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Impulse Control of Conditional McKean–Vlasov Jump Diffusions

Author

Listed:
  • Nacira Agram

    (KTH Royal Institute of Technology)

  • Giulia Pucci

    (KTH Royal Institute of Technology)

  • Bernt Øksendal

    (University of Oslo)

Abstract

In this paper, we consider impulse control problems involving conditional McKean–Vlasov jump diffusions, with the common noise coming from the $$\sigma $$ σ -algebra generated by the first components of a Brownian motion and an independent compensated Poisson random measure. We first study the well-posedness of the conditional McKean–Vlasov stochastic differential equations (SDEs) with jumps. Then, we prove the associated Fokker–Planck stochastic partial differential equation (SPDE) with jumps. Next, we establish a verification theorem for impulse control problems involving conditional McKean–Vlasov jump diffusions. We obtain a Markovian system by combining the state equation with the associated Fokker–Planck SPDE for the conditional law of the state. Then we derive sufficient variational inequalities for a function to be the value function of the impulse control problem, and for an impulse control to be the optimal control. We illustrate our results by applying them to the study of an optimal stream of dividends under transaction costs. We obtain the solution explicitly by finding a function and an associated impulse control, which satisfy the verification theorem.

Suggested Citation

  • Nacira Agram & Giulia Pucci & Bernt Øksendal, 2024. "Impulse Control of Conditional McKean–Vlasov Jump Diffusions," Journal of Optimization Theory and Applications, Springer, vol. 200(3), pages 1100-1130, March.
    • Handle: RePEc:spr:joptap:v:200:y:2024:i:3:d:10.1007_s10957-023-02370-6
    DOI: 10.1007/s10957-023-02370-6
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    References listed on IDEAS

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