{"id":"b2a027ef-911a-4180-a828-5df4eba1e5af","arxiv_id":"2507.11294","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Any Hawkes jump-diffusion with an integrable kernel can be approximated arbitrarily well by an augmented Markov jump-diffusion, and optimal control values converge under the same approximation.","lead":"This paper shows that a broad class of counting processes with history-dependent jump rates, called Hawkes processes, can be replaced by a slightly larger Markov process to any desired precision. That matters because it gives a practical route to optimal-control problems in finance, cyber-risk, and neuroscience where jump clustering and delayed excitation or inhibition are important.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.1's proof assumes WLOG f=0, but a running cost changes the value function; without a separate argument for f≠0 the control convergence claim is unproven as written.","rationale":"The reader's weakest_assumption is Assumption 1.3, a real scope restriction that the abstract's 'minimal integrability' wording downplays. I agree that Assumption 1.3 is load-bearing for the continuity estimates: without it, Theorem 2.4's proof cannot control the state-dependent intensity term with only L1 information. However, the sharpest internal flaw I find is in the control theorem. The proof of Theorem 4.1 explicitly declares f=0 'without loss of generality,' which is false: the value of a control problem with a non-zero running cost is not reducible to a terminal-cost problem by ignoring f, because f depends on the control and the state path. The proof then only establishes convergence for terminal costs, and the truncation step in Section 5.4 addresses only g. The paper's portfolio example in Section 4.3 has a running consumption term, so the gap directly affects the advertised application. This is not an objection to the Markov approximation itself, which is supported by a plausible contraction argument plus a density result. But as written, Theorem 4.1's statement for general running costs lacks proof. Because this is a fixable proof gap rather than a demonstrated falsehood, I do not change the reader's CONDITIONAL verdict; I would keep it conditional pending a repaired argument for f≠0 or an explicit restriction to Lipschitz running costs.","tokens_in":30656,"tokens_out":11583,"duration_ms":157924,"concrete_test":"Instantiate the framework with f≡1 and g≡0, with any dynamics satisfying Assumptions 1.1-1.3, e.g., the trivial dynamics X_t=x0. Then V0=T>0 by (4.3), but the f=0 reduction would predict V0=0. Re-running Section 5.4 with this f shows Step 1 bounds only E|g(X^n_T)-g(X_T)|=0 and never produces a bound on E∫_0^T |f(X^n_t)-f(X_t)|dt, so the proof cannot yield |V^n_0-V_0|→0 for running costs. A correct proof must add the auxiliary state Y_t=∫_0^t f(s,X_s,α_s)ds and handle its approximation; the test is whether such an augmented-state argument can be completed under the stated polynomial-growth assumptions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 5.4 the proof of Theorem 4.1 begins: 'For notational simplicity, we assume without loss of generality that f = 0.' This reduction is not valid. The value function in (4.3) is V0 = sup_α E[∫_0^T f(t,X_t,α_t)dt + g(X_T)]; setting f=0 changes the objective unless f is a constant independent of state and control, which is not assumed. Step 1 only proves E|g(X^n_T)-g(X_T)| ≤ ε for terminal costs, and Step 2 only truncates g. No term involving E∫|f(t,X^n_t,α_t)-f(t,X_t,α_t)|dt is ever estimated. Under the stated polynomial-growth condition on f, uniform convergence of the running cost under the same control α is not immediate; it requires additional regularity (e.g., Lipschitz continuity in x, uniformly in (t,a)) or a truncation argument that is not supplied. This gap is directly relevant to the paper's headline application: Section 4.3 is an infinite-horizon consumption problem with running utility U(ctXt), and its convergence argument invokes Theorem 4.1 for the finite-horizon running-cost value functions in (4.12) and (4.14). The theorem may be true and fixable by augmenting the state with Y_t=∫_0^t f(s,X_s,α_s)ds and proving convergence for the augmented terminal cost, but as written the proof does not establish the stated theorem for f≠0.