Advanced Search
Article Contents
Article Contents

L1 semigroup generation for Fokker-Planck operators associated to general Lévy driven SDEs

  • * Corresponding author: Espen R. Jakobsen

    * Corresponding author: Espen R. Jakobsen

E. R. Jakobsen is supported by the Toppforsk (research excellence) project Waves and Nonlinear Phenomena (WaNP), grant no. 250070 from the Research Council of Norway

Abstract Full Text(HTML) Related Papers Cited by
  • We prove a new generation result in $L^1$ for a large class of non-local operators with non-degenerate local terms. This class contains the operators appearing in Fokker-Planck or Kolmogorov forward equations associated with Lévy driven SDEs, i.e. the adjoint operators of the infinitesimal generators of these SDEs. As a byproduct, we also obtain a new elliptic regularity result of independent interest. The main novelty in this paper is that we can consider very general Lévy operators, including state-space depending coefficients with linear growth and general Lévy measures which can be singular and have fat tails.

    Mathematics Subject Classification: 47G20, 47D06, 47D07, 60H10, 35K10, 60G51.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] H. Abels, Pseudodifferential and Singular Integral Operators: An Introduction with Applications, De Gruyter Textbook, De Gruyter, 2012.
    [2] N. AlibaudS. Cifani and E. R. Jakobsen, Continuous dependence estimates for nonlinear fractional convection-diffusion equations, SIAM Journal on Mathematical Analysis, 44 (2012), 603-632.  doi: 10.1137/110834342.
    [3] D. Applebaum, Lévy Processes And Stochastic Calculus, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2004. doi: 10.1017/CBO9780511755323.
    [4] D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 607-694. 
    [5] V. Bally and L. Caramellino, On the distances between probability density functions, Electron. J. Probab., 19 (2014), 1-33.  doi: 10.1214/EJP.v19-3175.
    [6] O. E. Barndorff-Nielsen, Normal inverse gaussian distributions and stochastic volatility modelling, Scand. J. Stat., 24 (1997), 1-13.  doi: 10.1111/1467-9469.t01-1-00045.
    [7] R. F. Bass and D. A. Levin, Transition probabilities for symmetric jump processes, Trans. Amer. Math. Soc., 354 (2002), 2933-2953.  doi: 10.1090/S0002-9947-02-02998-7.
    [8] F. E. BenthK. H. Karlsen and K. Reikvam, Optimal portfolio management rules in a non-Gaussian market with durability and intertemporal substitution, Finance and Stochastics, 5 (2001), 447-467.  doi: 10.1007/s007800000032.
    [9] S. V. Bodnarchuk and A. M. Kulik, Conditions for the existence and smoothness of the distribution density for the Ornstein-Uhlenbeck process with Lévy noise, Teor. Imovir. Mat. Statyst., (2008), 21-35. 
    [10] V. I. Bogachev, N. V. Krylov, M. Röckner and S. V. Shaposhnikov, Fokker-Planck-Kolmogorov Equations, vol. 207, American Mathematical Society, Providence, RI, 2015. doi: 10.1090/surv/207.
    [11] K. Bogdan and T. Jakubowski, Estimates of heat kernel of fractional laplacian perturbed by gradient opera- tors, Comm. Math. Phys., 271 (2007), 179-198.  doi: 10.1007/s00220-006-0178-y.
    [12] J.-M. Bony, Principe du maximum dans les espaces de Sobolev, Comptes Rendus Acad. Sci. Paris, Série A, 265 (1967), 333-336 (French). 
    [13] B. P. Brooks, The coefficients of the characteristic polynomial in terms of the eigenvalues and the elements of an n × n matrix, Appl. Math. Lett., 19 (2006), 511-515.  doi: 10.1016/j.aml.2005.07.007.
    [14] Y. A. Butko, M. Grothaus and O. G. Smolyanov, Feynman formulae and phase space Feynman path integrals for tau-quantization of some Lévy-Khintchine type Hamilton functions, J. Math. Phys., 57 (2016), 023508, 22pp. doi: 10.1063/1.4940697.
    [15] J. Cai and H. Yang, On the decomposition of the absolute ruin probability in a perturbed compound poisson surplus process with debit interest, Annals of Operations Research, 212 (2014), 61-77.  doi: 10.1007/s10479-011-1032-y.
    [16] T. Cass, Smooth densities for solutions to stochastic differential equations with jumps, Stochastic Processes and their Applications, 119 (2009), 1416-1435.  doi: 10.1016/j.spa.2008.07.005.
    [17] L. Chen, The Numerical Path Integration Method for Stochastic Differential Equations, Ph. D. thesis, NTNU Norwegian University of Science and Technology, 2016.
    [18] L. ChenE. R. Jakobsen and A. Naess, On numerical density approximations of solutions of SDEs with unbounded coefficients, Adv. Comput. Math., 44 (2018), 693-721.  doi: 10.1007/s10444-017-9558-4.
