{"id":"f9cb969f-2269-4b76-9921-c7fb95059605","arxiv_id":"2411.18374","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors prove Lehoczky's and Malyutin's drawdown formulas via excursion theory and analyze the Markov process of the maximum at the first drawdown time.","lead":"This paper gives new mathematical proofs for formulas describing when a random process first falls a fixed distance below its running maximum, and how big that maximum is at that moment. It also extends the formulas to processes on intervals with a lower boundary, which matters for applications like risk management and option pricing.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the master formula (4.9) is the standard Pitman-Yor compensation formula, and the level-products in Lemma 4.7 are effectively countable products over disjoint excursions, so the central drawdown identities are not undermined.","rationale":"The reader identified the excursion-theoretic master formula (4.9) as the weakest assumption. I agree that it is load-bearing, but I do not see that it is insecure: the formula is supported by the peer-reviewed Pitman-Yor framework [23], and the paper's Brownian and reflected-Brownian examples are consistent. The specific technical worry about uncountable products in Lemma 4.7 is resolved by the fact that all but countably many excursion levels have zero duration, so the product manipulation is legitimate. No counterexample or internal contradiction appears in the main theorems. The secondary issues flagged by the reader, such as an unproved statement in Proposition 5.1 and the unshown equivalence of (3.10) with (3.9), are real but do not affect the central distributional identities; they justify a conditional acceptance only in the sense of requesting expositional fixes. Since my assessment does not change the reader's conditional verdict, the recommended verdict remains UNCHANGED.","tokens_in":19073,"tokens_out":28988,"duration_ms":268254,"concrete_test":"Independently re-derive formula (3.9) for Brownian motion with drift mu > 0 (Class 2a) by solving the corresponding boundary-value problem for E_x(exp(-alpha H_eta); D^-_{H_eta} < y) directly from the generator, without using excursion theory, and compare the resulting closed form with (3.9). A match would corroborate the master formula (4.9) in the transient regime; a mismatch would reveal a hidden assumption in the excursion-theoretic derivation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing objection to the central claim. The most delicate input is the master formula (4.9), which is indeed load-bearing, but it is not merely a citation to an unpublished preprint: it is the standard Pitman-Yor compensation formula for excursions below the running maximum of a one-dimensional diffusion, stated in [23] and conveniently specialized in [9]. The apparent uncountable products in Lemma 4.7 are formally acceptable: the levels z for which H_{z+}-H_z > 0 form a countable family of disjoint nonempty open intervals, so identity (4.25) applies to the countable subproduct of non-unit factors. Theorems 3.1, 3.6, and 3.7 reproduce the known Brownian results, and Example 3.10 for reflected Brownian motion is internally consistent with the formulas. The remaining issues are secondary expositional gaps, such as the proof of the full statement of Proposition 5.1, the claimed equivalence of (3.10) and (3.9), and several typos; these do not threaten the stated distributional identities.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper gives excursion-theoretic proofs of two classical drawdown results: Lehoczky's formula for the joint Laplace transform of the first drawdown time theta_delta and the running maximum M_theta_delta, and Malyutin's formula for the joint law of the first hitting time H_eta and the maximum drawdown D^-_{H_eta}. The formulas are extended to one-dimensional diffusions with a finite lower boundary. The authors also study the pure jump process (M_theta_delta)_{delta>=0}, derive its Markov generator and jump measure, and compare it with (D^-_{H_eta}). The main proofs use the point process of excursions below the running maximum, with intensity measure dS(y) n_y^-(de).","tokens_in":19238,"tokens_out":31844,"duration_ms":286027,"significance":"If correct, the paper provides a unified and transparent explanation for both formulas, extends them beyond the cases treated in [15] and [18], and reproduces known Brownian and reflected Brownian formulas in Examples 3.9 and 3.10. The central identities (3.1), (3.4), (3.6), and (3.9) are exact distributional statements with no fitted parameters, and the Brownian checks lend credibility to the main argument. The main caveats are that the load-bearing master formula (4.9) is cited rather than proved, and that several secondary results in Section 5 are asserted with details omitted. These issues do not appear to threaten the truth of the main theorems but should be addressed before publication.","major_comments":[{"comment":"The proofs of Lemmas 4.4, 4.6, 4.7 and of Theorems 3.1 and 3.7 all use