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A recursive formula for the $n^\text{th}$ survival function and the $n^\text{th}$ first passage time distribution for jump and diffusion processes. Applications to the pricing of $n^\text{th}$-to-default CDS

T0 review · 3 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A recursive formula gives the nth survivor's survival function for many-body Markov processes with killing boundaries.

desk verdict Two-coordinate core and examples are solid, but the many-coordinate theorem is asserted rather than proved; worth a referee, not a desk reject. read the letter →

arxiv 2509.02347 v2 pith:ZOYZXMW4 submitted 2025-09-02 math.PR cond-mat.stat-mechq-fin.PR

Alessio Lapolla This is my paper · ORCID
classification math.PRcond-mat.stat-mechq-fin.PR MSC 60J7060J2560J2760K3582C3191G20
keywords survivalfunctionfirstpassagetimeorderstatisticskillingboundarymany-bodyMarkovprocessnth-to-defaultCDSsinglefilediffusionmultivariatePoisson
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 compute, exactly, the survival function and first-passage-time distribution of the nth coordinate to be killed among N interacting Markovian coordinates in the presence of a killing boundary. It first proves a two-coordinate formula that splits the last survivor's survival function into the joint survival function plus integrals over the first kill time and the survivor's subsequent independent evolution. It then generalizes to N coordinates by enumerating all ordered kill paths on a graph of alive/dead states, expressing each path's contribution as a nested integral over the successive kill times. If correct, the result provides an order-statistics formula for first passage times of many-body jump and diffusion processes, with concrete applications to single-file diffusion and to pricing nth-to-default credit default swaps. The author notes the formula is cumbersome and, in the single-file diffusion example, identical to an existing reflection-principle result, but argues the recursive path structure exposes the non-Markovian, path-dependent nature of the problem.

What carries the argument

The central object is the directed graph Γ whose nodes are binary strings labeling alive (A) and dead (D) coordinates, and the path contributions I_g(t) (Definition 1, Eq. 44). Each I_g is a nested integral over the ordered kill times τ1 < ... < τn−1, built from the conditional density P^N_n of the surviving coordinates given the full kill history, the marginal first-passage densities F_{n+1}, ..., F_N of successively reduced systems, and the transition density of the final survivor. The path sum over G^n, the set of paths from the all-alive state to states with exactly n alive coordinates, carries the recursion in Theorem 2 (Eq. 45).

What would settle it

Take two correlated Brownian motions with a killing boundary and a post-kill drift for the survivor that depends on which coordinate was killed first (e.g., the survivor's drift is +1 if coordinate 1 dies first and −1 if coordinate 2 dies first). Compute the last-survivor survival function by Monte Carlo simulation and compare with the Theorem 1 formula evaluated numerically; a statistically significant discrepancy in the regime where the two kill times are close would show the factorization fails.

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Extended reading notes

Core claim

The central claim is that for N Markovian coordinates with a killing boundary, the survival function of the nth surviving coordinate satisfies the recursion S^n(t) = S^{n+1}(t) + sum_{g in G^n} I_g(t), where each I_g is a nested integral over the ordered kill times of a path through the graph of alive/dead configurations, involving the product of conditional densities of the survivor given the history and the marginal first-passage densities of successively reduced systems. The two-coordinate case (Theorem 1) is the base case, writing the last-survivor survival function as the joint survival function plus integrals over the first kill time of the survivor's subsequent independent evolution.

Load-bearing premise

The joint density of the ordered kill history is assumed to factorize into the product of marginal first-passage densities of successively reduced systems times a conditional density of the surviving coordinates, and the survivor's post-kill evolution is assumed to be Markovian with the reduced dynamics——this is the premise that carries the nested-integral formula.

Editorial extensions

If this is right

  • The recursion provides an exact, if combinatorially heavy, method to compute order statistics of first passage times for any Markovian many-body system with killing, without resorting to Monte Carlo simulation.
  • It yields semi-analytical pricing formulas for nth-to-default credit default swaps, with the fee and protection legs expressed directly through the survival functions S^{N-n+1}.
  • In the single-file diffusion example, the formula reproduces the known reflection-principle result, confirming the method on a nontrivial interacting diffusion.
  • The path-sum structure makes explicit how the kill history enters the dynamics, highlighting that the reduced survivor is Markovian only conditionally on the full ordered sequence of deaths.
  • The framework naturally handles simultaneous killings, as in the multivariate Poisson model, where multiple coordinates can hit the boundary at the same time.