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies jump-diffusions whose jump intensity is driven by a Hawkes process with a general integrable kernel. It proves an L1 continuity result for the state process with respect to the kernel (Theorem 2.4), then combines this with the density of sums of exponentials in L1 to obtain a Markov approximation: for any epsilon and horizon T one can replace the kernel by a finite sum of exponentials so that the corresponding state process satisfies E[sup_{0<=s<=T}|X_s - Xtilde_s|] <= epsilon (Theorem 3.1). This approximation is then applied to stochastic control: Theorem 4.1 claims convergence of value functions for controlled Hawkes jump-diffusions with running and terminal costs, and Section 4.3 presents a portfolio optimization example with log utility. The approximation argument is the core contribution; the control application, as written, has substantial gaps.","tokens_in":31054,"tokens_out":13018,"duration_ms":162237,"significance":"If Theorem 3.1 is correct, it gives a simple and broadly applicable route from Volterra-type Hawkes dynamics to Markovian dynamic programming and HJB methods, without restricting to completely monotone or otherwise special kernels. The explicit L1 kernel-continuity bounds with exponential dependence on the horizon are useful, and the proof strategy is elegant and largely self-contained. The paper also ships a numerical illustration that supports the plausibility of the approximation. However, the control part is currently not established as stated: the proof of Theorem 4.1 assumes away the running cost, the uniformity over controls is not proved, and the log-utility example does not satisfy the hypotheses of the theorem it invokes. These are fixable in a revision, but they materially limit the paper's claims as they stand.","major_comments":[{"comment":"The proof of Theorem 4.1 begins with 'without loss of generality f = 0', but f enters the value function (4.3) through the running cost, so this reduction changes the optimization unless f is constant in x and the control. Step 1 only controls E|g(X^n_T)-g(X_T)| and Step 2 only truncates g; no estimate is given for E[integral_0^T |f(t,X^n_t,alpha_t)-f(t,X_t,alpha_t)|dt]. Under the stated polynomial-growth assumption on f, such an estimate is not automatic and requires an additional Lipschitz or truncation argument. This gap propagates to Section 4.3, where Theorem 4.1 is invoked for finite-horizon running-cost problems (4.12) and (4.14). The theorem may be salvageable by augmenting the state with Y_t = integral_0^t f(s,X_s,alpha_s)ds and proving convergence for the augmented terminal cost, or by proving a uniform running-cost estimate, but as written the claimed convergence for f != 0 is unproved.","section":"Theorem 4.1 proof, Step 1"},{"comment":"Step 1 of the proof of Theorem 4.1 uses Theorem 2.4 to bound E|X^n_T - X_T| for an arbitrary control alpha, but the constants C1 and C2 in Theorem 2.4 depend on the Lipschitz constants and the stability margin of the coefficients, and therefore can depend on alpha. The definition of A in Section 4.1 only requires each alpha to satisfy Assumptions 1.1-1.3; it does not impose bounds uniform over alpha. Passing from E|g(X^n_T)-g(X_T)| <= epsilon for each fixed alpha to |V^n_0 - V_0| <= epsilon requires a choice of n that works uniformly over alpha in A. This uniformity is not established. A uniform version of the kernel-continuity bound, or a restriction of A to controls with uniform Lipschitz and stability constants, is needed.","section":"Section 4.3, Eqs. (4.12)-(4.14)"},{"comment":"The convergence argument for the log-utility example invokes Theorem 4.1 for the finite-horizon value functions V^epsilon_0 and V^{n,epsilon}_0, whose running payoff is f(t,x,(c,omega)) = e^{-rho t} log(c x). This f is neither continuous with polynomial growth in x uniformly in the control, nor Lipschitz in x uniformly in (t,a), so the hypotheses of Theorem 4.1 are not satisfied. In addition, the