{"id":"d15ad7cb-92aa-4093-aa27-651b9d245e3a","arxiv_id":"2607.10708","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Self-exciting PDMPs with endogenous intensity resetting admit unique invariant measures with explicit densities, enabling closed-form optimal intervention thresholds for exponential jumps.","lead":"The paper builds a new class of self-exciting piecewise-deterministic Markov processes with endogenous resetting and proves they are ergodic with an explicit invariant density. It then solves a long-run average cybersecurity control problem that chooses the intervention threshold to balance technology cost against average attack losses.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the reader's already-flagged finiteness assumption.","rationale":"The central claim is the existence/uniqueness of an explicit invariant measure for Y under a.s.-finite endogenous reset. The manuscript constructs the PDMP via Hille–Yosida, derives the generator, proves regeneration implies Harris recurrence (Theorem 4.1), and solves the resulting integro-differential balance equations case-by-case, obtaining recursive integral formulae that specialise cleanly for exponential jumps. All steps are standard continuous-time Markov-process arguments; the appendices supply the detailed ODE integrations and boundary matching. The only place the argument can fail is precisely when TA=∞ with positive probability, which the authors already characterise and illustrate. No hidden boundedness assumption, no circular use of ergodicity, and no algebraic inconsistency in the special-case control solution were found. Consequently the reader's CONDITIONAL verdict (high confidence, low correctness risk) already places the correct weight on the finiteness hypothesis; no adjustment is warranted.","tokens_in":27538,"tokens_out":596,"duration_ms":10075,"concrete_test":"Independently re-derive the density π(0,λ) of Theorem 4.2 from the generator balance (A.3)–(A.6) without using the integrating factor F; if the same power-law expression is recovered, the ODE step is solid. Separately, plug the exponential parameters of Figure 1 into the three-series test of Theorem 3.1 (or the mgf condition of Theorem 3.2) and confirm TA is a.s. finite, so that the plotted density is justified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest_assumption correctly isolates the load-bearing condition: Assumption 4.1 (P(TA<∞)=1 for all y) is necessary for Harris recurrence and the explicit densities of Theorems 4.2–4.4. The paper itself supplies the counter-example (Section 3, uniform jumps of size 2^{-n}) and the necessary/sufficient conditions (Theorems 3.1–3.3) under which the assumption holds or fails. No further internal gap, circularity, or derivation error appears in the generator construction (Theorem 2.1), the martingale decompositions, the recursive ODEs for the densities, or the closed-form exponential case (Theorem 5.1 and Proposition 5.1). The control problem is correctly reduced to the stationary mean of Λ via Corollary 2.1 and ergodicity. The only practical caveat is that applied users must verify the Cramér-type conditions before trusting the stationary formulae; this is already noted by the reader.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper constructs a piecewise-deterministic Markov process X=(N,Λ,C) with endogenous resetting of the intensity Λ at a threshold A, motivated by self-exciting cyber-attack models. Existence of the process is obtained via Hille–Yosida (Theorem 2.1). Under the assumption that the reset time TA is a.s. finite, Y=(N,Λ) is shown to be Harris recurrent with unique (up to scaling) invariant measure Π whose density is given by explicit recursive integral formulae (Theorems 4.2–4.4). For exponential jumps with λ o=β the density is closed-form (Theorem 5.1), and the long-run average control problem of choosing A is solved explicitly in a special case (Proposition 5.1).","tokens_in":27766,"tokens_out":1150,"duration_ms":12709,"significance":"The construction fills a genuine gap in the PDMP literature: classical Davis theory does not allow boundary-to-boundary transitions of the type required here (Remark 2.3). The regeneration structure is exploited cleanly via Kaspi–Mandelbaum to obtain both uniqueness and explicit densities without Lyapunov functions, which is a non-trivial technical contribution. The cyber-security control problem is reduced rigorously to the stationary mean of Λ (Corollary 2.1), and the exponential case yields a fully closed-form optimiser. These results are of interest both to pure PDMP theory and to applied cyber-risk modelling.","major_comments":[{"comment":"Assumption 4.1 (P(TA<∞)=1 for every starting point) is load-bearing for Harris recurrence and for the densities of Theorems 4.2–4.4. The paper correctly supplies necessary and