Reading between the lines

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

  • The recursive path structure could suggest approximate schemes for large N by truncating or averaging over paths in Γ, potentially reaching mean-field-like descriptions of absorbing particle systems.
  • If the factorization of the kill-history density fails for systems where the survivor's post-kill law depends on the identity rather than just the position of the killed coordinate, corrections to the formula would be needed; such cases may motivate generalized conditional densities.
  • The paper's emphasis on path dependence connects naturally to persistence exponents and extreme-value statistics of correlated processes, where the order of extreme events matters.
  • The CDS application could be extended to collateralized debt obligations or to models with default-dependent intensity term structures, as the paper itself suggests, while retaining the path-sum form.
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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

3 major / 3 minor

Summary. The paper claims a recursive formula for the survival function of the n-th surviving coordinate of an N-coordinate Markov process with a killing boundary. Theorem 1 treats the two-coordinate case, expressing the last-survivor survival function as the joint survival function plus integrals over the first-kill time and the survivor's position. Theorem 2 and Definition 1 extend this to N coordinates via a graph of killing paths, and applications are given to a bivariate Poisson process, a two-particle single-file diffusion, a trivariate Poisson process, and n-th-to-default CDS pricing. The examples are accompanied by Monte Carlo comparisons, and the single-file result is asserted to reproduce the known Locatelli formula.

Significance. If the general formula were correct, it would provide an exact order-statistics representation for first-passage problems in many-body Markov systems, with potential applications in physics and credit risk. The paper's strengths are its explicit analytic formulas for the Poisson and single-file examples and the Monte Carlo verification of those examples. The two-coordinate single-file result being consistent with an independent known result is a useful sanity check. However, the claimed generality for N≥3 is not established, and the main theorem is not supported by the proof supplied.

major comments (3)
  1. [§3, Definition 1, Eq. (44)] The path contribution I_g(t) integrates only over the final survivor position y and the kill times τ1,...,τ_{n-1}. For N≥3, the law of later kill times depends on the positions of the surviving coordinates at earlier kill times. For example, for three independent Brownian particles killed at the boundary, the second kill time distribution after the first kill is the first-passage time of a Brownian started from the conditional position at τ1, not from the original initial distribution. Eq. (44) uses F_{n+1}(τ_{n-1})...F_N(τ_1) as marginals from the original system, so it does not represent the ordered kill-time density. The one-sentence induction proof of Theorem 2 does not address this. The trivariate Poisson example is special because the independent exponential structure makes the factorization exact; it does not validate the general claim. This is a load-bearing gap for Eq. (45).
  2. [§2, Theorem 1, Eq. (7)] Eq. (7) states P(τ2=τm)=1−S2(t). This is not an identity: the left side is the time-independent probability that coordinate 2 is the first to be killed, while S2(t) was defined as the joint survival P(t≤τm). If S2(t) is instead read as the marginal survival of X2, the equality would still be false because P(τ2=τm) does not equal P(τ2≤t). The denominator in Eq. (8) should be P(τ2=τm), so the cancellation used to obtain Eq. (4) is invalid. The two-coordinate formula may be correct, as the examples suggest, but the proof as written is not rigorous.
  3. [§2.2, Eq. (37)] The statement that Eq. (37) is 'identical to the result reported in [27]' is not substantiated. Eq. (37) is an infinite series whose terms contain an integral with a ratio of infinite series 1/C(τ), whereas Eq. (40) is a closed-form expression. The agreement is only shown visually in Fig. 2. A numerical comparison of the two expressions should be provided, or the claim should be softened to 'numerically agrees with'.
minor comments (3)
  1. [§3.1, Eq. (50)] In the displayed computation for I(t), the second integral is written with lower limit 0 and upper limit τ, but the outer variable is t and the first integral is over τ. The second upper limit should be t.
  2. [§3.1, Eq. (49)] In the third displayed path, 'X0=1' should presumably be 'X1=1' to match the notation of the other paths.
  3. [§2, Eq. (7)] The notation S2(t) is used inconsistently: at the beginning of Section 2 it is defined as P(t≤τm), but in Eq. (7) it appears to denote a marginal survival probability. This ambiguity contributes to the proof error.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the recursive survival formula is a path-decomposition identity validated against independent benchmarks and Monte Carlo.

