{"id":"9f2b02fd-25f5-4991-aacb-f80142ebb0e6","arxiv_id":"1908.01068","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Ergodic control of compound-Poisson jump diffusions is shown to admit optimal stationary Markov controls and an HJB characterization under structural hypotheses weaker than near-monotonicity.","lead":"This paper proves ergodic control problems for jump diffusions with compound Poisson jumps admit optimal stationary controls characterized by a Hamilton-Jacobi-Bellman equation, even when running costs are not near-monotone. It matters because such problems arise in optimal scheduling of multiclass parallel server networks, where the new assumptions were previously thought to block the HJB approach.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Most load-bearing issue is the un-reduced role of Assumption 2.2: the HJB characterization in Theorem 5.2 depends on the coercive Lyapunov solution (2.9), and Remark 2.1 concedes this is not replaceable by stabilizability when the Lévy measure is unbounded.","rationale":"The paper flags the dependence of the HJB derivation on Assumption 2.2 in Remark 2.1 and verifies the assumption in the motivating queueing examples, so this is not a hidden flaw. However, the reader's weakest-assumption analysis is correct: for non-compactly supported Lévy measures, the vanishing-discount argument in Section 5 does not reduce (2.9) to the more primitive stabilizability hypothesis (2.10), and the bounds (5.5)-(5.6) rely on the full nonlocal Lyapunov inequality. A natural superpolynomial cost can make (2.9) unsolvable even when the model is stabilizable, which would leave the claimed general characterization inapplicable. That is a genuine scope limitation, not an internal inconsistency under the stated hypotheses. It supports the reader's CONDITIONAL verdict rather than demanding rejection.","tokens_in":23486,"tokens_out":30919,"duration_ms":325200,"concrete_test":"Test the claimed necessity in Remark 2.1: take d=1, b_hatv(x)=-x, σ=1, ν(dz)=e^{-z}1_{z≥0}dz (finite mass and second moment, unbounded support), and R_hatv(x)=e^x for x≥0 with a smooth extension. Show that any nonnegative coercive C^2 V solving (2.9) must grow at least exponentially with rate 1, while for such V the nonlocal term ∫_0∞ V(x+z)e^{-z}dz diverges, so (2.9) has no solution even though stabilizability and Assumption 2.1 are satisfied. Then attempt to prove the Lemma 5.1 bounds (5.5)-(5.6) directly from Assumption 2.1 and (2.10); if the bounds fail, Assumption 2.2 is essential and the HJB theorem is restricted exactly to the solvability class of (2.9).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Remark 2.1 is the key self-identified limitation. For non-compactly supported ν, the proof of Theorem 5.2 uses (2.9) to obtain the lower/upper bounds (5.5)-(5.6) in Lemma 5.1 and the local boundedness of ~I(V0+3V) in Lemma 5.2. These are exactly the estimates that replace the near-monotone attainment-in-a-compact-set argument of [3], so (2.9) is load-bearing for the vanishing-discount HJB derivation. The compact-support case can be reduced to (2.10) via [3, Theorem 3.7], but the non-compact case cannot, as the paper acknowledges. The assumption is also not merely qualitative: (2.9) requires a C^2 coercive V whose nonlocal increment is finite and compensates R_hatv outside a compact set, which implicitly imposes moment and growth matching between the running cost and ν. The Section 3 examples are polynomial with ∫|z|^m ν(dz)<∞ and verify (2.9), giving genuine but narrow support. The concern is not that the theorem is false under (2.9), but that the announced generality rests on an external Lyapunov hypothesis whose primitive status is unresolved for unbounded Lévy measures.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the ergodic control problem for d-dimensional controlled jump diffusions with compound Poisson jumps and nonnegative running costs, relaxing the near-monotone cost and uniform stability assumptions of earlier work. The model is set under two structural hypotheses: a Lyapunov inequality for V0 with a coercive comparison function F (Assumption 2.1) and a stabilizing stationary Markov control vhat with a coercive C^2 solution V to A^vhat V <= kappa 1_B0 - R_vhat (Assumption 2.2). The main results are: existence and x-independence of the optimal ergodic value rho* and an optimal stationary Markov control (Theorem 4.1); the alpha-discounted HJB equation with minimal nonnegative solutions (Theorem 5.1); the ergodic HJB equation min_u [A^u V* + R] = rho* a.e., with V*- in o(V0), obtained by vanishing discount, together with a characterization of optimal controls via pointwise minimization (Theorem 5.2); C^{2,r} regularity under polynomial growth Assumption 5.1 (Theorem 5.3); pathwise