{"id":"58ad5525-d107-40f8-83f3-93a0aff0dff3","arxiv_id":"2509.02347","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Recursive path-sum formulas give the survival function and first passage time distribution of the nth surviving coordinate of a Markov process with a killing boundary, applied to Poisson and single-file diffusion models and to nth-to-default CDS pricing.","lead":"This paper derives recursive formulas for the probability that the nth of several interacting random processes survives past a time t, and for the distribution of when the nth one dies. The author applies them to a two-particle diffusion problem and to pricing nth-to-default credit default swaps.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition 1 (Eq. 44) silently assumes the ordered kill times have independent marginal densities, omitting the required integration over survivor states at intermediate kill times; Theorem 2 is therefore underived for N≥3.","rationale":"The paper's central claim is Theorem 2/Definition 1: a recursive, path-sum formula for the nth-survival function of an N-coordinate Markov process with killing. For this to hold for general N, the expression for I_g(t) in Eq. (44) must be the exact contribution of a kill-history path. The strongest assumption in that expression is the factorization of the joint law of the ordered kill times and survivor positions into a product of marginal first-passage densities F_k times one conditional density at the final kill, with no integration over survivor states at earlier kills. In a strong Markov process, the state at τ_1 (all surviving coordinates) is needed to determine the law of τ_2; omitting that integral is correct only under a special independence/factorization property. The proof of Theorem 2 is a sketch, and Remark 2 does not supply the missing conditioning. The reader's verdict CONDITIONAL is appropriate: the two-coordinate theorem is supported by explicit calculation and Monte Carlo, and the trivariate Poisson examples work because the independent Poisson drivers make the factorization exact along each path. But the general diffusion/jump claim is not derived. The concrete test proposed—N=3 single-file diffusion—would settle whether the factorization fails for a genuinely coupled system. I agree with the reader's weak-assumption diagnosis; Eq. (7) is a separate concrete error in the proof of Theorem 1, but the load-bearing issue for the paper's main theorem is the kill-history factorization.","tokens_in":14020,"tokens_out":12953,"duration_ms":152412,"concrete_test":"Take N=3 hard-core Brownian particles in [0,1] with killing at 1 (single-file diffusion), initial uniform order. Implement Eq. (45) for S_1(t) using Definition 1's integrand for the two-kill paths, computing P^N_n and F_k as the paper's Eq. (44) prescribes. Compare with Brownian dynamics simulation (10^6 trajectories, dt=10^-6) over t∈[0.01,10]. If S_1(t) deviates beyond Monte Carlo error (≈10^-3), the missing integration over the survivor state at τ_1 is confirmed; if it matches, the factorization holds at least for this coupled system and the concern is refuted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eq. (44) defines the path contribution I_g(t) as an integral over the final survivor position y only, with the joint law of the kill history represented by the product F_{n+1}(τ_{n-1})···F_N(τ_1) times a conditional density P^N_n(y,τ_{n-1}|history). The proof of Theorem 2 is a one-sentence induction sketch, and Remark 2 does not define how the F_k depend on the preceding state. For a general Markov process with killing, after the first kill at τ_1 the surviving coordinates occupy a state Y_1∈R^{N-1}; the density of the second kill time τ_2 and the subsequent state is obtained by evolving from Y_1, so Y_1 must be integrated. Eq. (44) has no such intermediate integral. The factorization into marginals would require the transition kernel of the reduced system to be independent of Y_1 once conditioned on the kill event, which is not true for interacting diffusions (e.g., single-file diffusion with N≥3). The two-coordinate case (Theorem 1) is immune because only one kill precedes the final survivor; the trivariate Poisson examples are also special because the independent driving Poisson processes make the relevant kill-time product exact along each path. Thus the central claim is not established for the general jump/diffusion setting advertised.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14284,"tokens_out":8036,"duration_ms":92500,"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":[{"comment":"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).","section":"§3, Definition 1, Eq. (44)"},{"comment":"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.","section":"§2, Theorem 1, Eq. (7)"},{"comment":"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'.","section":"§2.2, Eq. (37)"}],"minor_comments":[{"comment":"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.","section":"§3.1, Eq. (50)"},{"comment":"In the third displayed path, 'X0=1' should presumably be 'X1=1' to match the notation of the other paths.","section":"§3.1, Eq. (49)"},{"comment":"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.","section":"§2, Eq. (7)"}],"recommendation":"reject","confidential_remarks":"The self-citations [21,22] are essential to the single-file example and the paper does not fully reproduce the definitions needed to verify Eq. (34). The manuscript's main claim about general N-coordinate processes is not supported by the derivation; a resubmission restricted to the two-coordinate result and the Poisson/exponential models could be viable, but the current version overclaims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short take: the two-coordinate result is fine and the examples are honest, but the many-coordinate theorem is asserted rather than proved, and the proof of the two-coordinate theorem contains a wrong equation. Still worth a referee, because the gap might be repairable.\n\nWhat is actually new: the graph-organized recursive formula (Theorem 2) for the nth survival function, which is not in the literature. The bivariate and trivariate Poisson closed forms are also new, and the check against Locatelli et al. for single-file diffusion is reassuring. I re-derived the key integrals (50) and (52) and they evaluate correctly. The author is candid about the method being more complicated than existing closed forms and slower than simulation, which is refreshing.