{"id":"ea16ece0-7a00-41ee-b01b-e8e93fedd0a5","arxiv_id":"1908.03183","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Nonnegative bounded-variation diffusion coefficients with jumps are shown to yield a unique solution for Holder-driven SDEs via the inverse of the integrated reciprocal diffusion.","lead":"This paper proves existence and uniqueness for one-dimensional stochastic differential equations whose diffusion coefficient can jump, as long as the driving noise is sufficiently irregular and spends little time near the jump points. The result gives an explicit solution formula, extending earlier work on fractional Brownian and Rosenblatt noise.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.1 overclaims global existence: σ(x)=1+x² and Y_t=t satisfy the stated assumptions, yet the proposed solution tan(t) ceases to exist at t=π/2.","rationale":"The reader's weakest assumption was the inverse-moment bound and the imported chain rule. I agree those are risky, but the most decisive internal problem is that no condition prevents the explicit solution from leaving the domain of Λ^{-1}. Since the paper aims at very general σ with only local integrability of 1/σ, Λ can be bounded; σ(x)=1+x² is the standard example. The deterministic driver Y_t=t satisfies Assumption 2.1 for suitable α, β, ε, so all hypotheses of Theorem 2.1 are met while the claimed solution is not globally defined. This is an easy fix—add a range/surjectivity condition or a local/explosion formulation—but as written the main theorem is false. The imported chain rule concern is important for a complete verification, but if a referee accepts [3], the range issue remains and independently forces a revision. Hence the recommendation is conditional acceptance with required amendments, not outright rejection: the counterexample is excluded by natural growth assumptions such as σ bounded below by a positive constant, or more generally Λ proper, and the intended examples are unaffected.","tokens_in":72,"tokens_out":21709,"duration_ms":425194,"concrete_test":"Set σ(x)=1+x², X0=0, Y_t=t, T=2. Verify Assumptions 2.1 and 2.2 as above (α=0.75, β=0.5, ε=0.1). Then evaluate the claimed formula: Λ(x)=arctan x, so X_t=tan(t), with a singularity at t=π/2. If the theorem is meant for all t∈[0,T], it fails; if meant only before explosion, the statement must add the stopping time inf{t: Z_t∉range(Λ)} and the uniqueness statements in Theorem 2.2 must be adjusted accordingly. This single check settles whether the missing range/explosion condition is real.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 2 states Theorem 2.1 for any σ satisfying Assumption 2.2 (locally BV, one-signed, 1/σ locally integrable) and any Hölder driver Y with Z_t = Λ(X0)+Y_t−Y0 satisfying Assumption 2.1. The proof applies the chain rule to Λ^{-1}, but Λ^{-1} is only defined on the range of Λ. Local integrability of 1/σ does not force Λ(R)=R. Counterexample: σ(x)=1+x², so Λ(x)=arctan x, range (−π/2,π/2); take X0=0, Y_t=t. For α=0.75, β=0.5, ε=0.1, (β+ε)/α=0.8<1, so sup_y ∫_0^T |t−y|^{-0.8}dt<∞, hence Assumption 2.1 holds, and Y is α-Hölder. The proposed solution is X_t=tan t, which is not a real-valued function on any interval containing π/2. Indeed dX=(1+X²)dt has no global real solution. Thus the theorem needs an explicit surjectivity/range condition such as Λ(R)=R, or it must be formulated up to the explosion time inf{t: Z_t∉range(Λ)}. This is a direct counterexample to the stated global existence claim, independent of the separate concern that Theorem 4.2 and Proposition 4.2 are imported from [3] without proofs.