optimality under an additional structural Assumption 6.1 (Theorem 6.1); and an approximate HJB construction yielding near-optimal controls that fix a stable control outside a large ball (Theorem 7.1, Corollary 7.1). The paper includes queueing-network examples, including W and V models, that satisfy the hypotheses.","tokens_in":23824,"tokens_out":8880,"duration_ms":82423,"significance":"The paper makes a solid contribution to ergodic control of jump diffusions by removing the near-monotonicity restriction on the running cost, which is essential for many-server queueing applications. The hypotheses are explicit, the main theorems are carefully stated, and the proof strategy, based on perturbing the running cost by a coercive term and passing to the vanishing-discount limit, is coherent. The paper is also honest about the role of its assumptions: Remark 2.1 explicitly states that Assumption 2.2 is not needed for existence but is crucial for the HJB derivation when the Levy measure is not compactly supported, and the examples verify the assumptions in nontrivial queueing models. The main limitations are presentation-level: the abstract's pathwise-optimality assertion omits Assumption 6.1, and the genuinely primitive character of Assumption 2.2 for unbounded Levy measures deserves more prominence. If the stated hypotheses are accepted, the central claims appear sound.","major_comments":[{"comment":"The abstract states that 'optimal stationary Markov controls are a.s. pathwise optimal' without the qualification that this requires Assumption 6.1 in addition to Assumptions 2.1 and 2.2. Theorem 6.1 is explicitly conditional on Assumption 6.1, which requires F = phi composed with V0 with concave increasing phi and boundedness of sigma and nabla V0/(1+phi composed with V0). As written, the abstract promises a result that is not proved under the hypotheses used for Theorems 4.1 and 5.2. Please qualify the pathwise-optimality statement in the abstract and in the introduction.","section":"Abstract and Section 6"},{"comment":"Remark 2.1 correctly notes that for non-compactly supported nu, Assumption 2.2 cannot be replaced by the stabilizability hypothesis (2.10) and is crucial in Section 5. Because Lemma 5.1 and Lemma 5.2 use (2.9) to obtain the bounds (5.5), (5.6) and the local boundedness of ~I(V0+3V), the HJB characterization in Theorem 5.2 rests on (2.9) as an additional primitive Lyapunov condition rather than as a consequence of stabilizability. The Section 3 examples are polynomial with finite m-th moments and verify (2.9), but they do not settle the general non-compact case. Please state this limitation more prominently in the abstract and introduction, and either prove or clearly delineate a class of primitive conditions on (b, sigma, nu, R) that imply (2.9).","section":"Remark 2.1 and Section 5"},{"comment":"The statement of Theorem 5.2(c) is ambiguous: it says V* is the unique solution (up to additive constant) to the equation min_u [A^u V + R] = rho a.e. with rho <= rho*. If rho is allowed to vary, the uniqueness statement is not literally true unless all such solutions have the same constant; if rho is meant to be a fixed constant, the inequality should be removed or explained. The proof only explicitly treats solutions of (5.22) with the constant rho*, and the comparison argument for solutions with smaller rho is not supplied. Please clarify the quantifier over rho and provide the argument or a reference.","section":"Theorem 5.2(c)"}],"minor_comments":[{"comment":"The proof of Lemma 5.2 twice references an equation '(5.7)', but no display with that number appears in the text; the intended target is the oscillation bound asserted in the lemma. Please renumber the displays or correct the references.","section":"Lemma 5.2"},{"comment":"The phrase 'tight when restricted to B(K x U)' is confusing; since K x U is not compact, the meaning should be 'tight when restricted to the Borel sigma-algebra of K x U' or, more simply, 'tight on compact subsets of K x U'. Please rephrase.","section":"Section 6, proof of Theorem 6.1"},{"comment":"The abstract contains a small typo, 'full characterizations', which should be 'full characterization'.","section":"Abstract"},{"comment":"The phrase 'evaluating (7.2) at v^epsilon_*' is imprecise; it should say 'applying Ito's formula to (7.2) under the control v^epsilon_*' or similar.","section":"Section 7, proof of Theorem 7.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a competent extension of the authors' own prior work, and several key estimates are deferred to [3], [4], and [7]. The main concern for the editor is that the abstract overstates the pathwise-optimality result by omitting Assumption 6.1, and that the load-bearing nature of Assumption 2.2 for non-compact Levy measures should be made more prominent. These are fixable within the manuscript's scope, and I do not see grounds for rejection. The paper would benefit from an explicit discussion of whether (2.9) can be verified from primitive stabilizability conditions when nu has unbounded support."