inequalities V^epsilon_0 <= V_0 and V^{n,epsilon}_0 <= V^n_0 used after (4.13) require the integrand e^{-rho t} U(c_t x_t) to be nonnegative; for U = log this can fail unless additional assumptions are imposed. The example therefore needs its own verification or a modified theorem that covers it.","section":"Section 4.3"}],"minor_comments":[{"comment":"The abstract promises 'minimal integrability conditions', but Assumption 1.3 is a structural dichotomy (bounded jump rate, or state-independent gamma and nu) that is used in every contraction estimate, starting at inequalities (5.1)-(5.2). This should be stated in the introduction as a substantive restriction, not presented as part of the integrability conditions.","section":"Abstract"},{"comment":"The sentence 'the univariate proofs extend to the multivariate SDE (1.2)' is an assertion rather than a proof. Since the formal statements are multivariate, it would be preferable either to give the matrix/spectral-radius extension explicitly or to state the main theorems in the univariate setting.","section":"Section 1"},{"comment":"The displayed approximating kernel phi^(3)(t) = -0.82 e^{-0.5t} + 0.58 e^{-t} + 1.39 e^{1.5t} contains a growing exponential e^{1.5t}; this cannot be a sum of decaying exponentials and is not in L1(R+). This is presumably a typo for e^{-1.5t}, but as printed it contradicts the admissibility condition for the approximating kernel.","section":"Section 3"},{"comment":"In the estimate for |V^M_0 - V_0|, the Markov inequality gives a bound of order M^{-1/2}, not M^{-1}; the displayed 'C/M' should be 'C M^{-1/2}'. The convergence conclusion is unaffected but the displayed rate is incorrect.","section":"Section 5.4"},{"comment":"There are several typographical and notational slips: 'Bukholder-Davis-Gundy' in Section 5.1, 'indepdendent' in Assumption 1.1, 'generaility' in Section 5.4, and Proposition 1.4 states E[sup_t |X_t|^p] <= +infinity, which should say that this expectation is finite.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The core Markov approximation result and its proof appear sound and useful; the referee report focuses on Section 4 because the control theorem and the application are not established as written. The paper is likely publishable after the control section is repaired, either by proving the missing running-cost and uniformity estimates or by narrowing the statements to the cases that are actually proved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: the paper proves a clean Markov approximation theorem for Hawkes jump-diffusions with general integrable non-monotone kernels, and then applies it to control. The approximation result (Theorem 3.1) is genuinely useful and the proof is largely sound. But the control convergence theorem has a real gap: the proof of Theorem 4.1 assumes f=0 without justification, and the running-cost case is not actually proved. The paper also overstates the generality in the abstract.\n\nWhat is new: the kernel-continuity estimate (Theorem 2.4) for non-monotone L1 kernels with explicit time-horizon dependence. The density of exponentials in L1 is classical, and the idea of exponential approximation appears in the completely monotone and Erlang-sum settings, but combining it with a direct continuity bound for general kernels is a nice, practical bridge to Markov control. The paper is honest about relying on external density results and on [25] for a second-moment bound.\n\nThe soft spots: first, Assumption 1.3 (either psi bounded above, or gamma and nu state-independent) is invoked throughout the contraction arguments and is not implied by integrability. The abstract's \"minimal integrability conditions\" is misleading; this assumption excludes models with unbounded state-dependent jump rates. Second, the proof of Theorem 4.1 explicitly says 'without loss of generality f=0' in Section 5.4. That is not a WLOG. The value function includes a running-cost integral; setting f=0 changes the objective. Step 1 only handles terminal cost g, Step 2 truncates g, and no term involving the running cost is ever estimated. This is a load-bearing issue for the advertised control application, since Section 4.3 invokes Theorem 4.1 for finite-horizon running-cost problems (4.12) and (4.14). The theorem might be fixable by augmenting the state with the running-cost integral, but as written the proof does not establish it. Third, the paper announces a multivariate framework and then restricts to the univariate case 'to reduce clutter'; the theorems are only proved in the univariate case. That is a gap between the claim and the proof.