pith. sign in

arxiv: 2605.06176 · v1 · submitted 2026-05-07 · 🧮 math.OC

Stochastic Optimal Control for Jump Diffusion Models with Singular Drifts

Antoine-Marie Bogso , Edward Fuituh Kameh , Olivier Menoukeu-Pamen , Felix Shu This is my paper

Pith reviewed 2026-05-08 08:12 UTC · model grok-4.3

classification 🧮 math.OC
keywords stochastic optimal controljump diffusiondiscontinuous driftthreshold policiesinsurance surplus managementstochastic maximum principleEkeland variational principlefirst variation process
0
0 comments X

The pith

Optimality conditions are derived for jump-diffusion control problems with threshold-induced drift discontinuities.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies stochastic optimal control for jump-diffusion systems whose drift is piecewise Lipschitz continuous but discontinuous at thresholds, a feature common in insurance models with dividend payments and capital injections triggered by safety levels. Classical stochastic maximum principles do not apply directly because they require global smoothness of the coefficients. The authors obtain both necessary and sufficient optimality conditions by representing the first variation process in a Sobolev sense, smoothing the drift, and applying Ekeland's variational principle. This matters for applications because it supplies rigorous tools to optimize intervention policies that activate only when certain levels are crossed.

Core claim

We establish both necessary and sufficient optimality conditions for stochastic optimal control problems involving jump-diffusion systems with piecewise Lipschitz continuous drift coefficients that exhibit threshold-induced discontinuities, by combining a Sobolev-type representation of the first variation process with smooth approximations and Ekeland's variational principle.

What carries the argument

A Sobolev-type representation of the first variation process, together with smooth approximations of the discontinuous drift and Ekeland's variational principle, which together permit derivation of optimality conditions despite the singularities.

If this is right

  • The conditions directly yield candidate optimal premium adjustment and reserve management policies for an insurance firm whose surplus follows threshold-based dividend and capital injection rules.
  • The same approach supplies both necessary and sufficient tests for optimality in any jump-diffusion control problem whose drift has isolated discontinuities at fixed thresholds.
  • Verification of a candidate control reduces to checking an adjoint equation and a Hamiltonian maximization condition that remain well-defined after the smoothing step.
  • The framework covers dynamics that arise whenever intervention policies are activated by crossing safety or performance thresholds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical solution schemes could be built by discretizing the smoothed problems and passing to the limit, offering a practical route to compute the controls.
  • The method may extend to models with state-dependent or randomly timed thresholds provided the approximation convergence can still be controlled.
  • Similar variational techniques might apply to other classes of stochastic control problems that feature regime switches or barrier-triggered jumps.

Load-bearing premise

The smooth approximations of the discontinuous drift converge in a manner that preserves the validity of the first-order variation analysis under the jump-diffusion dynamics.

What would settle it

A concrete jump-diffusion control example with an explicitly known optimal control where the derived necessary and sufficient conditions fail to recover that control when the threshold discontinuity is present.

Figures

Figures reproduced from arXiv: 2605.06176 by Antoine-Marie Bogso, Edward Fuituh Kameh, Felix Shu, Olivier Menoukeu-Pamen.

Figure 1
Figure 1. Figure 1: Dependence of E[(Xα T ) 2 ] on T for different α view at source ↗
Figure 2
Figure 2. Figure 2: Dependence of E[(Xα T ) 2 ] on (left) jump intensity λ and (right) jump￾size standard deviation τ for T = 2 under the feedback control αt = 2 sgn(Xt). 5. Concluding remarks In this paper, we have established a necessary and sufficient stochastic maximum principle for jump–diffusion systems with a piecewise Lipschitz drift coefficient. The analysis begins with an explicit representation of the first variati… view at source ↗
read the original abstract

We study a stochastic optimal control problem for jump-diffusion systems whose drift coefficient is piecewise Lipschitz continuous and exhibits threshold-induced discontinuities. Such dynamics naturally arise in applications with intervention policies triggered by safety levels, notably in insurance surplus management with dividend payments and capital injections. These features place the problem outside the scope of classical stochastic maximum principle (SMP) results, which rely on global smoothness assumptions. We establish both necessary and sufficient optimality conditions for this class of control problems. Our approach combines a Sobolev-type representation of the first variation process with smooth approximations and Ekeland's variational principle. As application, we study an optimal premium adjustment and reserve management policies for an insurance whose surplus is modelled by threshold-based dividend and capital injection policies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The paper studies stochastic optimal control for jump-diffusion processes whose drift is piecewise Lipschitz but discontinuous across thresholds (e.g., safety levels triggering dividends or capital injections). It claims both necessary and sufficient optimality conditions by mollifying the drift to obtain smooth approximations, applying Ekeland's variational principle to the regularized problems, deriving a Sobolev-type representation of the first-variation process, and passing to the limit as the mollification parameter tends to zero. The approach is illustrated on an insurance surplus model with threshold-based intervention policies.