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a solid extension of the authors' earlier ergodic HJB theory to costs that are not near-monotone, and the main theorems look correct to me. The genuinely new pieces are the non-near-monotone framework under Assumptions 2.1–2.2, the gradient estimate in Lemma 5.3, and the supersolution bound in Lemma 7.1 that drives the bounded-domain approximation. The C^{2,r} regularity result and the approximate HJB construction are useful tools for queueing applications, and the examples in Section 3 give real substance.\n\nThe paper earns credit for being explicit about its own limits. Remark 2.1 admits that Assumption 2.2 is not needed for existence, and that for non-compactly supported Lévy measure it is the load-bearing step in the HJB derivation. That is honest, but it does mean the announced generality rests on an external Lyapunov hypothesis whose primitive status is unresolved for unbounded Lévy measures. The compact-support case can be reduced to the weaker stabilizability hypothesis via the prior paper, so the practical gap is narrower than the abstract suggests.\n\nSoft spots are mostly presentation and packaging. The abstract states pathwise optimality without mentioning Assumption 6.1, which is a real overstatement. The proofs lean heavily on the authors' previous work [3]–[5], [7]; a referee will want those dependencies checked carefully, but that is normal for this line of research. There is an equation-number typo in the proof of Lemma 5.2. None of these are load-bearing errors.\n\nThe core derivation is coherent and the assumptions are explicit. I did not find circularity: the structural hypotheses are external, not defined in terms of the results. The citation pattern is largely self-referential but appropriate given that the framework was developed in the authors' earlier papers; that is not a flaw in itself.\n\nWho is this for? Stochastic control theorists working on HJB equations for jump diffusions, and queueing network analysts who need ergodic control of non-near-monotone costs. It deserves a serious referee; with minor revisions (abstract fix, typo, maybe a remark expanding on the status of Assumption 2.2) it can be a solid journal paper.","headline":"A credible, useful extension of ergodic HJB theory to non-near-monotone costs, with honest caveats; the abstract overstates pathwise optimality and Assumption 2.2 carries real weight.","tokens_in":24269,"tokens_out":2505,"would_cite":true,"duration_ms":23250,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93E20","60J75","35Q93","60J60","35F21","93E15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The ergodic control problem for jump diffusions is fully characterized by one HJB equation under broad structural hypotheses.","keywords":["controlled jump diffusions","compound Poisson process","ergodic control","Hamilton-Jacobi-Bellman equation","stationary Markov control","pathwise optimality","parallel server networks","spatial truncation"],"falsifier":"Construct a controlled jump diffusion with a finite non-compactly supported jump measure that satisfies Assumption 2.1 and has some stabilizing control with finite ergodic cost, but violates Assumption 2.2, and check whether the normalized discounted value functions $V_\\alpha(x)-V_\\alpha(0)$ remain uniformly bounded in oscillation on a fixed ball as $\\alpha\\downarrow 0$. If their oscillation diverges, Lemma 5.2 fails and the HJB convergence theorem cannot hold; alternatively, if a limiting $V_*$ exists but $V_*^-\\notin o(V_\\circ)$, the uniqueness and stochastic-representation part of Theorem 5.2 is refuted.","tokens_in":23237,"feed_emoji":"⚙️","tokens_out":12214,"duration_ms":109019,"temperature":0.7,"pith_summary":"This paper proves that the long-run average (ergodic) control problem for diffusions driven by Brownian motion plus compound Poisson jumps is fully solvable under two structural hypotheses on the drift, the jump measure, and the running cost. The hypotheses allow running costs that are not near-monotone (they need not grow to infinity in every direction) and dynamics that are not stable under every control, which is exactly the situation that arises in scheduling large parallel server networks. The paper establishes the existence of an optimal stationary Markov control, shows the optimal value $\\rho^*$ is a constant across starting states, and characterizes all optimal controls as the minimizers of the ergodic Hamilton-Jacobi-Bellman