the master formula (4.9), which is quoted from the arXiv preprint [9] with 'see also [23]'. Because this formula is the structural foundation of the excursion-theoretic argument, the paper should state it as a proposition and give a proof or an exact reference (e.g., a theorem number in Pitman-Yor [23]) that covers the Classes 1 and 2, including the reflecting lower-boundary case. Please also clarify how the formula treats the possible infinite excursion at the terminal maximum in the transient case (Class 2). I do not regard this as a correctness error: (4.9) is the standard compensation formula for excursions below the running maximum, and the Brownian examples in Section 3.4 are consistent with it. The request is for self-containedness and verifiability.","section":"Section 4, Eq. (4.9)"},{"comment":"The generator A_rho and the jump measure nu_{y,rho} in Proposition 5.4 and Remark 5.5 are original claimed results, but their derivations are omitted with the phrases 'fairly straightforward calculations' and 'we skip the details'. Proposition 5.1 similarly proves only (5.2) and leaves the full Markov kernel (5.1) to the reader. Please supply these derivations, or state precisely which of these statements are actually needed for the subsequent results. Without them, the analysis of (M_theta_delta) in Section 5 is not fully supported.","section":"Section 5, Propositions 5.4 and 5.7"}],"minor_comments":[{"comment":"The sentence 'it can be shown that (3.10) and (3.9) are equivalent' is unsupported; either supply the calculation or delete the claim, since (3.9) is already proven.","section":"Remark 3.8"},{"comment":"The equality {D^-_{H_eta}<delta} = {M_theta_delta>eta} is used without comment; since (3.4) and (3.6) imply M_theta_delta has a continuous distribution, the equality is correct, but it should be stated explicitly.","section":"Proof of Theorem 3.7"},{"comment":"There are several typos: 'refered' and 'litterature' in the Introduction, 'indentical' in Example 3.10, 'contnuity' in Section 5.1, and 'Comparision' in Section 5.2. In Lemma 4.4, 'The formulas (4.14) and (4.14) in case alpha=0' should refer to (4.13) and (4.14).","section":"Throughout"},{"comment":"The displayed formula for phi'_{rho+s}(0+) has an incorrect argument in the denominator: it should be S(y)-S(y-rho-s), not S(y)-S(y-rho+s). The subsequent integral uses the correct denominator, so this appears to be a typo.","section":"Remark 5.8"},{"comment":"Reflecting Brownian motion with drift 1 has scale function S(z)=1-e^{-2z} (up to constants), not S(z)=1-e^{-z}. Either the drift should be 1/2 or the scale function should be adjusted.","section":"Example 3.5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a solid contribution to the drawdown literature and fits the scope of math.PR. The main theorems appear correct, and the paper does not show signs of circularity or fitted parameters. The dependence on the unpublished preprint [9] should be tightened in the revision, and the omitted derivations in Section 5 should be supplied. I do not see a need to question the novelty relative to [10] and [28]; the paper's unified treatment of Lehoczky's and Malyutin's formulas is a genuine contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a solid paper with real content, but don't oversell the novelty. The main drawdown identities go back to Lehoczky, Taylor, Malyutin, and Brockwell. What's new is the unified excursion proof, the extension to diffusions with a finite lower boundary (Theorem 3.1), and the study of the Markov process (M_theta_delta). The proof of Theorem 3.7 via the same excursion machinery is elegant, and the connection (3.8) is the clean insight. The examples reproduce known Brownian and reflected-Brownian results, which is a good sanity check.\n\nI'm not worried about the reliance on the master formula (4.9). It is cited to Fitzsimmons' preprint, but it is the standard Pitman–Yor compensation formula for excursions below the running maximum. The stress-test note is right: the apparent uncountable products in Lemma 4.7 reduce to countable products because only countably many levels z have H_{z+}-H_z > 0. So the central distributional identities hold up.\n\nThe soft spots are secondary but real. Several claims are asserted without full derivation: the equivalence of (3.10) and (3.9) in Remark 3.8, the generator formula in Proposition 5.4 (details skipped), and the full statement of Proposition 5.1 (only the marginal version is proved). There are also typos and small slips. None of this threatens the main theorems, but an editor should ask for the gaps to be filled or clearly referenced, and for a careful proofreading pass.\n\nWho is this for? People working on drawdowns, excursion theory, or diffusion functionals. It is a useful paper to have in the literature because it puts known formulas on a unified path and extends them to a natural boundary case. I would cite it if I worked in this area. The paper deserves a serious referee: it is technically substantial, honest about the known status of the formulas, and the new parts are well motivated.