    [19] Z.-Q. ChenE. HuL Xie and X. Zhang, Heat kernels for non-symmetric diffusion operators with jumps, J. Differential Equations, 263 (2017), 6576-6634.  doi: 10.1016/j.jde.2017.07.023.
    [20] Z.-Q. Chen and T. Kumagai, A priori hölder estimate, parabolic harnack principle and heat kernel estimates for diffusions with jumps., Rev. Mat. Iberoam., 26 (2010), 551-589.  doi: 10.4171/RMI/609.
    [21] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman and Hall/CRC Financial Mathematics Series, CRC Press, 2004.
    [22] N. Dunford, J. T. Schwartz, W. G. Bade and R. G. Bartle, Linear Operators, Part Ⅰ: General Theory, Pure and Applied Mathematics. vol. 7, etc, New York; Groningen printed, 1958.
    [23] K. J. Engel and R. Nagel, et al., One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000.
    [24] L. C. Evans, Partial Differential Equations, Graduate studies in mathematics, American Mathematical Society, 2010. doi: 10.1090/gsm/019.
    [25] S. Fornaro and L. Lorenzi, Generation results for elliptic operators with unbounded diffusion coefficients in Lp- and Cb-spaces, Discrete and Continuous Dynamical Systems, 18 (2007), 747-772.  doi: 10.3934/dcds.2007.18.747.
    [26] M. G. Garroni and J. L. Menaldi, Second Order Elliptic Integro-Differential Problems, Chapman & Hall/CRC Research Notes in Mathematics Series, CRC Press, 2002. doi: 10.1201/9781420035797.
    [27] Ĭ. Ī. Gīhman and A. V. Skorohod, Stochastic Differential Equations, Springer-Verlag, New York-Heidelberg, 1972, Translated from the Russian by Kenneth Wickwire, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 72.
    [28] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001
    [29] F. Gimbert and P.-L. Lions, Existence and regularity results for solutions of second-order elliptic integro-differential operators, Ricerche di Matematica, 33 (1984), 315-358. 
    [30] S. Hiraba, Existence and smoothness of transition density for jump-type Markov processes: Applications of Malliavin calculus, Kodai Math. J., 15 (1992), 29-49.  doi: 10.2996/kmj/1138039525.
    [31] V. Kolokoltsov, Symmetric stable laws and stable-like jump-diffusions, Proc. London Math. Soc., 80 (2000), 725-768.  doi: 10.1112/S0024611500012314.
    [32] T. Komatsu and A. Takeuchi, On the smoothness of PDF of solutions to SDE of jump type, International Journal of Differential Equations and Applications, 2 (2001), 141-197. 
    [33] A. Kyprianou, W. Schoutens and P. Wilmott, Exotic Option Pricing and Advanced Lévy Models, Wiley, 2005.
    [34] P.-L. Lions, A remark on Bony maximum principle, Proc. Amer. Math. Soc., 88 (1983), 503-508.  doi: 10.1090/S0002-9939-1983-0699422-3.
    [35] T. Mikosch, Non-life Insurance Mathematics: An Introduction with the Poisson Process, Universitext, Springer Berlin Heidelberg, 2009. doi: 10.1007/978-3-540-88233-6.
    [36] B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Universitext, Springer Berlin Heidelberg, 2007. doi: 10.1007/978-3-540-69826-5.
    [37] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
    [38] G. B. Price, Bounds for determinants with dominant principal diagonal, Proceedings of the American Mathematical Society, 2 (1951), 497-502.  doi: 10.1090/S0002-9939-1951-0041093-2.
    [39] P. Protter, Stochastic Integration and Differential Equations, Stochastic Modelling and Applied Probability, Springer Berlin Heidelberg, 2005. doi: 10.1007/978-3-662-10061-5.
    [40] M. Schechter, Principles of Functional Analysis, Graduate studies in mathematics, American Mathematical Society, 2002.
    [41] W. Schoutens, Lévy Processes in Finance: Pricing Financial Derivatives, Wiley Series in Probability and Statistics, Wiley, 2003. doi: 10.1002/0470870230.
    [42] J. Wang, Sub-Markovian C0-semigroups generated by fractional laplacian with gradient perturbation, Integral Equations and Operator Theory, 76 (2013), 151-161.  doi: 10.1007/s00020-013-2055-3.
    [43] R. L. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis, Second edition. Pure and Applied Mathematics (Boca Raton). CRC Press, Boca Raton, FL, 2015.
    [44] W. Zhang and J. Bao, Regularity of very weak solutions for elliptic equation of divergence form, Journal of Functional Analysis, 262 (2012), 1867-1878.  doi: 10.1016/j.jfa.2011.11.027.
    [45] ________, Regularity of very weak solutions for nonhomogeneous elliptic equation, Communications in Contemporary Mathematics, 15 (2013), 1350012, 19pp. doi: 10.1142/S0219199713500120.
    [46] X. Zhang, Densities for SDEs driven by degenerate α-stable processes, Annals of Probability, 42 (2014), 1885-1910.  doi: 10.1214/13-AOP900.
  • 加载中

Article Metrics

HTML views(511) PDF downloads(199) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint