Asymptotic expansion of the transition density of the semigroup associated to a SDE driven by Lévy noise. (English) Zbl 1502.35171
Summary: In this work we consider a finite dimensional stochastic differential equation (SDE) driven by a Lévy noise \(L(t)=L_t\), \(t > 0\). The transition probability density \(p_t\), \(t > 0\) of the semigroup associated to the solution \(u_t, t\geqslant 0\) of the SDE is given by a power series expansion. The series expansion of \(p_t\) can be re-expressed in terms of Feynman graphs and rules. We will also prove that \(p_t\), \(t > 0\) has an asymptotic expansion in power of a parameter \(\beta > 0\), and it can be given by a convergent integral.
A remark on some applications will be given in this work.
A remark on some applications will be given in this work.
MSC:
| 35Q82 | PDEs in connection with statistical mechanics |
| 35C20 | Asymptotic expansions of solutions to PDEs |
| 81Q30 | Feynman integrals and graphs; applications of algebraic topology and algebraic geometry |
| 68T07 | Artificial neural networks and deep learning |
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
| 60G51 | Processes with independent increments; Lévy processes |
| 60J35 | Transition functions, generators and resolvents |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
Keywords:
stochastic differential equations; transition probability densities; Lévy processes; Feynman graphs and rules; Borel summability; neural networksReferences:
| [1] | S. Albeverio, L. Dipersio, E. Mastrogiacomo and B. Smii, A class of Lévy driven SDEs and their explicit invariant measures,Pot. Anal.45(2) (2016), 229-259. doi:10.1007/s11118-016-9544-3. · Zbl 1350.60049 |
| [2] | S. Albeverio, L. Dipersio, E. Mastrogiacomo and B. Smii, Explicit invariant measures for infinite dimensional SDE driven by Lévy noise with dissipative nonlinear drift I,Comm. Math. Sci.15(4) (2017), 957-983. doi:10.4310/CMS.2017.v15. n4.a3. · Zbl 1516.35579 |
| [3] | S. Albeverio, H. Gottschalk and J.-L. Wu, Convoluted generalized white noise, Schwinger functions and their analytic continuation to Wightman functions,Rev. Math. Phys8(6) (1996), 763-817. doi:10.1142/S0129055X96000287. · Zbl 0870.60038 |
| [4] | S. Albeverio, E. Lytvynov and A. Mahnig, A model of the term structure of interest rates based on Lévy fields,Stoch. Proc. Appl.114(2) (2004), 251-263. doi:10.1016/j.spa.2004.06.006. · Zbl 1151.91471 |
| [5] | S. Albeverio, V. Mandrekar and B. Rüdiger, Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise,Stochastic Process. Appl.119(3) (2009), 835-863. doi:10.1016/j.spa.2008.03. 006. · Zbl 1168.60014 |
| [6] | S. Albeverio, E. Mastrogiacomo and B. Smii, Small noise asymptotic expansions for stochastic PDE’s driven by dissipative nonlinearity and Lévy noise,Stoch. Proc. Appl.123(2013), 2084-2109. doi:10.1016/j.spa.2013.01.013. · Zbl 1291.35452 |
| [7] | S. Albeverio and B. Smii, Asymptotic expansions for SDE’s with small multiplicative noise,Stoch. Process. App.125(3) (2015), 1009-1031. doi:10.1016/j.spa.2014.09.009. · Zbl 1322.60082 |
| [8] | S. Albeverio and B. Smii, Borel summation of the small time expansion of some SDE’s driven by Gaussian white noise, Asymp. Anal.114(2019), 211-223. doi:10.3233/ASY-191525. · Zbl 1433.35405 |
| [9] | S. Albeverio, J.-L. Wu and T.S. Zhang, Parabolic SPDEs driven by Poisson white noise,Stoch. Proc. Appl.74(1998), 21-36. doi:10.1016/S0304-4149(97)00112-9. · Zbl 0934.60055 |
| [10] | O.E. Barndorff-Nielsen and N. Shephard, Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics,J. R. Stat. Soc. Ser. B Stat. Methodol.63(2) (2001), 167-241. doi:10.1111/1467-9868.00282. · Zbl 0983.60028 |
| [11] | A. Blake, P. Kohli and C. Rother,Markov Random Fields for Vision and Image Processing, The MIT Press, 2011. · Zbl 1236.68001 |
| [12] | S. Boyarchenko, I. Svetlana and Z. Sergi,Non-Gaussian Merton-Black-Scholes Theory, Vol. 9, World Scientific, Singapore, 2002. · Zbl 0997.91031 |
| [13] | Z. Brze´zniak and E. Hausenblas, Uniqueness in law of the Itô integral with respect to Lévy noise, in:Seminar Stoch. Anal., Random Fields and Appl., Vol. VI, Birkhauser, Basel, 2011, pp. 37-57. · Zbl 1278.60100 |
| [14] | K.L. Chung and M. Rao, General gauge theorem for multiplicative functional,Trans. Am. Math. Soc.306(1988), 819- 836. doi:10.1090/S0002-9947-1988-0933320-1. · Zbl 0647.60083 |
| [15] | R. Cont and P. Tankov,Financial Modelling with Jump Processes, Vol. 2, CRC Press, 2004. · Zbl 1052.91043 |