\n\nThe numerical example is illustrative, not a benchmark; the approximation coefficients and decay rate are free. That is fine for an illustration but should not be oversold.\n\nWho should read this: anyone working on control or simulation of Hawkes processes with non-exponential kernels. The approximation theorem itself deserves a serious referee; the control part needs a genuine fix for f≠0. My recommendation: send it to referees, but make clear that the running-cost gap and the univariate/multivariate mismatch must be addressed before publication. The core idea is solid and worth engaging with.","headline":"A useful Markov approximation theorem for general Hawkes kernels, but the control convergence proof has a real gap: the WLOG f=0 reduction in Theorem 4.1 is invalid.","tokens_in":31543,"tokens_out":2847,"would_cite":true,"duration_ms":31185,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G55","93E20","45D05","60H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that every Hawkes jump-diffusion with a general integrable kernel can be approximated on a finite horizon by a Markov jump-diffusion driven by exponential-sum kernels, with explicit error bounds and convergent control…","keywords":["Hawkes process","jump-diffusion SDE","Markov approximation","sums of exponentials","Volterra-type dynamics","stochastic control","kernel continuity","dynamic programming"],"falsifier":"One concrete test: take the same SDE with a state-dependent jump amplitude $\\gamma(x)$ and an unbounded jump-rate map $\\psi(x)=x$, so that Assumption 1.3 fails, and compute $\\mathbb{E}[\\sup_{0\\le s\\le T}|X_s-\\tilde X_s|]$ for a sequence of exponential-sum kernels $\\phi_n$ with $\\|\\phi_n-\\phi\\|_1\\to 0$. If the $L^1$ closeness between $X$ and $\\tilde X$ fails to go to zero, or if the constants $C_1,C_2$ in Theorem 2.4 blow up as $\\psi$'s Lipschitz constant grows, the central approximation claim collapses in that regime.","tokens_in":30462,"feed_emoji":"🧮","tokens_out":6813,"duration_ms":71270,"temperature":0.7,"pith_summary":"Volterra-type Hawkes jump-diffusions, whose jump intensity is driven by a general integrable kernel, are in general path-dependent and escape classical Markov control methods. This paper shows that any such kernel can be replaced, to arbitrary precision on a finite time horizon, by a sum of exponentials; the resulting process becomes a Markov jump-diffusion with auxiliary state variables. The state processes are shown to be continuous in the kernel in an $L^1$ sense, with an explicit error bound that grows exponentially in the horizon. Consequently, optimal control values of the original problem are approximated by values of Markov control problems, with error controlled by the kernel approximation error.","feed_headline":"Any integrable Hawkes kernel can be swapped for exponentials","feed_subtitle":"This unlocks dynamic programming for Volterra-type Hawkes control problems with explicit error bounds.","key_machinery":"The load-bearing construction is the replacement of the kernel by a finite sum of exponentials $\\phi_n(t)=\\sum_{k=1}^n \\eta_k e^{-\\beta_k t}$. For such kernels the intensity takes the form $\\lambda_t = \\lambda_\\infty + \\psi\\big(\\sum_{k=1}^n \\eta_k \\xi^k_t\\big)$ with $\\xi^k_t = \\int_0^{t-}\\int e^{-\\beta_k(t-s)} b(y)\\nu(s,X_{s-})\\mathbf{1}_{\\theta\\le\\lambda_s}\\Pi(ds,d\\theta,dy)$, and the flow property of exponentials gives the linear dynamics $d\\xi^k_t = -\\beta_k\\xi^k_t\\,dt + \\text{ jump term}$. The augmented process $(X,\\xi^1,\\dots,\\xi^n)$ is Markov with an explicit infinitesimal generator. Two facts carry the argument: sums of exponentials are dense in $L^1(\\mathbb R_+)$ for any fixed $\\beta>0$ (not only for completely monotone kernels), and Hawkes jump-diffusions are continuous in their kernel, via Theorem 2.4, $\\mathbb{E}[\\sup |\\tilde X-X|] \\le C_1\\|\\tilde\\phi-\\phi\\|_1 e^{C_2 T}$, proved by a contraction argument using Assumption 1.3. The same exponential dynamics give an $O(nT)$ simulation cost instead of $O(T^2)$ for the original Volterra SDE.","core_discovery":"The paper's central claim is Theorem 3.1: under Lipschitz regularity and a stability condition on the Hawkes kernel, for any $\\varepsilon>0$ and horizon $T$, a solution $(X,\\lambda)$ of a Hawkes jump-diffusion with a general integrable (possibly non-monotone) kernel $\\phi$ is within $\\varepsilon$ in $\\mathbb{E}[\\sup_{0\\le s\\le T}|X_s-\\tilde X_s|]$ of a $(n{+}1)$-dimensional Markov jump-diffusion $(\\tilde X,\\xi^1,\\dots,\\xi^n)$ whose kernel is a linear combination of exponentials. The estimate is quantitative: $\\mathbb{E}[\\sup |\\tilde X-X|] \\le C_1\\|\\tilde\\phi-\\phi\\|_1 e^{C_2 T}$, and it rests on an $L^1$ continuity theorem for the state process in the kernel. For stochastic control, Theorem 4.1 shows that the value of the Volterra-type control problem is the limit of the Markov value functions, and that with Lipschitz costs the error $|V^n_0 - V_0|$ is bounded by $\\|\\phi_n-\\phi\\|_1 C_1 e^{C_2 T}$. The result therefore converts a class of path-dependent control problems into standard Hamilton-Jacobi-Bellman problems with auxiliary dimensions.","pith_inferences":["Beyond the paper: the same exponential-sum lifting could make rough-volatility-type kernels (integrable power kernels, not necessarily completely monotone) accessible to Markovian approximation on finite horizons, since only $L^1$ integrability and the stability condition are used.","Beyond the paper: the approximation does not optimise over the decay parameter $\\beta$; a testable extension is to select $\\beta$ from the kernel's effective tail length and compare the resulting $L^1$ error and the number of exponentials needed in practice.","Beyond the paper: the continuity-in-kernel theorem suggests an alternative numerical route, directly discretising the kernel with piecewise-exponential fits rather than global exponential sums, with the same error control once Assumption 1.3 holds.","Beyond the paper: for state-dependent intensities, the paper's $L^1$ contraction method indicates the error bound may be improvable from exponential to polynomial in $T$ when the baseline intensity $\\lambda_\\infty$ has additional contraction properties; that improvement is not claimed in the paper."],"forward_implications":["Markov approximation is available for non-monotone kernels too, so delayed excitation and inhibition patterns, such as refractory periods in neuronal networks, fall inside the dynamic programming framework.","Numerical simulation of the original Volterra Hawkes SDE costs $O(T^2)$; simulating the exponential-sum system costs $O(nT)$, a strict improvement for large horizons.","The approximation error is explicit: $\\mathbb{E}[\\sup |X-\\tilde X|] \\le C_1\\|\\tilde\\phi-\\phi\\|_1 e^{C_2 T}$, so the number of exponentials $n$ needed for a target precision is determined by the kernel approximation problem.","Optimal control values converge: $|V^n_0 - V_0| \\le \\|\\phi_n-\\phi\\|_1 C_1 e^{C_2 T}$ under Lipschitz costs, giving a quantitative HJB-based route to nearly optimal strategies for Volterra-type Hawkes control problems.","The value function of the Markov problem solves an HJB equation in $n+1$ dimensions, so existing numerical dynamic programming machinery, including neural-network approximations, applies."],"supporting_citations":[{"why":"Introduces the Hawkes self-exciting point process that motivates the dynamics studied here.","marker":"[36]"},{"why":"Supplies the density of finite sums of exponentials in $L^p(\\mathbb R_+)$, the approximation ingredient in Theorem 3.1.","marker":"[39]"},{"why":"Provides the $L^1$ approximation by exponential polynomials used in the proof of Theorem 3.1.","marker":"[6]"},{"why":"Exponential-kernel Hawkes portfolio optimization problem that the paper generalizes