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Local correlation entropy
L1 semigroup generation for Fokker-Planck operators associated to general Lévy driven SDEs
Norwegian University of Science and Technology, NO-7491, Trondheim, 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.
References:
[1] |
H. Abels,
Pseudodifferential and Singular Integral Operators: An Introduction with Applications, De Gruyter Textbook, De Gruyter, 2012. |
[2] |
N. Alibaud, S. 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. Benth, K. 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. Chen, E. 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. Chen, E. Hu, L 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. Alibaud, S. 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. Benth, K. 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. Chen, E. 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. Chen, E. Hu, L 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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