equation. It then proves that optimal stationary controls are almost surely pathwise optimal, and that near-optimal policies can be built by solving the HJB equation on a sufficiently large bounded domain while fixing a stable control outside it.","feed_headline":"Ergodic control of jump diffusions solved by one HJB equation","feed_subtitle":"A general Lyapunov hypothesis supplies existence, optimality, pathwise optimality, and near-optimal policies.","key_machinery":"The load-bearing object is the ergodic Hamilton-Jacobi-Bellman (HJB) equation governed by the integro-differential generator $A^u\\phi(x)=a_{ij}(x)\\partial_{ij}\\phi(x)+\\tilde b_i(x,u)\\partial_i\\phi(x)+\\int_{\\mathbb{R}^d}(\\phi(x+y)-\\phi(x))\\,\\nu(dy)$, where $\\nu$ is the finite measure recording the sizes and rates of the compound Poisson jumps. The argument runs through the vanishing-discount method: the $\\alpha$-discounted value functions solve $\\min_u[A^u V_\\alpha+R]=\\alpha V_\\alpha$, and two Lyapunov inequalities (Assumptions 2.1 and 2.2) provide the oscillation bound and the control of the negative part of $V_\\alpha$ needed to extract a locally $C^{1,\\rho}$ limit $V_*$. A scaling-based gradient estimate upgrades solutions to $C^{2,r}$ regularity under polynomial-growth hypotheses, and a concave-transform trick on the Lyapunov function turns the ergodic HJB estimate into tightness of the random empirical measures, which is the step that yields pathwise optimality.","core_discovery":"Under Assumptions 2.1 and 2.2, the ergodic control problem for the jump diffusion admits an optimal stationary Markov control, the optimal value $\\rho^*$ is constant in the initial state, and the normalized $\\alpha$-discounted value functions $V_\\alpha(x)-V_\\alpha(0)$ converge to a function $V_*$ that solves the ergodic HJB equation $\\min_{u\\in U}[A^u V_*(x)+R(x,u)]=\\rho^*$ almost everywhere, with $V_*^-\\in o(V_\\circ)$ and $V_*(0)=0$. A stationary Markov control is optimal if and only if it selects the minimizer in this equation almost everywhere, and $V_*$ is the unique solution up to an additive constant in that class. Under Assumption 6.1 every average-cost optimal stationary Markov control is also optimal for the pathwise ergodic criterion, meaning it minimizes the almost-sure limsup of the average running cost. Under Assumption 7.1, fixing a stable control outside a large ball and solving the HJB equation on the ball yields controls whose ergodic cost is within any prescribed tolerance of $\\rho^*$ once the ball is large enough.","pith_inferences":["The bounded-domain approximation suggests a concrete numerical scheme: solve the semilinear HJB equation on a ball with an exterior stabilizing control fixed, then take the radius large; the paper proves near-optimality but does not quantify the needed radius.","The scaling gradient estimate appears portable: any second-order integro-differential equation with finite jump measure and polynomial data should inherit the same gradient growth, which may simplify regularity proofs in other jump-process control problems.","The negative-part bound $V_*^-\\in o(V_\\circ)$ is a natural replacement for near-monotonicity; one could test whether the same condition yields uniqueness of ergodic HJB solutions for other non-coercive cost structures, such as costs that vanish on lower-dimensional sets.","Because the structural assumptions are abstracted from heavily loaded many-server queueing networks, pathwise optimality implies that scheduling policies chosen by HJB minimizers are optimal along almost every sample path of the limiting diffusion, strengthening the usual expectation-based guarantees."],"forward_implications":["Solving the ergodic HJB equation gives both the optimal value and an if-and-only-if verification criterion for stationary Markov controls, so optimal scheduling policies can be certified by one PDE.","Every average-cost optimal stationary Markov control is pathwise optimal under Assumption 6.1, so policies that minimize expected long-run cost also minimize almost-sure long-run cost.","Near-optimal policies can be computed by solving the HJB equation on a large bounded ball with a fixed stable control outside; the cost gap shrinks to zero as the ball grows.","Under polynomial growth of the data, solutions of both the discounted and the ergodic HJB equations are $C^{2,r}$ on all of $\\mathbb{R}^d$, so classical elliptic regularity applies to the value functions.","The results recover the earlier near-monotone theory as a special case and also cover uniformly stable dynamics and dynamics that are transient under some controls."],"supporting_citations":[{"why":"Supplies the near-monotone ergodic-control framework, the stochastic calculus formula for rough jump kernels, and the