\n\nRecommendation: send to peer review, not desk-reject. Ask for minor-to-moderate revisions to fill the sketched proofs and fix typos.","headline":"Solid excursion-theoretic treatment of drawdown identities; known formulas in new light, plus a finite-boundary extension and a clean analysis of the running-maximum-before-drawdown process.","tokens_in":19858,"tokens_out":2779,"would_cite":true,"duration_ms":25486,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J60","60J65","60G40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves exact drawdown formulas for one-dimensional diffusions by excursion theory, extending them to diffusions with a lower boundary.","keywords":["drawdown","running maximum","first drawdown time","maximum drawdown","excursion theory","one-dimensional diffusion","Poisson point process","scale function"],"falsifier":"For a diffusion with explicit scale function, evaluate the right side of (3.4) and compare with Monte Carlo simulation of θ_δ and M_{θ_δ}; the scale function S(z) = 1 − exp(−e^z) in Example 3.4 is a natural test case because the paper itself shows the drawdown time is finite with probability strictly less than 1, so the tail formula makes a sharp prediction about the conditional distribution of the maximum.","tokens_in":18809,"feed_emoji":"📉","tokens_out":6839,"duration_ms":58870,"temperature":0.7,"pith_summary":"The paper sets out to prove exact formulas for drawdowns of regular one-dimensional diffusions: the joint distribution of the first time the process falls δ below its running maximum and of the maximum reached before that time, and the joint distribution of the first hitting time of a level and the maximum drawdown before that hitting time. The results are Lehoczky's formula and Malyutin's formula, both extended to diffusions with a finite lower boundary. The paper's route is excursion theory rather than discrete approximation: the excursions below the running maximum form a Poisson point process, and the drawdown identities fall out of that structure. If the claims are right, the formulas hold exactly for two broad classes of diffusions, including geometric Brownian motion.","feed_headline":"Excursions below the max give exact drawdown distributions","feed_subtitle":"Lehoczky's and Malyutin's formulas are reproved and extended to diffusions with a lower boundary.","key_machinery":"The machinery is the Poisson point process of excursions below the running maximum. As the process climbs, each new maximum level y spawns an excursion below y; the collection is a Poisson point process with intensity dS(y) n^y_−(de), where S is the scale function and n^y_− is the Itô excursion law restricted to paths staying below y. Formula (4.9), the master formula, converts sums over these excursions into integrals against that intensity. The drawdown size δ enters by asking whether an excursion at level y reaches y−δ, and the functions b_α and c_α are the corresponding excursion-law expectations.","core_discovery":"The central claim is that for a regular one-dimensional diffusion in Class 1 or Class 2, the joint Laplace transform of the first drawdown time θ_δ and the maximum M_{θ_δ} is given explicitly in Theorem 3.1 by an integral against the scale function, with tail probability P_x(M_{θ_δ} > y) = exp(−∫_{x∨(δ+l)}^y dS(z)/(S(z) − S(z − δ))). The same excursion-theoretic machinery gives, in Theorem 3.7, the joint law of the first hitting time H_η and the maximum drawdown before that time. The key structural identity is that the event that the maximum drawdown before H_η is at most δ coincides with the event that the maximum before the first drawdown time of size δ reaches η. This is the content of Theorems 3.1, 3.6 and 3.7.","pith_inferences":["A natural test of the method is to extend it to the state-dependent drawdown time θ_φ with threshold φ(M_t); the paper notes this is a straightforward adaptation, but a fully written proof for nonconstant φ would settle whether the excursion intensity carries the whole argument.","The identity {D_{H_η}^− ≤ δ} = {M_{θ_δ} ≥ η} suggests that any distributional theorem proved for one of these processes transfers to the other, so the explicit semigroup for (M_{θ_δ}) could in principle be derived from known semigroup results for (D_{H_η}^−) in broader diffusion classes.","In transient Class 2 diffusions the drawdown time can be infinite with positive probability, as Example 3.4 shows, so financial applications should condition on the drawdown occurring before interpreting the Laplace transform as an ordinary expectation."],"forward_implications":["For every diffusion in Class 1 the first drawdown time of size δ is finite almost surely, so the formulas describe a genuine stopping time with tail probability given by (3.4).","The same excursion computation yields the joint law of the first hitting time H_η and the maximum drawdown before H_η, not merely the marginal law of the drawdown.","The additive-functional version (3.14) extends the formulas to diffusions killed at a rate g, with the functions