November 2018, 38(11): 5735-5763. doi: 10.3934/dcds.2018250

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

Norwegian University of Science and Technology, NO-7491, Trondheim, Norway

* Corresponding author: Espen R. Jakobsen

Received  January 2018 Revised  June 2018 Published  August 2018

Fund Project: 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

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.

Citation: Linghua Chen, Espen R. Jakobsen. L1 semigroup generation for Fokker-Planck operators associated to general Lévy driven SDEs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5735-5763. doi: 10.3934/dcds.2018250
References:
[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.

show all references

References:
[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.

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