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Density in small time at accessible points for jump processes

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  • Picard, Jean

Abstract

We consider a process Yt which is the solution of a stochastic differential equation driven by a Lévy process with an initial condition Y0 = y0. We assume conditions under which Yt has a smooth density for any t > 0. We consider a point y that the process can reach with a finite number of jumps from y0, and prove that, as t tends to 0, the density at this point is of order t[Gamma] for some [Gamma] = [Gamma](y0, y). Some applications to the potential analysis of the process are given.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:67:y:1997:i:2:p:251-279
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    Cited by:

    1. Jorge González Cázares & Aleksandar Mijatović, 2022. "Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation," Finance and Stochastics, Springer, vol. 26(4), pages 671-732, October.
    2. Ishikawa, Yasushi & Kunita, Hiroshi & Tsuchiya, Masaaki, 2018. "Smooth density and its short time estimate for jump process determined by SDE," Stochastic Processes and their Applications, Elsevier, vol. 128(9), pages 3181-3219.
    3. Paweł Sztonyk, 2010. "Estimates of Tempered Stable Densities," Journal of Theoretical Probability, Springer, vol. 23(1), pages 127-147, March.
    4. Picard, Jean & Savona, Catherine, 2000. "Smoothness of harmonic functions for processes with jumps," Stochastic Processes and their Applications, Elsevier, vol. 87(1), pages 69-91, May.
    5. 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.
    6. Jorge Gonz'alez C'azares & Aleksandar Mijatovi'c, 2020. "Simulation of the drawdown and its duration in L\'{e}vy models via stick-breaking Gaussian approximation," Papers 2011.06618, arXiv.org, revised Mar 2021.

    More about this item

    Keywords

    60J75 60H07;

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