Probability Surveys

Stochastic differential equations with jumps

Richard F. Bass

Full-text: Open access

Abstract

This paper is a survey of uniqueness results for stochastic differential equations with jumps and regularity results for the corresponding harmonic functions.

Article information

Source
Probab. Surveys, Volume 1 (2004), 1-19.

Dates
First available in Project Euclid: 8 November 2004

Permanent link to this document
https://projecteuclid.org/euclid.ps/1099928647

Digital Object Identifier
doi:10.1214/154957804100000015

Mathematical Reviews number (MathSciNet)
MR2095564

Zentralblatt MATH identifier
1189.60114

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.) 60J75: Jump processes

Keywords
stochastic differential equations jumps martingale problems pathwise uniqueness Harnack inequality harmonic functions Dirichlet forms

Citation

Bass, Richard F. Stochastic differential equations with jumps. Probab. Surveys 1 (2004), 1--19. doi:10.1214/154957804100000015. https://projecteuclid.org/euclid.ps/1099928647


Export citation

References

  • Albeverio, Sergio; Song, Shiqi. Closability and resolvent of Dirichlet forms perturbed by jumps. Potential Anal. 2 (1993), no. 2, 115--130.
  • Applebaum, David; Tang, Fuchang. Stochastic flows of diffeomorphisms on manifolds driven by infinite-dimensional semimartingales with jumps. Stochastic Process. Appl. 92 (2001), no. 2, 219--236.
  • Applebaum, David. Lévy processes and stochastic calculus, Cambridge Univ. Press, May 2004, to appear.
  • Bass, Richard F. Uniqueness in law for pure jump Markov processes. Probab. Theory Related Fields 79 (1988), no. 2, 271--287.
  • Bass, Richard F. Occupation time densities for stable-like processes and other pure jump Markov processes. Stochastic Process. Appl. 29 (1988), no. 1, 65--83.
  • Bass, Richard F. Probabilistic techniques in analysis. Probability and its Applications (New York). Springer-Verlag, New York, 1995.
  • Bass, Richard F. Stochastic differential equations driven by symmetric stable processes. Séminaire de Probabilités, XXXVI, 302--313, Lecture Notes in Math., 1801, Springer, Berlin, 2003. \cmpcite1 971 592
  • Bass, Richard F. Stochastic calculus for discontinuous processes, http://www.math.uconn.edu/$\sim$bass/scdp.pdf
  • Bass, Richard F. General theory of processes, http://www.math.uconn.edu/$\sim$bass/gtp.pdf
  • Bass, Richard F.; Burdzy, Krzysztof; Chen, Zhen-Qing. Stochastic differential equations driven by stable processes for which pathwise uniqueness fails, Stoch. Proc. & their Applic., to appear, http://www.math.uconn.edu/$\sim$bass/unstab.pdf
  • Bass, Richard F.; Kassmann, Moritz. Harnack inequalities for non-local operators of variable order, Trans. Amer. Math. Soc., to appear, http://www.math.uconn.edu/$\sim$bass/mosernl.pdf
  • Bass, Richard F.; Levin, David A. Harnack inequalities for jump processes. Potential Anal. 17 (2002), no. 4, 375--388.
  • Bass, Richard F.; Levin, David A. Transition probabilities for symmetric jump processes. Trans. Amer. Math. Soc. 354 (2002), no. 7, 2933--2953.
  • Chen, Zhen-Qing. Multidimensional symmetric stable processes. Korean J. Comput. Appl. Math. 6 (1999), no. 2, 227--266.
  • Chen, Zhen-Qing; Kumagai, Takashi. Heat kernel estimates for stable-like processes on d-sets. Stoch. Proc. & their Appl. 108 (2003), no. 1, 27--62.
  • Çinlar, E.; Jacod, J. Representation of semimartingale Markov processes in terms of Wiener processes and Poisson random measures. Seminar on Stochastic Processes, 1981, pp. 159--242, Birkhäuser, Boston, Mass., 1981.
  • Dellacherie, Claude; Meyer, P.-A. Probabilités et potentiel. Chapitres I à IV. Edition entièrement refondue. Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. XV. Actualités Scientifiques et Industrielles, No. 1372. Hermann, Paris, 1975.
  • Dellacherie, Claude; Meyer, P.-A. Probabilités et potentiel. Chapitres V à VIII. Théorie des martingales. Revised edition. Actualités Scientifiques et Industrielles, 1385. Hermann, Paris, 1980.
  • Fujiwara, Tsukasa. Stochastic differential equations of jump type on manifolds and Lévy flows. J. Math. Kyoto Univ. 31 (1991), no. 1, 99--119.