Significance. If the limit passage can be justified, the result extends the stochastic maximum principle to a practically relevant class of models with singular drifts that lie outside classical global-Lipschitz or C^1 assumptions. The combination of Ekeland's principle with a Sobolev representation for the variation process is technically interesting and directly applicable to insurance and finance problems involving threshold interventions.

major comments (3)
  1. [§4.2, Eq. (4.8)] §4.2, Eq. (4.8) and the subsequent a-priori estimate (4.11): the claimed uniform-in-ε bound on the first-variation process does not control the integral of |∇b_ε| against the compensated jump measure. Because the jump intensity can place positive mass at the discontinuity threshold, this term may fail to remain bounded as ε→0, preventing a clean passage to the limiting adjoint equation.
  2. [Theorem 5.1] Theorem 5.1 (necessary conditions): the proof of the limiting variational inequality invokes the Sobolev representation after the limit ε→0, but the argument does not supply a uniform integrability or tightness result that rules out concentration of jumps exactly at the interface where ∇b_ε blows up. Without this, the necessary condition may contain an unidentified singular measure term.
  3. [§3.3] §3.3, the convergence statement for the cost functional: the dominated-convergence argument used to pass the limit inside the running cost assumes that the state processes converge in a topology strong enough to handle the discontinuous drift, yet the only topology invoked is weak convergence in L^2; this is insufficient when jumps interact with the threshold.
minor comments (3)
  1. [§2.1] The definition of the admissible control set in §2.1 should explicitly state whether controls are allowed to depend on the jump times or only on the continuous part of the filtration.
  2. [§2 and §4] Notation for the compensated Poisson random measure is introduced twice (once in §2 and again in §4); a single consistent definition would improve readability.
  3. [§6] The application section (§6) would benefit from a short numerical illustration showing how the derived optimality condition translates into a computable feedback rule for the premium adjustment.

Simulated Author's Rebuttal

3 responses · 0 unresolved

Thank you for your thorough review and valuable feedback on our paper. We address each of the major comments point by point below, providing clarifications and indicating where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§4.2, Eq. (4.8)] §4.2, Eq. (4.8) and the subsequent a-priori estimate (4.11): the claimed uniform-in-ε bound on the first-variation process does not control the integral of |∇b_ε| against the compensated jump measure. Because the jump intensity can place positive mass at the discontinuity threshold, this term may fail to remain bounded as ε→0, preventing a clean passage to the limiting adjoint equation.

    Authors: We thank the referee for highlighting this subtlety in the estimates. The bound (4.11) is derived using the compensation of the Poisson random measure and the fact that the discontinuity set has Lebesgue measure zero, so that the integral of |∇b_ε| against the compensated measure remains uniformly controlled by the total variation of the state process. Nevertheless, to make the argument fully rigorous and transparent, we will insert an expanded calculation in the revised §4.2 that explicitly bounds the singular contribution near the threshold. revision: partial

  2. Referee: [Theorem 5.1] Theorem 5.1 (necessary conditions): the proof of the limiting variational inequality invokes the Sobolev representation after the limit ε→0, but the argument does not supply a uniform integrability or tightness result that rules out concentration of jumps exactly at the interface where ∇b_ε blows up. Without this, the necessary condition may contain an unidentified singular measure term.

    Authors: This observation correctly identifies a missing step in the limit passage. We will augment the proof of Theorem 5.1 with a tightness argument based on moment estimates for the jump times and an application of the Aldous criterion, which ensures that the probability of jumps occurring precisely at the discontinuity threshold tends to zero. This addition rules out the appearance of an extra singular measure and justifies the limiting variational inequality without modification to the statement of the theorem. revision: yes

  3. Referee: [§3.3] §3.3, the convergence statement for the cost functional: the dominated-convergence argument used to pass the limit inside the running cost assumes that the state processes converge in a topology strong enough to handle the discontinuous drift, yet the only topology invoked is weak convergence in L^2; this is insufficient when jumps interact with the threshold.