to nonexponential kernels.","marker":"[4]"},{"why":"Empirical evidence that a non-exponential kernel $\\eta t e^{-\\beta t}$ fits cybersecurity data, motivating general kernels.","marker":"[15]"},{"why":"Companion work supplies moment bounds and functional approximation lemmas used in Appendices A and B.","marker":"[25]"},{"why":"Contraction argument for diffusion SDEs with jumps adapted in the proof of Proposition 1.4.","marker":"[32]"},{"why":"Reference for dynamic programming and HJB theory for Markov jump-diffusions used in Section 4.","marker":"[43]"}],"fun_headline_variants":["Any integrable Hawkes kernel can be swapped for exponentials","Approximate Hawkes kernels with exponentials for Markov control","Exponential swaps make Hawkes jump-diffusions Markov tractable","Hawkes kernels approximated by exponentials for optimal control","Reduce Hawkes jump-diffusions to Markov form via exponentials"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The result requires Assumption 1.3: either the jump-rate function $\\psi$ is bounded above, or the jump-amplitude $\\gamma$ and the impact function $\\nu$ do not depend on the state variable $X$. Every contraction estimate in the kernel-continuity theorem uses this to control the term involving $|\\tilde\\lambda-\\lambda|$; without it, the $L^1$ continuity of $X$ in the kernel, the basis of the Markov approximation, is not established.","fun_headline_variants_meta":{"raw":{"variants":["Any integrable Hawkes kernel can be swapped for exponentials","Approximate Hawkes kernels with exponentials for Markov control","Exponential swaps make Hawkes jump-diffusions Markov tractable","Hawkes kernels approximated by exponentials for optimal control","Reduce Hawkes jump-diffusions to Markov form via exponentials"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000175,"raw_usage":{"total_tokens":1258,"prompt_tokens":891,"completion_tokens":367,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":507,"completion_tokens_details":{"reasoning_tokens":282}},"tokens_in":507,"tokens_out":367,"duration_ms":4908,"temperature":1.0,"reasoning_tokens":282,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:15:06.216584+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete test: take the same SDE with a state-dependent jump amplitude $\\gamma(x)$ and an unbounded jump-rate map $\\psi(x)=x$, so that Assumption 1.3 fails, and compute $\\mathbb{E}[\\sup_{0\\le s\\le T}|X_s-\\tilde X_s|]$ for a sequence of exponential-sum kernels $\\phi_n$ with $\\|\\phi_n-\\phi\\|_1\\to 0$. If the $L^1$ closeness between $X$ and $\\tilde X$ fails to go to zero, or if the constants $C_1,C_2$ in Theorem 2.4 blow up as $\\psi$'s Lipschitz constant grows, the central approximation claim collapses in that regime.","supporting_citations":[{"cited_title":"Approximation with sums of exponentials in Lp[0, ∞)","cited_arxiv_id":null,"evidence_quote":"Supplies the density of finite sums of exponentials in $L^p(\\mathbb R_+)$, the approximation ingredient in Theorem 3.1."},{"cited_title":"A note on L1-approximations by exponential polynomials and Laguerre exponential polynomials","cited_arxiv_id":null,"evidence_quote":"Provides the $L^1$ approximation by exponential polynomials used in the proof of Theorem 3.1."},{"cited_title":"Portfolio choice in markets with contagion","cited_arxiv_id":null,"evidence_quote":"Exponential-kernel Hawkes portfolio optimization problem that the paper generalizes to nonexponential kernels."},{"cited_title":"Multivariate Hawkes pro- cess for cyber insurance","cited_arxiv_id":null,"evidence_quote":"Empirical evidence that a non-exponential kernel $\\eta t e^{-\\beta t}$ fits cybersecurity data, motivating general kernels."},{"cited_title":"Functional approximation of the marked Hawkes risk process","cited_arxiv_id":null,"evidence_quote":"Companion work supplies moment bounds and functional approximation lemmas used in Appendices A and B."},{"cited_title":"Applied stochastic control of jump diffusions","cited_arxiv_id":null,"evidence_quote":"Reference for dynamic programming and HJB theory for Markov jump-diffusions used in Section 4."}],"review_version":1}