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Existence of densities for multi-type continuous-state branching processes with immigration

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  • Friesen, Martin
  • Jin, Peng
  • Rüdiger, Barbara

Abstract

Let X be a multi-type continuous-state branching process with immigration on state space R+d. Denote by gt, t≥0, the law of X(t). We provide sufficient conditions under which gt has, for each t>0, a density with respect to the Lebesgue measure. Such density has, by construction, some Besov regularity. Our approach is based on a discrete integration by parts formula combined with a precise estimate on the error of the one-step Euler approximations of the process. As an auxiliary result, we also provide a criterion for the existence of densities of solutions to a general stochastic equation driven by Brownian motions and Poisson random measures, whose coefficients are Hölder continuous and might be unbounded.

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  • Friesen, Martin & Jin, Peng & Rüdiger, Barbara, 2020. "Existence of densities for multi-type continuous-state branching processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5426-5452.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:9:p:5426-5452
    DOI: 10.1016/j.spa.2020.03.012
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    References listed on IDEAS

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    1. Duhalde, Xan & Foucart, Clément & Ma, Chunhua, 2014. "On the hitting times of continuous-state branching processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 124(12), pages 4182-4201.
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    3. Filipović, Damir & Mayerhofer, Eberhard & Schneider, Paul, 2013. "Density approximations for multivariate affine jump-diffusion processes," Journal of Econometrics, Elsevier, vol. 176(2), pages 93-111.
    4. Christa Cuchiero & Damir Filipovi'c & Eberhard Mayerhofer & Josef Teichmann, 2009. "Affine processes on positive semidefinite matrices," Papers 0910.0137, arXiv.org, revised Apr 2011.
    5. Picard, Jean, 1997. "Density in small time at accessible points for jump processes," Stochastic Processes and their Applications, Elsevier, vol. 67(2), pages 251-279, May.
    6. Aurélien Alfonsi, 2015. "Affine Diffusions and Related Processes: Simulation, Theory and Applications," Post-Print hal-03127212, HAL.
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    Cited by:

    1. Martin Friesen & Peng Jin, 2022. "Volterra square-root process: Stationarity and regularity of the law," Papers 2203.08677, arXiv.org, revised Oct 2022.

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