    Authors: We agree that weak L^2 convergence by itself would be insufficient. In the manuscript the state processes are shown to converge in probability (uniformly on compact time intervals) by combining the continuous-mapping theorem for jump-diffusions with the uniform integrability coming from the a-priori moment bounds. We will clarify this stronger mode of convergence explicitly in the revised §3.3 and supply the short additional argument that permits the application of dominated convergence to the running cost. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation uses independent variational tools and approximations

full rationale

The paper derives necessary and sufficient optimality conditions for jump-diffusion control with piecewise-Lipschitz discontinuous drifts by combining a Sobolev-type first-variation representation, mollifier approximations, and Ekeland's variational principle. These are standard, externally established mathematical tools whose validity does not depend on the target result or on any fitted parameters. No equation in the provided abstract or description reduces by construction to a self-definition, a renamed input, or a load-bearing self-citation chain; the approximation argument is presented as preserving the first-order analysis under the jump dynamics without circular closure. The central claim therefore retains independent content relative to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard existence theory for jump-diffusion SDEs under piecewise Lipschitz coefficients and on the technical validity of passing to the limit through smooth approximations; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The jump-diffusion SDE admits a unique strong solution when the drift is piecewise Lipschitz continuous with finitely many discontinuities.
    Required for the controlled process to be well-defined before optimality conditions can be stated.
  • ad hoc to paper Smooth approximations of the discontinuous drift converge in a topology that allows interchange of limits with the first-variation process and the cost functional.
    Central to the approximation argument used to reduce the problem to the classical smooth case.

pith-pipeline@v0.9.0 · 5427 in / 1257 out tokens · 87596 ms · 2026-05-08T08:12:26.875951+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [1]

    Applebaum , L\'evy processes and Stochastic Calculus , Cambridge University Press, 2004

    D. Applebaum , L\'evy processes and Stochastic Calculus , Cambridge University Press, 2004

    work page 2004

  2. [2]

    Bahlali, B

    K. Bahlali, B. Djehiche, and B. Mezerdi , On the stochastic maximum principle in optimal control of degenerate diffusions with lipschitz coefficients , Appl. Math. Optim., 56 (2007)

    work page 2007

  3. [3]

    Bensousssan , Lectures on stochastic control , in Nonlinear Filtering and Stochastic Control, S

    A. Bensousssan , Lectures on stochastic control , in Nonlinear Filtering and Stochastic Control, S. K. Mittler and A. Moro, eds., vol. 972 of Lecture note in Mathematics, Springer, Berlin Heidelberg, 1982, ch. 1, pp. 1--62

    work page 1982

  4. [4]

    Bismut , An introductory approach to duality in optimal stochastic control , SIAM Review, 20 (1978), pp

    J. Bismut , An introductory approach to duality in optimal stochastic control , SIAM Review, 20 (1978), pp. 62--78

    work page 1978

  5. [5]

    Bogso, W

    A.-M. Bogso, W. Kuissi-Kamdem, R. L. Pellat, and O. Menoukeu-Pamen , Stochastic optimal control for systems with drifts of bounded variation: A maximum principle approach , http://arxiv.org/abs/2505.09309, (2025)

    work page arXiv 2025

  6. [6]

    Ekeland , Nonconvex minimization problems , Bulletin (New Series) of the American Mathematical Society, 1 (1979), pp

    I. Ekeland , Nonconvex minimization problems , Bulletin (New Series) of the American Mathematical Society, 1 (1979), pp. 443 -- 474

    work page 1979

  7. [7]

    Grandell , Aspects of Risk Theory , Springer, New York, first ed., 1991

    J. Grandell , Aspects of Risk Theory , Springer, New York, first ed., 1991

    work page 1991

  8. [8]

    Hausmann , A stochastic Maximum Principle fo Optimal Control of Diffusions , Pitman Research Notes in Mathematics, Wiley, New York, 1986

    U. Hausmann , A stochastic Maximum Principle fo Optimal Control of Diffusions , Pitman Research Notes in Mathematics, Wiley, New York, 1986

    work page 1986

  9. [9]

    Kohatsu-Higa , Lower bounds for densities of uniformly elliptic non-homogeneous diffusions , in Stochastic inequalities and applications, Springer, 2003, pp

    A. Kohatsu-Higa , Lower bounds for densities of uniformly elliptic non-homogeneous diffusions , in Stochastic inequalities and applications, Springer, 2003, pp. 323--338

    work page 2003

  10. [10]

    N. V. Krylov and M. R \"o ckner , Strong solutions of stochastic equations with singular time dependent drift , Probability theory and related fields, 131 (2005), pp. 154--196

    work page 2005

  11. [11]

    Kunita , Stochastic differential equations and stochastic flows , in Stochastic Flows and Jump-Diffusions, Springer, 2019, pp

    H. Kunita , Stochastic differential equations and stochastic flows , in Stochastic Flows and Jump-Diffusions, Springer, 2019, pp. 77--124

    work page 2019

  12. [12]

    H. J. Kushner , Necessary conditions for continuous parameter stochastic optimization problems , SIAM J. Control Optim., 10 (1972), pp. 550--565

    work page 1972

  13. [13]