empirical-measure lemmas that this paper extends.","marker":"[3]"},{"why":"Introduces the structural hypotheses behind Assumption 2.1 and the convex-analytic existence argument for optimal stationary Markov controls.","marker":"[4]"},{"why":"Provides the multiclass multi-pool network model and the leaf-elimination computations used to verify Assumptions 2.1 and 2.2 in the examples.","marker":"[5]"},{"why":"Establishes ergodicity and transience criteria for the jump-driven SDEs, used to show examples can be non-uniformly stable yet still satisfy the assumptions.","marker":"[6]"},{"why":"Supplies textbook machinery for existence via ergodic occupation measures, verification theorems, and resolvent-density arguments.","marker":"[7]"},{"why":"Gives the sample-path optimality technique for continuous diffusions that the paper adapts to jump diffusions with an added nonlocal estimate.","marker":"[10]"},{"why":"The companion application to scheduling multiclass many-server queues with service interruptions, which motivates the approximate-HJB construction.","marker":"[15]"},{"why":"Provides the elliptic regularity, local maximum principle, and maximum-principle estimates used in Lemmas 5.2 and 5.3 and Theorem 5.3.","marker":"[25]"}],"fun_headline_variants":["One HJB equation solves ergodic control of jump diffusions","General structural hypothesis yields optimal jump-diffusion control","Ergodic control for jump diffusions: HJB solves it all","Pathwise-optimal ergodic control via a single HJB equation","Jump diffusion ergodic control tamed by one HJB equation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Assumption 2.2: there exists a stable control $\\hat v$ and a coercive $C^2$ function $V$ such that $A^{\\hat v}V(x)\\le \\kappa\\mathbf 1_{B_\\circ}(x)-R_{\\hat v}(x)$ on all of $\\mathbb{R}^d$. This Lyapunov stability hypothesis is what makes the derivation of the ergodic HJB equation work when the jump measure has non-compact support; the paper's Remark 2.1 concedes it is not needed for existence, where the weaker condition that some control has finite ergodic cost already suffices.","fun_headline_variants_meta":{"raw":{"variants":["One HJB equation solves ergodic control of jump diffusions","General structural hypothesis yields optimal jump-diffusion control","Ergodic control for jump diffusions: HJB solves it all","Pathwise-optimal ergodic control via a single HJB equation","Jump diffusion ergodic control tamed by one HJB equation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000438,"raw_usage":{"total_tokens":2231,"prompt_tokens":959,"completion_tokens":1272,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":575,"completion_tokens_details":{"reasoning_tokens":1185}},"tokens_in":575,"tokens_out":1272,"duration_ms":9595,"temperature":1.0,"reasoning_tokens":1185,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:25:03.593510+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a controlled jump diffusion with a finite non-compactly supported jump measure that satisfies Assumption 2.1 and has some stabilizing control with finite ergodic cost, but violates Assumption 2.2, and check whether the normalized discounted value functions $V_\\alpha(x)-V_\\alpha(0)$ remain uniformly bounded in oscillation on a fixed ball as $\\alpha\\downarrow 0$. If their oscillation diverges, Lemma 5.2 fails and the HJB convergence theorem cannot hold; alternatively, if a limiting $V_*$ exists but $V_*^-\\notin o(V_\\circ)$, the uniqueness and stochastic-representation part of Theorem 5.2 is refuted.","supporting_citations":[{"cited_title":"Arapostathis, L","cited_arxiv_id":null,"evidence_quote":"Supplies the near-monotone ergodic-control framework, the stochastic calculus formula for rough jump kernels, and the empirical-measure lemmas that this paper extends."},{"cited_title":"Arapostathis and G","cited_arxiv_id":null,"evidence_quote":"Provides the multiclass multi-pool network model and the leaf-elimination computations used to verify Assumptions 2.1 and 2.2 in the examples."},{"cited_title":"Arapostathis, Some new results on sample path optimality in ergodic contro l of diffusions , IEEE Trans","cited_arxiv_id":null,"evidence_quote":"Gives the sample-path optimality technique for continuous diffusions that the paper adapts to jump diffusions with an added nonlocal estimate."},{"cited_title":"Arapostathis, G","cited_arxiv_id":null,"evidence_quote":"The companion application to scheduling multiclass many-server queues with service interruptions, which motivates the approximate-HJB construction."},{"cited_title":"Gilbarg and N","cited_arxiv_id":null,"evidence_quote":"Provides the elliptic regularity, local maximum principle, and maximum-principle estimates used in Lemmas 5.2 and 5.3 and Theorem 5.3."}],"review_version":1}