| [16] | Ph. Courrège, Sur la forme intégro-différentielle des opérateurs deCk∞(Rn)dansC(Rn)satisfaisant au principe du maximum”, Sém. Théorie du potentiel (1965/66) Exposé 2. |
| [17] | N.G. De Bruijn,Asymptotic Methods in Analysis, North-Holland, Amsterdam, 1958, Zbl. 82,42 (2nd ed. 1975, 3rd ed., New York: Dover 1981). · Zbl 0082.04202 |
| [18] | E. Eberlein and K. Glau, Variational solutions of the pricing PIDEs for European options in Lévy models,Appl. Math. Finance(2014). · Zbl 1395.91497 |
| [19] | M. Freidlin,Markov Processes and Differential Equations, Asymptotic Problems, Birkhäuser, Basel, 1996. · Zbl 0863.60049 |
| [20] | I.I. Gihman and A.V. Skorohod,Stochastic Differential Equations, Springer-Verlag, Berlin-Heidelberg. New York, 1972. · Zbl 0242.60003 |
| [21] | H. Gottschalk and B. Smii, How to determine the law of the solution to a SPDE driven by a Lévy space-time noise, Journal of Mathematical Physics43(2007), 1-22. · Zbl 1137.60332 |
| [22] | H. Gottschalk, B. Smii and H. Thaler, The Feynman graph representation of general convolution semigroups and its applications to Lévy statistics,Journal of Bernoulli Society14(2) (2008), 322-351. doi:10.3150/07-BEJ106. · Zbl 1158.60361 |
| [23] | R.L. Graham, D.E. Knth and O. Patashnik,Concrete Mathematics, Addison-Wesley, 1994. · Zbl 0836.00001 |
| [24] | P. Hartman and A. Wintner, On the infinitisimal generators of integral convolutions,Am. J. Math.64(1942), 273-298. doi:10.2307/2371683. · Zbl 0063.01951 |
| [25] | V. Knopova, On the Feynman-Kac semigroup for some Markov processes,Mod. Stoch: Theor. App.2(2015), 107-129. doi:10.15559/15-VMSTA26. · Zbl 1352.60110 |
| [26] | V. Knopova and R.L. Schilling, A note on the existence of transition probability densities of Lévy processes,Foru. Math. 25(2013), 125-149. · Zbl 1269.60050 |
| [27] | A. Løkka, B. Øksendal and F. Proske, Stochastic partial differential equations driven by Lévy space-time white noise, Ann. Appl. Prob.14(2004), 1506-1528. doi:10.1214/105051604000000413. · Zbl 1053.60069 |
| [28] | L. Lovász,Combinatorial Problems and Exercises, 2nd edn, North Holland, Amsterdam, Netherlands, 1993. · Zbl 0785.05001 |
| [29] | G. Marsaglia and J.C.W. Marsaglia, A new derivation of Stirling’s approximation ofn!,Amer. Math. Monthly97(1990), 826-829. doi:10.1080/00029890.1990.11995666. · Zbl 0786.05007 |
| [30] | T. Meyer-Brandis and F. Proske, Explicit representation of strong solutions of SDEs driven by infinite dimensional Lévy processes,J. Theor. Prob.23(2010), 301-314. doi:10.1007/s10959-009-0226-6. · Zbl 1197.60054 |
| [31] | D. Mugnolo, Gaussian estimates for a heat equation on a network,Netw. Heter. Media2(2007), 55-79. doi:10.3934/nhm. 2007.2.55. · Zbl 1142.35349 |
| [32] | D. Mugnolo and S. Romanelli, Dynamic and generalized Wentzell node conditions for network equations,Math. Meth. Appl. Sciences30(2007), 681-706. doi:10.1002/mma.805. · Zbl 1163.34039 |
| [33] | B. Øksendal,Stochastic Differential Equations: An Introduction with Applications, 6th edn, Springer-Verlag, Berlin Heidelberg, 2010. |
| [34] | K.I. Sato,Lévy Processes and Infinite Divisibility, Cambridge University Press, 1999. · Zbl 0973.60001 |
| [35] | L.S. Schulman,Techniques and Applications of Path Integrations, John Wiley and Sons, New York, 1981. · Zbl 0587.28010 |
| [36] | J.-P. Serre,Trees, Springer, Berlin, 1980. · Zbl 0548.20018 |
| [37] | T. Shiga, A recurrence criterion for Markov processes of Ornstein-Uhlenbeck type,Prob. Th. Rel. Fields.85(1990), 425-447. doi:10.1007/BF01203163. · Zbl 0677.60088 |
| [38] | B. Smii, A linked cluster theorem of the solution of the generalized Burger equation,Applied Mathematical Sciences. (Ruse)6(1) (2012), 21-38. · Zbl 1248.05040 |
| [39] | B. Smii, A large diffusion expansion for the transition function of Lévy Ornstein-Uhlenbeck processes,Appl. Math. Inf. Sci.10(4) (2016), 1-8. doi:10.18576/amis/100434. |
| [40] | A.D. Sokal, An improvement of Watson’s theorem on Borel summability,J. Math. Phys.21(2) (1980), 261-263. doi:10. 1063/1.524408. · Zbl 0441.40012 |
| [41] | R. Song, Two-sided estimates on the density of the Feynman-Kac semigroups of stable-like processes,Elec. J. Prob.11 (2006), 146-161. doi:10.1214/EJP.v11-308. · Zbl 1112.60059 |
| [42] | C. Turchetti,Stochastic Models of Neural Networks, IOS Press, 2004 · Zbl 1058.68098 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
