Anticipating linear stochastic differential equations driven by a Lévy process

Jorge A. Leon (Departamento de Control Automatico Cinvestav-IPN)
David Márquez-Carreras (Universitat de Barcelona)
Josep Vives (Universitat de Barcelona)

Abstract


In this paper we study the existence of a unique solution for  linear stochastic differential equations driven by a Lévy process, where the initial condition and the coefficients are random and not necessarily adapted to the underlying filtration. Towards this end, we  extend the method based on Girsanov transformation on Wiener space and developped by Buckdahn [7] to the canonical Lévy space, which is introduced in [25].

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Pages: 1-26

Publication Date: October 5, 2012

DOI: 10.1214/EJP.v17-1910

References

  • Alós, Elisa; León, Jorge A.; Vives, Josep. An anticipating Itô formula for Lévy processes. ALEA Lat. Am. J. Probab. Math. Stat. 4 (2008), 285--305. MR2456970
  • Applebaum, David; Siakalli, Michailina. Asymptotic stability of stochastic differential equations driven by Lévy noise. J. Appl. Probab. 46 (2009), no. 4, 1116--1129. MR2582710
  • Behme, Anita; Lindner, Alexander; Maller, Ross. Stationary solutions of the stochastic differential equation $dV_ t=V_ {t^ -}dU_ t+dL_ t$ with Lévy noise. Stochastic Process. Appl. 121 (2011), no. 1, 91--108. MR2739007
  • Bennett, Jonathan; Wu, Jiang-Lun. Stochastic differential equations with polar-decomposed Lévy measures and applications to stochastic optimization. Front. Math. China 2 (2007), no. 4, 539--558. MR2346435
  • Bojdecki, T.: Teoría General de Procesos e Integración Estocástica. Aportaciones Matemáticas de la Sociedad Matemática Mexicana, 2004. MR1647035
  • Buckdahn, Rainer. Anticipative Girsanov transformations and Skorohod stochastic differential equations. Mem. Amer. Math. Soc. 111 (1994), no. 533, viii+88 pp. MR1219706
  • Buckdahn, Rainer. Linear Skorohod stochastic differential equations. Probab. Theory Related Fields 90 (1991), no. 2, 223--240. MR1128071
  • Buckdahn, Rainer. Transformations on the Wiener space and Skorohod-type stochastic differential equations. Seminarberichte [Seminar Reports], 105. Humboldt Universität, Sektion Mathematik, Berlin, 1989. 110 pp. MR1033989
  • Cont, Rama; Tankov, Peter. Financial modelling with jump processes. Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL, 2004. xvi+535 pp. ISBN: 1-5848-8413-4 MR2042661
  • Federico, Salvatore; Øksendal, Bernt Karsten. Optimal stopping of stochastic differential equations with delay driven by Lévy noise. Potential Anal. 34 (2011), no. 2, 181--198. MR2754970
  • Ikeda, Nobuyuki; Watanabe, Shinzo. Stochastic differential equations and diffusion processes. Second edition. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. xvi+555 pp. ISBN: 0-444-87378-3 MR1011252
  • Itô, Kiyosi. On a stochastic integral equation. Proc. Japan Acad. 22, (1946). nos. 1-4, 32--35. MR0036958
  • Jien, Yu-Juan; Ma, Jin. Stochastic differential equations driven by fractional Brownian motions. Bernoulli 15 (2009), no. 3, 846--870. MR2555202
  • Jing, Shuai; León, Jorge A. Semilinear backward doubly stochastic differential equations and SPDEs driven by fractional Brownian motion with Hurst parameter in $(0,1/2)$. Bull. Sci. Math. 135 (2011), no. 8, 896--935. MR2854758
  • Kunita, Hiroshi. Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms. Real and stochastic analysis, 305--373, Trends Math., Birkhäuser Boston, Boston, MA, 2004. MR2090755
  • Jacod, Jean. The Euler scheme for Lévy driven stochastic differential equations: limit theorems. Ann. Probab. 32 (2004), no. 3A, 1830--1872. MR2073179
  • León, Jorge A.; Solé, Josep L.; Vives, Josep. A pathwise approach to backward and forward stochastic differential equations on the Poisson space. Stochastic Anal. Appl. 19 (2001), no. 5, 821--839. MR1857898
  • León, Jorge A.; Tudor, Constantin. Chaos decomposition of stochastic bilinear equations with drift in the first Poisson-Itô chaos. Statist. Probab. Lett. 48 (2000), no. 1, 11--22. MR1767606
  • León, Jorge A.; Ruiz de Chávez, J.; Tudor, C. Strong solutions of anticipating stochastic differential equations on the Poisson space. Bol. Soc. Mat. Mexicana (3) 2 (1996), no. 1, 55--63. MR1395911
  • Nualart, David. The Malliavin calculus and related topics. Second edition. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. xiv+382 pp. ISBN: 978-3-540-28328-7; 3-540-28328-5 MR2200233
  • Picard, Jean. Transformations et équations anticipantes pour les processus de Poisson. (French) [Anticipating transformations and equations for Poisson processes] Ann. Math. Blaise Pascal 3 (1996), no. 1, 111--123. MR1397328
  • Privault, Nicolas. Linear Skorohod stochastic differential equations on Poisson space. Stochastic analysis and related topics, V (Silivri, 1994), 237--253, Progr. Probab., 38, Birkhäuser Boston, Boston, MA, 1996. MR1396334
  • Protter, Philip E. Stochastic integration and differential equations. Second edition. Applications of Mathematics (New York), 21. Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2004. xiv+415 pp. ISBN: 3-540-00313-4 MR2020294
  • Rubenthaler, Sylvain. Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process. Stochastic Process. Appl. 103 (2003), no. 2, 311--349. MR1950769
  • Sato, K. I.: Lévy processes and infinitely divisible distributions. phCambridge, 1999. MR1739520
  • Solé, Josep Lluís; Utzet, Frederic; Vives, Josep. Canonical Lévy process and Malliavin calculus. Stochastic Process. Appl. 117 (2007), no. 2, 165--187. MR2290191
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