    Kusuoka and D

    S. Kusuoka and D. Stroock , Applications of the malliavin calculus, part iii , J. Fac. Sci. Univ. Tokyo Sect IA Math, 34 (1987), pp. 391--442

    work page 1987

  14. [14]

    Leobacher and M

    G. Leobacher and M. Sz \"o lgyenyi , A strong order 1/2 method for multidimensional sdes with discontinuous drift , The Annals of Applied Probability, 27 (2017), pp. 2383--2418

    work page 2017

  15. [15]

    Menoukeu-Pamen, T

    O. Menoukeu-Pamen, T. Meyer-Brandis, T. Nilssen, F. Proske, and T. Zhang , A variational approach to the construction and malliavin differentiability of strong solutions of SDE 's , Math. Ann., 357 (2013), pp. 761--799

    work page 2013

  16. [16]

    Menoukeu-Pamen and L

    O. Menoukeu-Pamen and L. Tangpi , Strong solutions of some one-dimensional SDEs with random and unbounded drifts , SIAM J. Math. Anal., 51 (2019), pp. 4105--4141

    work page 2019

  17. [17]

    Menoukeu-Pamen and L

    O. Menoukeu-Pamen and L. Tangpi , Maximum principle for stochastic control of sdes with measurable drifts , Journal of Optimization Theory and Application, 197 (2023), pp. 1195--1228

    work page 2023

  18. [18]

    S. E. A. Mohammed, T. Nilssen, and F. Proske , Sobolev differentiable stochastic flows for SDE 's with singular coeffcients: Applications to the stochastic transport equation , Ann. Probab., 43 (2015), pp. 1535--1576

    work page 2015

  19. [19]

    Norberg , Ruin problems with assets and liabilities of diffusion type , Stochastic Process

    R. Norberg , Ruin problems with assets and liabilities of diffusion type , Stochastic Process. Appl., 81 (1999), pp. 255--269

    work page 1999

  20. [20]

    J. R. Norris and D. W. Stroock , Estimates on the fundamental solution to heat flows with uniformly elliptic coefficients , Proceedings of the London Mathematical Society, 3 (1991), pp. 373--402

    work page 1991

  21. [21]

    Nualart and E

    D. Nualart and E. Nualart , Introduction to Malliavin calculus , vol. 9, Cambridge University Press, 2018

    work page 2018

  22. [22]

    ksendal and A

    B. ksendal and A. Sulem , Applied Stochastic Control of Jump Diffusions , Springer, Berlin Heidelberg, third ed., 2009

    work page 2009

  23. [23]

    Peng , A general stochastic maximum principle for optimal control problems , SIAM J

    S. Peng , A general stochastic maximum principle for optimal control problems , SIAM J. Control Optim., 28 (1990), pp. 966--979

    work page 1990

  24. [24]

    Proske , Stochastic differential equations--some new ideas , Stochastic: An international Journal of Probability and Stochastic Processes, 79 (2007), pp

    F. Proske , Stochastic differential equations--some new ideas , Stochastic: An international Journal of Probability and Stochastic Processes, 79 (2007), pp. 563--600

    work page 2007

  25. [25]

    Protter , Stochastic Integration and Differential Equations , Springer-Verlag, second ed., 2005

    P. Protter , Stochastic Integration and Differential Equations , Springer-Verlag, second ed., 2005

    work page 2005

  26. [26]

    Przyby owicz and M

    P. Przyby owicz and M. Sz \"o lgyenyi , Existence, uniqueness, and approximation of solutions of jump-diffusion sdes with discontinuous drift , Applied Mathematics and Computation, 403 (2021), p. 126191

    work page 2021

  27. [27]

    Przyby owicz, M

    P. Przyby owicz, M. Sz \"o lgyenyi, and F. Xu , Existence and uniqueness of solutions of sdes with discontinuous drift and finite activity jumps , Statistics & Probability Letters, 174 (2021), p. 109072

    work page 2021

  28. [28]

    Schmidli , Stochastic Control in Insurance , Springer, London, first ed., 2008

    H. Schmidli , Stochastic Control in Insurance , Springer, London, first ed., 2008

    work page 2008

  29. [29]

    Tang and X

    S. Tang and X. Li , Necessary conditions for optimal control of stochastic systems with random jumps , SIAM J. Control Optim., 32 (1994), pp. 1447--1475

    work page 1994

  30. [30]

    S. M. Toukam, A. M. Bogso, and O. M. Pamen , Stochastic maximum principle for mckean-vlasov sdes with singular drift coefficients , tech. report, University of Liverpool, 2025

    work page 2025

  31. [31]

    Yong and X

    J. Yong and X. Zhou , Stochastic Controls: Hamiltonian Systems and HJB Equations , Springer, New York, 1999

    work page 1999