  • Fujiwara, Tsukasa; Kunita, Hiroshi. Stochastic differential equations of jump type and Lévy processes in diffeomorphisms group. J. Math. Kyoto Univ. 25 (1985), no. 1, 71--106.
  • Fukushima, Masatoshi; Ōshima, Yōichi; Takeda, Masayoshi. Dirichlet forms and symmetric Markov processes. de Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 1994.
  • He, Sheng Wu; Wang, Jia Gang; Yan, Jia An. Semimartingale theory and stochastic calculus. CRC Press, Boca Raton, FL, 1992.
  • Hoh, Walter. The martingale problem for a class of pseudo-differential operators. Math. Ann. 300 (1994), no. 1, 121--147.
  • Hoh, Walter. Pseudodifferential operators with negative definite symbols and the martingale problem. Stochastics Stochastics Rep. 55 (1995), no. 3-4, 225--252.
  • Hoh, Walter. Pseudo differential operators with negative definite symbols of variable order. Rev. Mat. Iberoamericana 16 (2000), no. 2, 219--241.
  • Hoh, Walter. Perturbations of pseudodifferential operators with negative definite symbol. Appl. Math. Optim. 45 (2002), no. 3, 269--281.
  • Hoh, Walter; Jacob, Niels. Some Dirichlet forms generated by pseudo differential operators. Bull. Sci. Math. 116 (1992), no. 3, 383--398.
  • Hoh, Walter; Jacob, Niels. Pseudo-differential operators, Feller semigroups and the martingale problem. Stochastic processes and optimal control (Friedrichroda, 1992), 95--103, Stochastics Monogr., 7, Gordon and Breach, Montreux, 1993.
  • Jacob, Niels. Further pseudodifferential operators generating Feller semigroups and Dirichlet forms. Rev. Mat. Iberoamericana 9 (1993), no. 2, 373--407.
  • Jacob, Niels. Non-local (semi-) Dirichlet forms generated by pseudodifferential operators. Dirichlet forms and stochastic processes (Beijing, 1993), 223--233, de Gruyter, Berlin, 1995.
  • Jacob, Niels. Pseudo-differential operators and Markov processes. Mathematical Research, 94. Akademie Verlag, Berlin, 1996.
  • Jacob, Niels. Pseudo differential operators and Markov processes. Vol. I. Fourier analysis and semigroups. Imperial College Press, London, 2001.
  • Jacob, Niels. Pseudo differential operators & Markov processes. Vol. II. Generators and their potential theory. Imperial College Press, London, 2002. \cmpcite1 917 230
  • Jacob, Niels; Leopold, Hans-Gerd. Pseudo-differential operators with variable order of differentiation generating Feller semigroups. Integral Equations Operator Theory 17 (1993), no. 4, 544--553.
  • Jacob, Niels; Schilling, René L. Lévy-type processes and pseudodifferential operators. Lévy processes, 139--168, Birkhäuser, Boston, MA, 2001.
  • Jacod, Jean. Calcul stochastique et problèmes de martingales. Lecture Notes in Mathematics, 714. Springer, Berlin, 1979.
  • Janicki, A.; Michna, Z.; Weron, A. Approximation of stochastic differential equations driven by $\alpha$-stable Lévy motion. Appl. Math. (Warsaw) 24 (1996), no. 2, 149--168.
  • Kolokoltsov, Vassili. Symmetric stable laws and stable-like jump-diffusions. Proc. London Math. Soc. (3) 80 (2000), no. 3, 725--768.
  • Kolokoltsov, Vassili. On Markov processes with decomposable pseudo-differential generators, preprint, http://www.ima.umn.edu/prob-pde/reprints-preprints/kolokoltsov/fel.pdf
  • Komatsu, Takashi. Markov processes associated with certain integro-differential operators. Osaka J. Math. 10 (1973), 271--303.
  • Komatsu, Takashi. On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations of jump type. Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), no. 8, 353--356.
  • Komatsu, Takashi. Markov processes associated with pseudodifferential operators. Probability theory and mathematical statistics (Tbilisi, 1982), 289--298, Lecture Notes in Math., 1021, Springer, Berlin, 1983.
  • Komatsu, Takashi. On the martingale problem for generators of stable processes with perturbations. Osaka J. Math. 21 (1984), no. 1, 113--132.
  • Komatsu, Takashi. On stable-like processes. Probability theory and mathematical statistics (Tokyo, 1995), 210--219, World Sci. Publishing, River Edge, NJ, 1996.