sufficient conditions (Theorems 3.1–3.3) and a counter-example (Section 3). For the control problem of Section 5, however, the reader is left without a practical check that the chosen A and the Exp(θ) jumps satisfy the Cramér-type condition of Theorem 3.2. A short remark or corollary verifying that, under Assumption 5.1, E[TA]<∞ for every A>β would close this gap and make the explicit optimiser of Proposition 5.1 fully rigorous.","section":null},{"comment":"In the proof of Theorem 4.2 (Appendix A.2) the singular measure bπ' is asserted to be supported only at {β}. While the argument via separation of measures is standard, the subsequent claim that limλ↑β(β−λ)π(n,λ)=0 (needed for integrability) relies on an induction that is only sketched. A one-line verification that the induction base holds for the explicit π(0,·) of (A.7) would remove any residual doubt about the construction of the probability measure.","section":null}],"minor_comments":[{"comment":"Page 1 and throughout: several references carry future dates (IBM 2025, NCSC 2025, WEF 2026, UK Cyber Action Plan 2026). These should be checked for consistency with the arXiv submission date or replaced by the latest publicly available versions.","section":null},{"comment":"Equation (2.11) and the subsequent generator of Y (4.1): the notation Gℓ(n,(A−λ)-) for the left limit is used without a formal definition; a short sentence after (2.12) would help.","section":null},{"comment":"Figure 1 caption: the parameter values β=1, A=2, θ=0.1, α=1.1 produce a density that appears to explode near β; a brief comment on the integrable singularity (cf. (A.54)) would aid the reader.","section":null},{"comment":"Proposition 5.1: the lengthy algebraic expression for A* (5.7) is hard to verify by hand. Supplying a short Mathematica/SymPy notebook or a numerical check against the first-order condition would increase reproducibility.","section":null},{"comment":"Typographical: “formulates and solves a control problem about the tractable case” (end of Introduction) should read “formulates and solves a control problem for the tractable case”; “looses” (p. 5) should be “losses”.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is solid and the technical contribution is real. The only reason I do not recommend outright acceptance is the minor but load-bearing verification gap around Assumption 4.1 in the exponential case; once that is closed the paper is ready. Fit for a probability journal with applied flavour is excellent."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new piece is a PDMP with endogenous boundary-to-boundary resetting (intensity hits A and is sent straight to λ_o). That sits outside classical Davis theory and outside the Evans–Majumdar exogenous-resetting literature, and they make the distinction carefully in Remark 2.3. Under the a.s.-finite-reset assumption they get Harris recurrence via the regeneration structure and then write down the invariant densities of Y=(N,Λ) by solving the generator equation, with the singular mass at the mean-reversion level handled properly. The exponential-jump case (λ_o=β) collapses to a closed form involving a Kummer function and an explicit optimal threshold A* for the long-run average cost. That is real, usable output.\n\nWhat they do well: the Hille–Yosida argument for the semigroup is standard but complete; the martingale decompositions for N, Λ and C are clean; the recursive integral formulae for the densities (Theorems 4.2–4.4) and the special-case density (Theorem 5.1) are derived without circularity; the control problem reduces correctly to the stationary mean of Λ via the ergodic theorem. The ruin-theory flavour of the conditions for E[T_A]<∞ is natural and they supply both necessary/sufficient criteria and a counter-example when the jumps are too small.\n\nThe soft spot is exactly the one the reader flagged: everything rests on Assumption 4.1 (P(T_A<∞)=1 for every starting point). If the intensity jumps are small relative to A the process never regenerates and the stationary formulae collapse. The paper itself tells you when that happens, so it is not a hidden flaw, but applied users who pick parameters without checking the Cramér-type conditions will get nonsense. That is a practical caveat, not a mathematical hole.