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Drawdowns of diffusions

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Pith's one-line read This paper proves exact drawdown formulas for one-dimensional diffusions by excursion theory, extending them to diffusions with a lower boundary.

desk verdict 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. read the letter →

arxiv 2411.18374 v1 pith:TMTHEUXZ submitted 2024-11-27 math.PR

Paavo Salminen , Pierre Vallois This is my paper · ORCID
classification math.PR MSC 60J6060J6560G40
keywords drawdownrunningmaximumfirsttimeexcursiontheoryone-dimensionaldiffusionPoissonpointprocessscalefunction
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

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).

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 (2)
  1. [Section 4, Eq. (4.9)] 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.
  2. [Section 5, Propositions 5.4 and 5.7] 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.
minor comments (5)
  1. [Remark 3.8] 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.
  2. [Proof of Theorem 3.7] 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.
  3. [Throughout] 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).
  4. [Remark 5.8] 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.
  5. [Example 3.5] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: drawdown identities are derived from the excursion-theoretic master formula (4.9) and internal pathwise equivalences; self-citations are ancillary.

full rationale

The central derivation chain is self-contained in the relevant sense. Theorem 3.1 is obtained by combining the Poisson point process master formula (4.9) with excursion-law evaluations in Lemma 4.4 and the integral-equation computation of lambda in Lemma 4.7; the master formula is cited to Fitzsimmons [9] and Pitman-Yor [23], external sources not authored by the present authors, and it is a standard compensation formula rather than an assumption of the drawdown formulas. Theorem 3.7 follows from Lemma 4.7 and the identity lambda(y;x)=E_x(exp(-alpha H_y); M_theta_delta > y), while Theorem 3.6 is obtained from (3.4) through the pathwise equivalence (3.8), {D^-_H_eta <= delta} = {M_theta_delta >= eta}, which is an internal identity rather than a hidden assumption. No parameter is fitted, no target formula is used as an input, and the self-citations to the authors' earlier work [28] and [29] concern ancillary Markov-process facts in Section 5.2 and standard excursion-law background; they are not load-bearing for the main distributional identities. Hence there is no circular step.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central results rest on standard one-dimensional diffusion theory (Ito-McKean formulas for hitting probabilities and Green functions) and on the Ito excursion theory, especially the master formula (4.9) for excursions below the running maximum. No free parameters are fitted. The only regularity restrictions are smoothness of the scale function and speed measure, chosen for simplicity. No invented entities are introduced.

assumptions (4)
  • domain assumption X is a regular one-dimensional diffusion in the sense of Ito-McKean with continuous paths, not killed inside its interval I, with continuously differentiable scale function S and continuously differentiable speed measure density.
    Section 2 establishes this standing framework and says 'For simplicity, it is assumed throughout the paper'. It restricts the class but is a regularity condition, not an extra physical postulate.
  • standard math The master formula (4.9): for y > x, the excursions (y, xi_y) below the running maximum form a Poisson point process with intensity dS(y) n_y_minus(de).
    Section 4.2 cites Fitzsimmons [9] and Pitman-Yor [23]. This is the central structural input used in Lemmas 4.4, 4.6, 4.7 and in the final proofs of Theorems 3.1 and 3.7.
  • standard math The hitting-time and resolvent formulas (2.1), (2.4) express E_x(exp(-alpha H_y)) and the Green function in terms of the fundamental solutions phi_alpha, psi_alpha and the Wronskian w_alpha.
    Section 2 quotes these from Ito-McKean [13]; they are used to derive formulas (4.15), (4.16), and to evaluate the excursion-law quantities in Lemma 4.4.
  • domain assumption The diffusion belongs to Class 1 or 2 as defined in Section 2, with r = +infinity and specified boundary behavior; in particular, geometric Brownian motion with drift is included when the parameters make it recurrent with S(+infinity)=+infinity or transient with S(+infinity)<+infinity.
    Theorems 3.1, 3.6, 3.7 are stated only for these classes, and the proof of Proposition 3.3 uses the class assumption to show theta_delta is finite.

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Pith. "Pith review of Drawdowns of diffusions." pith.science (2026). https://pith.science/paper/TMTHEUXZ

@misc{pith2026241118374,
  author       = {Pith},
  title        = {Pith review of: Drawdowns of diffusions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TMTHEUXZ}},
  note         = {Machine review of arXiv:2411.18374}
}
read the original abstract

In this paper we give excursion theoretical proofs of Lehoczky's formula (in an extended form allowing a lower bound for the underlying diffusion) for the joint distribution of the first drawdown time and the maximum before this time, and of Malyutin's formula for the joint distribution of the first hitting time and the maximum drawdown before this time. It is remarkable -- but there is a clean explanation -- that the excursion theoretical approach which we developed first for Lehoczky's formula provides also a proof for Malyutin's formula. Moreover, we discuss some generalizations and analyze the pure jump process describing the maximum before the first drawdown time when the size of the drawdown is varying

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