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies one-dimensional SDEs of the form dX_t = σ(X_t)dY_t, where Y is a Hölder continuous process of order α > 1/2 (e.g. fractional Brownian motion with H > 1/2 or the Rosenblatt process) and σ is a locally bounded-variation, one-signed function, possibly discontinuous, with 1/σ locally integrable. The main results are Theorem 2.1, which proposes the explicit solution X_t = Λ^{-1}(Λ(X_0)+Y_t−Y_0) with Λ(x)=∫ 1/σ, and Theorem 2.2, which gives uniqueness up to the first hitting time τ = inf{t : σ(X_t)=0} for solutions satisfying an inverse-distance moment condition (Assumption 2.1). The proof combines a pathwise generalized Stieltjes integral (Section 3) with an integration theory for discontinuously evaluated processes (Section 4), several key ingredients of which are imported from the companion paper [3].","tokens_in":15290,"tokens_out":11135,"duration_ms":116184,"significance":"If the range issue identified below is fixed, the paper would provide an elegant and quite general existence/uniqueness theory for one-dimensional SDEs with discontinuous coefficients, going substantially beyond the earlier results of [6] and [9] and covering natural drivers beyond fractional Brownian motion. The explicit solution formula, the absence of fitted parameters, and the treatment of examples such as the Cantor-function coefficient (Example 2.4) are clear strengths. The main obstacles are a missing global range condition in Theorem 2.1 and the fact that several load-bearing results in Section 4 are stated without proofs and are only deferred to [3].","major_comments":[{"comment":"The global existence claim is false as stated. Take σ(x)=1+x^2, X_0=0, Y_t=t, and any T>π/2. Then σ satisfies Assumption 2.2, and Z_t=Λ(X_0)+Y_t−Y_0=t satisfies Assumption 2.1 for α∈(1/2,1) (choose β∈(1−α,α) and ε>0 so that (β+ε)/α<1; then sup_y∫_0^T |t−y|^{-(β+ε)/α}dt<∞). The proposed formula gives X_t=Λ^{-1}(t)=tan t, which is undefined at t=π/2, and the ODE dX_t=(1+X_t^2)dt has no global real solution. The theorem must either add an explicit condition that Λ(R)=R, equivalently ∫^∞ 1/σ = ∫_{−∞} 1/σ = ∞, or restate existence only up to the explosion time inf{t : Λ(X_0)+Y_t−Y_0 ∉ range(Λ)}.","section":"Section 2, Theorem 2.1; Section 5.1"},{"comment":"The proofs of Proposition 4.2 and Theorem 4.2 are omitted ('we omit the details'), yet both are load-bearing: Theorem 2.1 applies Theorem 4.2 to Λ^{-1}, and Proposition 4.2 is used in Proposition 5.2 to establish uniqueness. The manuscript states that these follow from [3] after modifications, but the modifications are precisely what is needed to accommodate Assumption 2.1 and the case where f(X_0+) need not exist. Please provide the full proofs or, at minimum, a detailed statement of the modifications; a short deferral to the companion paper is not sufficient for the central chain rule on which the main theorems rest.","section":"Section 4, Proposition 4.2 and Theorem 4.2"},{"comment":"The proof of Lemma 4.2 explicitly establishes only boundedness and right-continuity of the displayed functions, with left-continuity relegated to 'the rest of the proof follows as in [3]'. Since the lemma states continuity and is used to justify Proposition 4.2, please include the left-continuity argument or give a precise statement of the corresponding result in [3] that covers this case.","section":"Section 4, Lemma 4.2"}],"minor_comments":[{"comment":"The symbol Z_t is used both for the process Λ(X_0)+Y_t−Y_0 (in Theorem 2.1) and for the solution Λ^{-1}(Λ(X_0)+Y_t−Y_0) (in Corollaries 2.1 and 2.2); please use different letters to avoid confusion.","section":"Section 2, Corollaries 2.1 and 2.2"},{"comment":"The proof writes sup_z∫_0^T |Z_t−z|^{-β/α}dt, whereas Assumption 2.1 contains an expectation and an exponent −(β+ε)/α; the reduction should be made explicit, especially for random drivers.","section":"Section 5.1, proof of Theorem 2.1"},{"comment":"The definition of τ_ε reads inf{t : σ(X_s) ≤ ε}; the variable in the condition should be t, not s.","section":"Section 5, Proposition 5.1"},{"comment":"The phrase 'satisfying 2.1' should read 'satisfying Assumption 2.1'.","section":"Corollary 2.2"}],"recommendation":"major_revision","confidential_remarks":"The counterexample to Theorem 2.1 is decisive and must be addressed before publication. The reliance on the companion paper [3] is acceptable in principle, but the manuscript should be more self-contained about the chain rule and the approximation results it imports. The paper fits the scope of the journal and the main method is worth publishing after a careful revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is the first general attempt at existence and uniqueness for one-dimensional SDEs dX_t = σ(X_t)dY_t where σ is locally BV and can jump, and Y is Hölder with α > 1/2. The construction via Λ(x) = ∫ 1/σ is classical for smooth σ, and the authors push it through the W^{θ,1} integration spaces from [3], covering fBM with H > 1/2 and the Rosenblatt process. The examples generalizing [6] and [9] are real progress. The writing is honest about what is imported from the companion paper.\n\nBut there is a direct counterexample to Theorem 2.1. Take σ(x) = 1 + x² and Y_t = t. Then Λ(x) = arctan x, whose range is bounded. Choose α = 0.75, β = 0.5, ε = 0.1. Assumption 2.1 holds for Z_t = t because ∫_0^T |t−y|^{−0.8}dt is bounded uniformly in y. All other assumptions are satisfied, but the claimed solution is X_t = tan t, which explodes at π/2. The theorem asserts a real-valued solution on the whole interval [0,T]; it fails for T > π/2. The missing condition is that Z_t must stay in the range of Λ, or the statement must be formulated up to the first exit time. This is not a technicality; it is the difference between global existence and a finite-time blow-up.\n\nSecondary issue: the load-bearing approximation result (Prop 4.2) and chain rule (Thm 4.2) are imported from [3] with \"we omit the details.\" That is a serious reliance on a companion paper, though [3] is published and the arguments are plausibly routine. It is not fatal, but it makes verification slow.\n\nIf these two issues are corrected—adding the range condition and either proving or precisely citing the deferred machinery—the paper is a solid contribution. As it stands, the main theorem overclaims, and the counterexample should be acknowledged and addressed. I would recommend sending to peer review because the method and scope are valuable and the flaw is repairable. A good referee will catch the same issue immediately; the authors should fix it before publication.","headline":"A promising method for discontinuous diffusions with Hölder drivers, but the main existence theorem is false as stated because it ignores the range of the Lamperti transform.","tokens_in":15867,"tokens_out":2897,"would_cite":false,"duration_ms":26788,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65C30","60H05","60G22","26A33"],"pacs":[],"model":"deepseek-v4-flash","headline":"For one-dimensional SDEs driven by Hölder noises, discontinuous diffusion coefficients still yield explicit solutions via a time change.","keywords":["stochastic differential equation","fractional calculus","Hölder continuity","discontinuity","bounded variation","fractional Brownian motion","pathwise integration","Rosenblatt process"],"falsifier":"Refutation would require a driver satisfying Assumption 2.1 and a coefficient $\\sigma$ satisfying Assumption 2.2 for which the candidate process fails (2.1). The simplest check is $Y_t=t^\\alpha$ with a two-valued $\\sigma$: the pathwise integral is an ordinary Lebesgue integral and the candidate is explicit, so the identity can be evaluated exactly; a mismatch would refute the construction. Separately, a driver with a plateau makes the inverse-distance integral $\\sup_y \\mathbb{E}\\int_0^T |X_t-y|^{-(\\beta+\\varepsilon)/\\alpha}dt$ diverge at the flat level, confirming that the variability condition is essential.","tokens_in":14788,"feed_emoji":"","tokens_out":15534,"duration_ms":158751,"temperature":0.7,"pith_summary":"The paper establishes that one-dimensional equations $dX_t=\\sigma(X_t)\\,dY_t$, driven by a Hölder-continuous process $Y$ of order $\\alpha>1/2$, admit solutions even when $\\sigma$ is discontinuous. The assumptions on $\\sigma$ are mild: it must keep one sign, be of locally bounded variation, and have a locally integrable reciprocal. The solution is explicit: $X_t=\\Lambda^{-1}(\\Lambda(X_0)+Y_t-Y_0)$, where $\\Lambda(x)=\\int_a^x \\frac{dy}{\\sigma(y)}$. Uniqueness is proved up to the first time $\\tau$ at which $\\sigma(X_t)=0$, within a class of processes satisfying a sufficient-variability condition on the driver. This matters because it