\n\nThe soft spots are in the proofs, not the examples. Eq. (7) says P(τ2=τm)=1−S2(t). That is not a probability: the left side is a constant, the right side is a function of t. The renewal structure of Theorem 1 is standard, but the proof as written does not support it. More importantly, Definition 1 in Eq. (44) defines each path contribution as an integral over the final survivor position only, with the kill history represented by a product of marginal first-passage densities. For N≥3, after the first kill the surviving coordinates occupy a state, and the distribution of the next kill time depends on that state. That state needs to be integrated over. The formula as written skips those intermediate integrals. It works for the trivariate Poisson because the independent driving processes make the product exact; it is not justified for interacting diffusions. The paper's own remark that the single-file result is identical to Locatelli et al. actually exposes the point: in two dimensions there is only one survivor state to integrate over, which is exactly why Theorem 1 holds. The CDS section is a simple application of an equivalent intensity model, with no new financial content.\n\nBottom line: a serious referee should look at this, mainly to determine whether the missing intermediate-state integrals can be added to Eq. (44) or whether the theorem needs to be restricted. As it stands, I would not rely on the general claim.","headline":"Two-coordinate core and examples are solid, but the many-coordinate theorem is asserted rather than proved; worth a referee, not a desk reject.","tokens_in":14864,"tokens_out":4104,"would_cite":false,"duration_ms":46607,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J70","60J25","60J27","60K35","82C31","91G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"A recursive formula gives the nth survivor's survival function for many-body Markov processes with killing boundaries.","keywords":["survival function","first passage time","order statistics","killing boundary","many-body Markov process","nth-to-default CDS","single file diffusion","multivariate Poisson process"],"falsifier":"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.","tokens_in":13725,"feed_emoji":"🎲","tokens_out":3525,"duration_ms":42188,"temperature":0.7,"pith_summary":"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.","feed_headline":"Exact formula orders first-passage deaths of many-body Markov processes","feed_subtitle":"Nested kill-time integrals give the nth survivor's survival function, with applications to single-file diffusion and CDS pricing.","key_machinery":"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).","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Locatelli et al. supply the reflection-principle result (Eq. 40) that the paper's single-file diffusion formula (Eq. 37) is shown to reproduce.","marker":"[27]"},{"why":"Marshall and Olkin's multivariate exponential distribution is the basis for the trivariate Poisson model with killing barrier at 1, used in the CDS pricing example.","marker":"[31]"},{"why":"Giesecke's exponential intensity model provides the equivalent framework for multi-name CDS valuation and the fee-leg and protection-leg formulas.","marker":"[6]"},{"why":"Lapolla and Godec's BetheSF method supplies the exact eigenfunction expansion of the single-file propagator used in the diffusion example.","marker":"[22]"},{"why":"Kawamura's trivariate Poisson structure gives the joint probability mass function used in the many-coordinate example.","marker":"[17]"},{"why":"Holgate's bivariate Poisson distribution is the foundational model for the two-coordinate jump example.","marker":"[12]"}],"fun_headline_variants":["Recursive formula gives nth survivor's survival in many-body processes","Nested integrals price nth-to-default CDS from Markov dynamics","Exact recursion for nth first-passage times in jump and diffusion processes","Many-body killing barrier: recursion for nth survivor's distribution"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Recursive formula gives nth survivor's survival in many-body processes","Nested integrals price nth-to-default CDS from Markov dynamics","Exact recursion for nth first-passage times in jump and diffusion processes","Many-body killing barrier: recursion for nth survivor's distribution"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000329,"raw_usage":{"total_tokens":1635,"prompt_tokens":667,"completion_tokens":968,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":411,"completion_tokens_details":{"reasoning_tokens":896}},"tokens_in":411,"tokens_out":968,"duration_ms":8791,"temperature":1.0,"reasoning_tokens":896,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T11:39:23.199654+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Active Brownian particles escaping a channel in single file","cited_arxiv_id":null,"evidence_quote":"Locatelli et al. supply the reflection-principle result (Eq. 40) that the paper's single-file diffusion formula (Eq. 37) is shown to reproduce."},{"cited_title":"Marshall and Ingram Olkin","cited_arxiv_id":null,"evidence_quote":"Marshall and Olkin's multivariate exponential distribution is the basis for the trivariate Poisson model with killing barrier at 1, used in the CDS pricing example."},{"cited_title":"A Simple Exponential Model for Dependent Defaults","cited_arxiv_id":null,"evidence_quote":"Giesecke's exponential intensity model provides the equivalent framework for multi-name CDS valuation and the fee-leg and protection-leg formulas."},{"cited_title":"BetheSF: Efficient computation of the exact tagged- particle propagator in single-file systems via the Bethe eigenspectrum","cited_arxiv_id":null,"evidence_quote":"Lapolla and Godec's BetheSF method supplies the exact eigenfunction expansion of the single-file propagator used in the diffusion example."},{"cited_title":"The structure of trivariate Poisson distribution","cited_arxiv_id":null,"evidence_quote":"Kawamura's trivariate Poisson structure gives the joint probability mass function used in the many-coordinate example."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Holgate's bivariate Poisson distribution is the foundational model for the two-coordinate jump example."}],"review_version":1}