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Derivative Formulae and Gradient Estimates for Stochastic Hamiltonian Systems with Jumps

  • Hua Zhang
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Abstract

This paper is concerned with stochastic Hamiltonian systems driven by Lévy processes. By employing the lent particle method developed by Bouleau and Denis in the framework of Malliavin calculus with jumps, we establish an explicit Bismut--Elworthy--Li type derivative formula for the associated Markov semigroup, as well as corresponding gradient estimates. The main novelty lies in extending the known results for Brownian-motion-driven degenerate systems to the nonlocal setting with jumps, where the presence of a pure jump noise and the degeneracy of the coefficients pose substantial difficulties. Our approach provides a systematic treatment of such nonlocal degenerate operators and fills a gap in the literature on derivative formulae for jump-type stochastic Hamiltonian systems.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026derivative,
  title = {{Derivative Formulae and Gradient Estimates for Stochastic Hamiltonian Systems with Jumps}},
  author = {Hua Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25771},
  eprint = {2609.25771},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25771},
}
RIS
TY  - UNPB
TI  - Derivative Formulae and Gradient Estimates for Stochastic Hamiltonian Systems with Jumps
AU  - Hua Zhang
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.25771
UR  - https://arxiv.org/abs/2609.25771
AB  - This paper is concerned with stochastic Hamiltonian systems driven by Lévy processes. By employing the lent particle method developed by Bouleau and Denis in the framework of Malliavin calculus with jumps, we establish an explicit Bismut--Elworthy--Li type derivative formula for the associated Markov semigroup, as well as corresponding gradient estimates. The main novelty lies in extending the known results for Brownian-motion-driven degenerate systems to the nonlocal setting with jumps, where the presence of a pure jump noise and the degeneracy of the coefficients pose substantial difficulties. Our approach provides a systematic treatment of such nonlocal degenerate operators and fills a gap in the literature on derivative formulae for jump-type stochastic Hamiltonian systems.
ER  -