extends the classical Lipschitz theory of SDEs to jump-type diffusion coefficients and to drivers such as fractional Brownian motion with $H>1/2$ and the Rosenblatt process.","feed_headline":"Time change solves SDEs with discontinuous diffusion coefficients","feed_subtitle":"For fractional Brownian and similar drivers, the solution is the inverse of an integral built from 1/σ.","key_machinery":"The central object is the increasing map $\\Lambda(x)=\\int_a^x \\frac{dy}{\\sigma(y)}$ and its inverse $\\Lambda^{-1}$. Formally $d\\Lambda(X_t)=\\sigma(X_t)^{-1}dX_t=dY_t$, so the candidate solution is $\\Lambda^{-1}(\\Lambda(X_0)+Y_t-Y_0)$, and the proof shows that $(\\Lambda^{-1})'=\\sigma$ and that $y\\mapsto\\sigma(\\Lambda^{-1}(y))$ is of locally bounded variation. The second ingredient is a pathwise integral defined through fractional Weyl–Marchaud derivatives and controlled by Gagliardo seminorms; Assumption 2.1 supplies the integrability of $|X_t-y|^{-(\\beta+\\varepsilon)/\\alpha}$ that makes discontinuous functions of $X$ tractable.","core_discovery":"Theorem 2.1 states that if the candidate process $Z_t=\\Lambda(X_0)+Y_t-Y_0$ satisfies the inverse-distance integrability bound and $\\sigma$ satisfies the one-signed locally-bounded-variation condition, then $X_t=\\Lambda^{-1}(Z_t)$ is a solution to (2.1). Theorem 2.2 states that every solution satisfying the same integrability condition is unique on $[0,\\tau]$, where $\\tau=\\inf\\{t:\\sigma(X_t)=0\\}$, and that $\\tau$ itself is uniquely determined; if $\\sigma$ never vanishes, uniqueness holds in that class for all times. Together the theorems give the first general existence-and-uniqueness statement for one-dimensional SDEs driven by Hölder noises with discontinuous diffusion coefficients.","pith_inferences":["Because the construction never uses Markov or martingale structure, the solution is a deterministic functional of the driver path; this suggests a pathwise simulation method: apply $\\Lambda^{-1}$ directly to a simulated driver instead of discretizing the SDE.","The inverse-distance condition is likely the real boundary of the method: drivers with flat stretches or paths that concentrate near particular levels fall outside, and non-uniqueness after $\\sigma$ hits zero is left open, so the theorem should be read as a statement about the class it explicitly defines.","A natural test of the same mechanism would be equations with drift, $dX_t=b(X_t)\\,dt+\\sigma(X_t)\\,dY_t$; the paper does not treat these, and $\\Lambda$ would no longer remove $X$ from the driving term, so the integrability condition would need to be reworked."],"forward_implications":["For every driver satisfying the variability condition—fractional Brownian motion with $H>1/2$, the Rosenblatt process, stationary processes with bounded densities—the SDE has a solution for every one-signed locally bounded-variation $\\sigma$ with locally integrable reciprocal.","If $\\sigma$ is bounded away from zero, the solution is unique in the class of processes satisfying Assumption 2.1; if $\\sigma$ can vanish, uniqueness is guaranteed at least up to the first hitting time of the zero set.","The earlier two-valued discontinuous coefficient and the power-type coefficient $\\sigma(x)=|x|^\\gamma$ are recovered as special cases, and in the existence part the extra restrictions on $\\gamma$ tied to the Hurst parameter disappear.","Examples built from the Cantor function plus a positive constant become uniquely solvable, showing that highly non-smooth but locally bounded-variation coefficients fit the framework."],"supporting_citations":[{"why":"Supplies the pathwise integration theory for discontinuously evaluated stochastic processes, including the Gagliardo-seminorm estimates and the inverse-distance condition used in Theorem 4.1.","marker":"[3]"},{"why":"Treats the special two-valued discontinuous diffusion coefficient that the present theorems generalize to arbitrary one-signed locally bounded-variation $\\sigma$.","marker":"[6]"},{"why":"Gives the bounded-variation composition criterion used to show $y\\mapsto\\sigma(\\Lambda^{-1}(y))$ is of