φ_{α,g} and ψ_{α,g} replacing φ_α and ψ_α.","The process δ ↦ M_{θ_δ} is Markov with explicit transition probabilities and generator, and the same transition structure governs η ↦ D_{H_η}^−.","For reflected Brownian motion, θ_δ has Laplace transform 1/cosh²(δ√(2α)), meaning it has the law of the sum of two independent hitting times of level δ."],"supporting_citations":[{"why":"Supplies the master formula (4.9) for the Poisson point process of excursions below the maximum.","marker":"[9]"},{"why":"Expresses b_α and c_α through the excursion law and offers a prior excursion-theoretic proof of Lehoczky's formula.","marker":"[10]"},{"why":"Establishes the excursion approach for one-dimensional diffusions used in the master formula.","marker":"[23]"},{"why":"Gives the original Lehoczky formula for the joint Laplace transform of θ_δ and M_{θ_δ}, which Theorem 3.1 generalizes to a lower boundary.","marker":"[15]"},{"why":"Gives Malyutin's joint distribution formula, which Theorem 3.7 extends.","marker":"[18]"},{"why":"Provides Brockwell's Brownian-with-drift maximum drawdown result, the case the paper generalizes.","marker":"[4]"},{"why":"Gives the semigroup of the process (D_{H_η}^−) and the relation used in Section 5.","marker":"[28]"},{"why":"Supplies the one-dimensional diffusion theory: scale function, speed measure, resolvent and fundamental solutions.","marker":"[13]"},{"why":"Provides handbook facts on Brownian motion and diffusions used in the examples, including reflected Brownian motion.","marker":"[3]"}],"fun_headline_variants":["Excursion theory yields exact drawdown distributions","Lehoczky and Malyutin drawdown formulas via excursions","Excursion proofs for drawdown and max distributions in diffusions","Drawdown times and maxima: exact laws from excursions","Two classic drawdown formulas reproved by excursions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument assumes the master formula (4.9): that the excursions below the running maximum form a Poisson point process with intensity dS(y) n^y_−(de) for the stated classes; if that structural fact fails, the calculations in Section 4 that produce the drawdown formulas lose their justification.","fun_headline_variants_meta":{"raw":{"variants":["Excursion theory yields exact drawdown distributions","Lehoczky and Malyutin drawdown formulas via excursions","Excursion proofs for drawdown and max distributions in diffusions","Drawdown times and maxima: exact laws from excursions","Two classic drawdown formulas reproved by excursions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000804,"raw_usage":{"total_tokens":3482,"prompt_tokens":844,"completion_tokens":2638,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":460,"completion_tokens_details":{"reasoning_tokens":2559}},"tokens_in":460,"tokens_out":2638,"duration_ms":17443,"temperature":1.0,"reasoning_tokens":2559,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:16:48.810892+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a diffusion with explicit scale function, evaluate the right side of (3.4) and compare with Monte Carlo simulation of θ_δ and M_{θ_δ}; the scale function S(z) = 1 − exp(−e^z) in Example 3.4 is a natural test case because the paper itself shows the drawdown time is finite with probability strictly less than 1, so the tail formula makes a sharp prediction about the conditional distribution of the maximum.","supporting_citations":[{"cited_title":"Excursions Above the Minimum for Diffusions","cited_arxiv_id":"1308.5189","evidence_quote":"Supplies the master formula (4.9) for the Poisson point process of excursions below the maximum."},{"cited_title":"A Proof of Lehoczky's Theorem on Drawdowns","cited_arxiv_id":"2209.04695","evidence_quote":"Expresses b_α and c_α through the excursion law and offers a prior excursion-theoretic proof of Lehoczky's formula."},{"cited_title":"Pitman and M","cited_arxiv_id":null,"evidence_quote":"Establishes the excursion approach for one-dimensional diffusions used in the master formula."},{"cited_title":"Lehoczky","cited_arxiv_id":null,"evidence_quote":"Gives the original Lehoczky formula for the joint Laplace transform of θ_δ and M_{θ_δ}, which Theorem 3.1 generalizes to a lower boundary."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives Malyutin's joint distribution formula, which Theorem 3.7 extends."},{"cited_title":"Brockwell","cited_arxiv_id":null,"evidence_quote":"Provides Brockwell's Brownian-with-drift maximum drawdown result, the case the paper generalizes."},{"cited_title":"Salminen and P","cited_arxiv_id":null,"evidence_quote":"Gives the semigroup of the process (D_{H_η}^−) and the relation used in Section 5."},{"cited_title":"Itô and H.P","cited_arxiv_id":null,"evidence_quote":"Supplies the one-dimensional diffusion theory: scale function, speed measure, resolvent and fundamental solutions."},{"cited_title":"Borodin and P","cited_arxiv_id":null,"evidence_quote":"Provides handbook facts on Brownian motion and diffusions used in the examples, including reflected Brownian motion."}],"review_version":1}