Ct^{q/\\alpha}$ used in Lemma 2.4."},{"cited_title":"Applebaum, L´ evy Processes and Stochastic Calculus, Cambr idge Univer- sity Press, 2nd Ed., 2009","cited_arxiv_id":null,"evidence_quote":"Provides the Ito formula for Lévy-type SDEs used to prove positivity and moment estimates."},{"cited_title":"Zhang, Derivative formulas and gradient estimates for SDEs driven by α-stable processes, Stochastic Process","cited_arxiv_id":null,"evidence_quote":"Supplies the maximal inequality for $\\alpha$-stable stochastic integrals used to bound the jump term in the convergence proof."},{"cited_title":"Willett, J","cited_arxiv_id":null,"evidence_quote":"Gives the nonlinear Gronwall inequality used to close the error bounds in Lemmas 4.1 and Theorem 4.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Inspires the positivity-preserving projection device and provides an implicit positivity-preserving scheme for the jump-extended CIR process."},{"cited_title":"Mikuleviˇ cius, F","cited_arxiv_id":null,"evidence_quote":"Establishes strong convergence orders for Euler approximations of multiplicative stable-noise SDEs, the baseline this paper extends to an explicit positivity-preserving construction."}],"review_version":1}