\n\nThis is for people who work on PDMPs, self-exciting point processes, or cyber-risk modelling and who want an explicit stationary object rather than another abstract existence theorem. The math is solid, the citations are appropriate, and there is no load-bearing circularity. I would send it to referees; the finiteness assumption just needs to be highlighted more loudly for the applied audience.","headline":"Solid, explicit PDMP construction with endogenous resetting and a clean special-case control solution; the only real caveat is the already-flagged finiteness assumption on reset times.","tokens_in":28373,"tokens_out":598,"would_cite":true,"duration_ms":9190,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J25","60G55","93E20","91B30"],"pacs":[],"model":"grok-4.5","headline":"A new class of self-exciting Markov processes with endogenous resetting has an explicit invariant density and yields an optimal cyber-intervention threshold.","keywords":["piecewise-deterministic Markov process","endogenous resetting","self-exciting intensity","Harris recurrence","invariant measure","cyber-risk control","long-run average cost"],"falsifier":"Take the concrete counter-example of Section 3 (uniform jumps of size at most 2^{-n} and A larger than the sum of all possible jumps): if the intensity path never reaches A, the empirical occupation measure of (N, Λ) fails to converge to the claimed Π.","tokens_in":28438,"feed_emoji":"🔐","tokens_out":622,"duration_ms":7212,"temperature":0.7,"pith_summary":"The paper builds a piecewise-deterministic Markov process that tracks the number of cyber attacks, their stochastic intensity, and cumulative loss, with intensity jumps that are self-exciting and with an endogenous reset that fires the moment intensity hits a chosen threshold A. Under conditions resembling the Cramér–Lundberg condition of ruin theory, the process regenerates almost surely, is Harris recurrent, and admits a unique (up to scaling) invariant measure whose density is written out by recursive integral formulae. Because the long-run average loss is then simply the stationary mean intensity times mean jump size, the firm’s problem of choosing A reduces to a one-dimensional calculus exercise that balances the cost of a more sensitive detection posture against that average intensity. When intensity jumps are exponential the density becomes closed-form, the stationary mean intensity is elementary, and the optimal A can be displayed explicitly.","feed_headline":"Cyber-intensity process resets endogenously and has closed-form law","feed_subtitle":"Explicit stationary density turns the optimal detection threshold into a one-dimensional calculus problem","key_machinery":"The endogenous resetting construction together with the Harris-recurrence argument of Kaspi–Mandelbaum: regeneration at the first hitting time of the intensity threshold supplies both uniqueness of the invariant measure and an explicit integral formula for its density.","core_discovery":"Under the assumption that the endogenous reset time TA is almost surely finite, the two-dimensional process Y = (N, Λ) is Harris recurrent and possesses a unique (up to scaling) invariant measure Π whose density is given explicitly by the recursive integral expressions of Theorems 4.2–4.4; for exponential jumps the density and the optimal intervention threshold are available in closed form.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Endogenous resets give cyber intensity process unique explicit stationary density","Piecewise deterministic Markov process with resets is ergodic under finite TA","Self-exciting cyber attack model admits closed-form invariant measure for exponential jump","Optimal intervention threshold for resettable intensity reduces to calculus problem","Harris recurrent two-dimensional process Y has recursive integral density formulas"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The whole invariant-measure construction collapses if intensity jumps are too small relative to the chosen threshold, so that the process never hits the reset boundary.","fun_headline_variants_meta":{"raw":{"variants":["Endogenous resets give cyber intensity process unique explicit stationary density","Piecewise deterministic Markov process with resets is ergodic under finite TA","Self-exciting cyber attack model admits closed-form invariant measure for exponential jumps","Optimal intervention threshold for resettable intensity reduces to calculus problem","Harris recurrent two-dimensional process Y has recursive integral density formulas"]},"model":"grok-4.5","effort":"low","cost_usd":0.004064,"raw_usage":{"total_tokens":1156,"prompt_tokens":622,"num_sources_used":0,"completion_tokens":90,"cost_in_usd_ticks":40640000,"prompt_tokens_details":{"text_tokens":622,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":444,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":622,"tokens_out":90,"duration_ms":4669,"temperature":1.0,"reasoning_tokens":444,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T09:50:50.969331+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Take the concrete counter-example of Section 3 (uniform jumps of size at most 2^{-n} and A larger than the sum of all possible jumps): if the intensity path never reaches A, the empirical occupation measure of (N, Λ) fails to converge to the claimed Π.","supporting_citations":[],"review_version":1}