full rationale

The claimed derivation is a decomposition of the event that exactly n coordinates survive. Theorem 1 (Eq. 4) conditions on the first kill time; the object P^2_1(y,s|X2∈∂R,s)F2(s) is by definition the joint density of the kill time and survivor position, so the formula does not use the target survival function as an input. Theorem 2 (Eqs. 44–45) recursively expresses S^n through S^{n+1} plus path contributions, with base case S^N taken as given; this is a downward recursion, not a circular definition. The examples are checked against Monte Carlo, and the single-file result is explicitly identical to the independent formula of Locatelli et al. [27]. The two self-citations [21,22] provide the eigenfunction machinery for the specific single-file example, but the recursion formula itself does not reduce to those papers and is not calibrated on them. The main weakness is rigor, not circularity: the proof of Theorem 2 is a one-sentence induction sketch, and Definition 1 (Eq. 44) omits the required integrations over survivor positions at intermediate kill times, so the general N≥3 statement is underived as written. The conclusion also admits that Eq. (37) is 'infinitely slower and more complicated' than [27], which is a limitation of usefulness, not circularity of derivation. No fitted parameter is renamed as a prediction and no uniqueness or ansatz claim is imported via self-citation.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard Markov assumptions (existence of propagators, finite inaccessible hitting times, strong Markov property) plus one unproven factorization: marginal first-passage densities of reduced systems multiply into the joint law of the ordered kill history (Eqs. 8 and 44). The model parameters in the examples (intensities, M, r, T, box geometry) are illustrative inputs, not fitted values. No new physical entities are introduced; the graph Gamma is a bookkeeping device.

free parameters (3)
  • Poisson intensities (bivariate: lambda1, lambda2, lambda12; trivariate: lambdai, lambdaij) = lambda1=1, lambda2=2, lambda12=0.8, M=5 (Fig 1); lambda1=1.2, lambda2=0.5, lambda3=3.3, lambda12=1.4, lambda13=3.1, lamb
    Illustrative model inputs chosen for the figures and table, not fitted to data. The recursive formula itself is independent of their values.
  • CDS pricing inputs r, T, payment frequency = r=0.02, T=5, semiannual fee payments
    Standard market-style inputs for a numerical illustration. No market data are used, so the Table 1 spreads are illustrative only.
  • Single-file diffusion coefficient and box geometry = diffusion coefficient 1, box (0,1) with reflecting boundary at 0 and killing boundary at 1
    Chosen model normalization ('unitary diffusion coefficient', Section 2.2); results scale trivially with these choices.
assumptions (5)
  • domain assumption All transition densities (P^1_i and P^2_i), conditional densities P^2_i(x,t|Xj in dR, t), and marginal first passage densities exist and are computable analytically or numerically.
    Stated before Theorem 1: 'All these functions are assumed to exist and to be computable analytically or at least numerically.' This is a heavy assumption; it excludes many interacting systems whose reduced-system propagators are intractable.
  • domain assumption All hitting times exist, are finite, and are inaccessible stopping times; coordinates are killed instantaneously upon hitting the boundary.
    Section 2: 'We always assume that all hitting times exist, are finite, and are inaccessible stopping times under the filtration Ft.' Simultaneous kills are only absorbed into the joint survival term S^2(t) in Eq. (4).
  • ad hoc to paper The density of the kill history factorizes: P^2_1(y,s|X2 in dR, s) F2(s) is the joint density of (first kill at s, survivor position y), and in general the product F_(n+1)(tau_(n-1)) ... F_N(tau_1) with P^N_n(...) is the joint density of the ordered kill history.
    This is the load-bearing structure of Theorems 1 and 2 (Eqs. 8 and 44, Remark 2). It is asserted, not proven. The proof of Theorem 1 even states the invalid identity P(tau2 = taum) = 1 - S2(t) in Eq. (7). The examples satisfy it only because of the exponential structure of Poisson processes and the strong Markov property of Brownian motion.
  • standard math The two-particle single-file propagator is the Bethe-ansatz eigenfunction representation of Lapolla and Godec [22] with an ordering operator (Eqs. 25-32), and the first passage density F^2(t) comes from Lapolla [21] (Eq. 36).
    Section 2.2 leans on the author's own prior work. [22] is a methods paper with released code (BetheSF), so the support is independent of this derivation, but it is self-citation and unverified in this preprint.
  • domain assumption For the single-file example the initial positions are iid uniform in the box, so the ordered particles map to independent particles and only the leftmost/rightmost ordering matters.
    Section 2.2: 'we consider a (arguably) more natural initial condition, where the two initial positions are drawn from the uniform distribution U.' The comparison with Locatelli et al. [27] uses the same condition.