  • Komatsu, Takashi. Uniform estimates for fundamental solutions associated with non-local Dirichlet forms. Osaka J. Math. 32 (1995), no. 4, 833--860.
  • Kunita, Hiroshi. Stochastic differential equations with jumps and stochastic flows of diffeomorphisms. Itô's stochastic calculus and probability theory, 197--211, Springer, Tokyo, 1996.
  • Le Gall, J.-F. Applications du temps local aux équations différentielles stochastiques unidimensionnelles. Séminaire de Probabilités, XVII, 15--31, Lecture Notes in Math., 986, Springer, Berlin, 1983.
  • Lepeltier, J.-P.; Marchal, B. Problème des martingales et équations différentielles stochastiques associées à un opérateur intégro-différentiel. Ann. Inst. H. Poincaré Sect. B (N.S.) 12 (1976), no. 1, 43--103.
  • Lyons, Terry J. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998), no. 2, 215--310.
  • Meyer, P.-A. Un cours sur les intégrales stochastiques. Séminaire de Probabilités, X (Seconde partie: Théorie des intégrales stochastiques, Univ. Strasbourg, Strasbourg, année universitaire 1974/1975), pp. 245--400. Lecture Notes in Math., Vol. 511, Springer, Berlin, 1976.
  • Nakao, S. On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations. Osaka J. Math. 9 (1972), 513--518.
  • Negoro, Akira. Stable-like processes: construction of the transition density and the behavior of sample paths near $t=0$. Osaka J. Math. 31 (1994), no. 1, 189--214.
  • Negoro, Akira; Tsuchiya, Masaaki. Convergence and uniqueness theorems for Markov processes associated with Lévy operators. Probability theory and mathematical statistics (Kyoto, 1986), 348--356, Lecture Notes in Math., 1299, Springer, Berlin, 1988.
  • Pragarauskas, G.; Zanzotto, P. A. On one-dimensional stochastic differential equations with respect to stable processes. Lithuanian Math. J. 40 (2000), no. 3, 277--295
  • Protter, Philip. Stochastic integration and differential equations, 2nd ed. Applications of Mathematics, 21. Springer-Verlag, Berlin, 2004.
  • Rong, Situ. On solutions of backward stochastic differential equations with jumps and applications. Stochastic Process. Appl. 66 (1997), no. 2, 209--236.
  • Skorokhod, A. V. Studies in the theory of random processes. Translated from the Russian by Scripta Technica, Inc. Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965
  • Song, Renming; Vondracek, Zoran. Harnack inequality for some classes of Markov processes, Math. Z., to appear, http://www.math.uiuc/$\sim$rsong/harnack-rev.pdf
  • Stroock, Daniel W. Diffusion processes associated with Lévy generators. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32 (1975), no. 3, 209--244.
  • Tanaka, Hiroshi; Tsuchiya, Masaaki; Watanabe, Shinzo. Perturbation of drift-type for Lévy processes. J. Math. Kyoto Univ. 14 (1974), 73--92.
  • Tsuchiya, Masaaki. On a small drift of Cauchy process. J. Math. Kyoto Univ. 10 (1970), 475--492.
  • Tsuchiya, Masaaki. On some perturbations of stable processes. Proceedings of the Second Japan-USSR Symposium on Probability Theory (Kyoto, 1972), pp. 490--497. Lecture Notes in Math., Vol. 330, Springer, Berlin, 1973.
  • Tsuchiya, Masaaki. Lévy measure with generalized polar decomposition and the associated SDE with jumps. Stochastics Stochastics Rep. 38 (1992), no. 2, 95--117.
  • Uemura, Toshihiro. On some path properties of symmetric stable-like processes for one dimension. Potential Anal. 16 (2002), no. 1, 79--91.
  • von Weizsäcker, Heinrich; Winkler, Gerhard. Stochastic integrals. An introduction. Friedr. Vieweg & Sohn, Braunschweig, 1990.
  • Williams, David R. E. Path-wise solutions of stochastic differential equations driven by Lévy processes. Rev. Mat. Iberoamericana 17 (2001), no. 2, 295--329.
  • Yamada, Toshio; Watanabe, Shinzo. On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11 1971 155--167.
  • Zanzotto, P. A. On solutions of one-dimensional stochastic differential equations driven by stable Lévy motion. Stochastic Process. Appl. 68 (1997), no. 2, 209--228.
  • Zanzotto, Pio Andrea. On stochastic differential equations driven by a Cauchy process and other stable Lévy motions. Ann. Probab. 30 (2002), no. 2, 802--825.