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Weak Dirichlet processes with jumps

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  • Bandini, Elena
  • Russo, Francesco

Abstract

This paper develops systematically the stochastic calculus via regularization in the case of jump processes. In particular one continues the analysis of real-valued càdlàg weak Dirichlet processes with respect to a given filtration. Such a process is the sum of a local martingale and an adapted process A such that [N,A]=0, for any continuous local martingale N. Given a function u:[0,T]×R→R, which is of class C0,1 (or sometimes less), we provide a chain rule type expansion for u(t,Xt) which stands in applications for a chain Itô type rule.

Suggested Citation

  • Bandini, Elena & Russo, Francesco, 2017. "Weak Dirichlet processes with jumps," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 4139-4189.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:12:p:4139-4189
    DOI: 10.1016/j.spa.2017.04.001
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    References listed on IDEAS

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    1. Bandini, Elena & Fuhrman, Marco, 2017. "Constrained BSDEs representation of the value function in optimal control of pure jump Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1441-1474.
    2. Gozzi, Fausto & Russo, Francesco, 2006. "Weak Dirichlet processes with a stochastic control perspective," Stochastic Processes and their Applications, Elsevier, vol. 116(11), pages 1563-1583, November.
    3. Gozzi, Fausto & Russo, Francesco, 2006. "Verification theorems for stochastic optimal control problems via a time dependent Fukushima-Dirichlet decomposition," Stochastic Processes and their Applications, Elsevier, vol. 116(11), pages 1530-1562, November.
    4. Cristina Girolami & Giorgio Fabbri & Francesco Russo, 2014. "The covariation for Banach space valued processes and applications," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(1), pages 51-104, January.
    5. Russo, Francesco & Vallois, Pierre, 1995. "The generalized covariation process and Ito formula," Stochastic Processes and their Applications, Elsevier, vol. 59(1), pages 81-104, September.
    6. Errami, Mohammed & Russo, Francesco, 2003. "n-covariation, generalized Dirichlet processes and calculus with respect to finite cubic variation processes," Stochastic Processes and their Applications, Elsevier, vol. 104(2), pages 259-299, April.
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    Cited by:

    1. Bruno Bouchard & Gr'egoire Loeper & Xiaolu Tan, 2021. "A $C^{0,1}$-functional It\^o's formula and its applications in mathematical finance," Papers 2101.03759, arXiv.org.
    2. Bouchard, Bruno & Loeper, Grégoire & Tan, Xiaolu, 2022. "A ℂ0,1-functional Itô’s formula and its applications in mathematical finance," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 299-323.
    3. Bruno Bouchard & Grégoire Loeper & Xiaolu Tan, 2022. "A C^{0,1}-functional Itô's formula and its applications in mathematical finance," Post-Print hal-03105342, HAL.
    4. Bruno Bouchard & Grégoire Loeper & Xiaolu Tan, 2021. "A C^{0,1}-functional Itô's formula and its applications in mathematical finance," Working Papers hal-03105342, HAL.

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