locally bounded variation.","marker":"[7]"},{"why":"Establishes Young-type SDE results for power nonlinearities $\\sigma(x)=|x|^\\gamma$ that Example 2.3 extends under weaker conditions.","marker":"[9]"},{"why":"Provides the fractional-calculus and generalized Stieltjes-integral framework for equations driven by fractional Brownian motion.","marker":"[13]"},{"why":"Defines the generalized Stieltjes-integral representation (3.1) used throughout the paper.","marker":"[16]"},{"why":"Supplies the convergence of discrete approximations to Hölder functions used in the proof of Proposition 5.1.","marker":"[17]"}],"fun_headline_variants":["Time change yields existence and uniqueness for discontinuous diffusion SDEs","Inverse time change solves SDEs with discontinuous diffusion","Existence and uniqueness for SDEs with discontinuous diffusions","First general result: SDEs with discontinuous diffusions solved","Inverse integral method tames discontinuous SDE diffusions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the relevant process does not linger near any fixed level: expectations of inverse powers of $|X_t-y|$, integrated over time, must be finite uniformly in $y$. If that fails, the pathwise integral of $\\sigma(X)$ against $Y$ and the chain rule that produces the solution formula are not justified.","fun_headline_variants_meta":{"raw":{"variants":["Time change yields existence and uniqueness for discontinuous diffusion SDEs","Inverse time change solves SDEs with discontinuous diffusion","Existence and uniqueness for SDEs with discontinuous diffusions","First general result: SDEs with discontinuous diffusions solved","Inverse integral method tames discontinuous SDE diffusions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000985,"raw_usage":{"total_tokens":4096,"prompt_tokens":779,"completion_tokens":3317,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":395,"completion_tokens_details":{"reasoning_tokens":3233}},"tokens_in":395,"tokens_out":3317,"duration_ms":29180,"temperature":1.0,"reasoning_tokens":3233,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:22:43.637245+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Refutation would require a driver satisfying Assumption 2.1 and a coefficient $\\sigma$ satisfying Assumption 2.2 for which the candidate process fails (2.1). The simplest check is $Y_t=t^\\alpha$ with a two-valued $\\sigma$: the pathwise integral is an ordinary Lebesgue integral and the candidate is explicit, so the identity can be evaluated exactly; a mismatch would refute the construction. Separately, a driver with a plateau makes the inverse-distance integral $\\sup_y \\mathbb{E}\\int_0^T |X_t-y|^{-(\\beta+\\varepsilon)/\\alpha}dt$ diverge at the flat level, confirming that the variability condition is essential.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the pathwise integration theory for discontinuously evaluated stochastic processes, including the Gagliardo-seminorm estimates and the inverse-distance condition used in Theorem 4.1."},{"cited_title":"Garzon, J.A","cited_arxiv_id":null,"evidence_quote":"Treats the special two-valued discontinuous diffusion coefficient that the present theorems generalize to arbitrary one-signed locally bounded-variation $\\sigma$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the bounded-variation composition criterion used to show $y\\mapsto\\sigma(\\Lambda^{-1}(y))$ is of locally bounded variation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes Young-type SDE results for power nonlinearities $\\sigma(x)=|x|^\\gamma$ that Example 2.3 extends under weaker conditions."},{"cited_title":"Nualart and A","cited_arxiv_id":null,"evidence_quote":"Provides the fractional-calculus and generalized Stieltjes-integral framework for equations driven by fractional Brownian motion."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the generalized Stieltjes-integral representation (3.1) used throughout the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the convergence of discrete approximations to Hölder functions used in the proof of Proposition 5.1."}],"review_version":1}