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Pith. "Pith review of A recursive formula for the $n^\text{th}$ survival function and the $n^\text{th}$ first passage time distribution for jump and diffusion processes. Applications to the pricing of $n^\text{th}$-to-default CDS." pith.science (2026). https://pith.science/paper/ZOYZXMW4

@misc{pith2026250902347,
  author       = {Pith},
  title        = {Pith review of: A recursive formula for the $n^\textth$ survival function and the $n^\textth$ first passage time distribution for jump and diffusion processes. Applications to the pricing of $n^\textth$-to-default CDS},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZOYZXMW4}},
  note         = {Machine review of arXiv:2509.02347}
}
abstract

We derive some rather general, but complicated, formulae to compute the survival function and the first passage time distribution of the $n^\text{th}$ coordinate of a many-body stochastic process in the presence of a killing barrier. First we will study the case of two coordinates and then we will generalize the results to three or more coordinates. Even if the results are difficult to implement, we will provide examples of their use applying them to a physical system, the single file diffusion, and to the financial problem of pricing a $n^\text{th}$-to-default credit default swap ($n^\text{th}$-CDS)

Figures

Figures reproduced from arXiv: 2509.02347 by the authors.

Figure 1
Figure 1. Comparison between a simulation (dots obtained averaging over 10, 000 realization) and the theory (solid lines) for a bivariate Poisson model with λ1 = 1, λ2 = 2, λ12 = 0.8, and M = 5. The red lines show the result for S 2 (t) (Eq. (17)) while the blue ones for S 1 (t)(Eq. (19)). theta) (25) P 2 (x1, x2, t|x01, x02, 0) = 2Θ(x2−x1) X∞ k1,k2=0 ϕk1 (x1)ϕk2 (x2)ϕk1 (x01)ϕk2 (x02)e−Λk1,k2 t ; and ϕk(x) = 2√ 2 cos  (2k +… view at source ↗
Figure 2
Figure 2. Comparison between a Brownian dynamics sim￾ulation (100, 000 realizations and a time step of 10−6 ) of a single file of two elements and the theoretical results. In par￾ticular we show the survival function of the leftmost particle in blue and of the rightmost in red. The dots are obtained using the simulation’s results while the solid lines using our formulas. The black dashed line has been obtained using the resul… view at source ↗
Figure 3
Figure 3. The graph Γ relative to the trivariate Poisson model in Section 3.1. A labels the alive coordinates and D the killed ones. Remark 2. In Eq. (44) FN denotes the first passage time distribution de￾scribing the first coordinate being killed, FN−1 denotes the first passage time distribution describing the second coordinate being killed in the original sys￾tem, that is the first coordinate to be killed in a system of N −… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Comparison between a Monte Carlo simulation (10, 000 realizations) and the theoretical survival functions of two trivariate Poisson processes. S 3 (t) in green, S 2 (t) in red, and S 1 (t) in blue. The dots are obtained using the simulation’s results while the solid li…

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