Stochastic differential equations with jumps

Probability Surveys

Volumn 1, Issue 1, 2004, Pages 1-19

  Bass, Richard F ^a  

^a UNIVERSITY OF CONNECTICUT   (United States)

Author keywords

Dirichlet forms; Harmonic functions; Harnack inequality; Jumps; Martingale problems; Pathwise uniqueness; Stochastic differential equations

Indexed keywords

EID: 27744498401     PISSN: None     EISSN: 15495787     Source Type: Journal    
DOI: 10.1214/154957804100000015     Document Type: Article

References (69)

1. Albeverio, Sergio; Song, Shiqi. Closability and resolvent of Dirichlet forms perturbed by jumps. Potential Anal. 2 (1993), no. 2, 115-130. MR1246745
2. Applebaum, David; Tang, Fuchang. Stochastic ows of diffeomorphisms on manifolds driven by infinite-dimensional semimartingales with jumps. Stochastic Process. Appl. 92 (2001), no. 2, 219-236. MR1817587
3. Applebaum, David. Lévy processes and stochastic calculus, Cambridge Univ. Press, May 2004, to appear.
4. Bass, Richard F. Uniqueness in law for pure jump Markov processes. Probab. Theory Related Fields 79 (1988), no. 2, 271-287. MR958291
5. 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. MR952820
6. Bass, Richard F. Probabilistic techniques in analysis. Probability and its Applications (New York). Springer-Verlag, New York, 1995. MR1329542
7. 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. (CMP 1 971 592) MR1971592
8. Bass, Richard F. Stochastic calculus for discontinuous processes, http://www.math.uconn.edu/~bass/scdp.pdf
9. Bass, Richard F. General theory of processes, http://www.math.uconn.edu/~bass/gtp.pdf
10. 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/~bass/unstab.pdf
11. 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/~bass/mosernl.pdf
12. Bass, Richard F.; Levin, David A. Harnack inequalities for jump processes. Potential Anal. 17 (2002), no. 4, 375-388. MR1918242
13. Bass, Richard F.; Levin, David A. Transition probabilities for symmetric jump processes. Trans. Amer. Math. Soc. 354 (2002), no. 7, 2933-2953. MR1895210
14. Chen, Zhen-Qing. Multidimensional symmetric stable processes. Korean J. Comput. Appl. Math. 6 (1999), no. 2, 227-266. MR1687746
15. Chen, Zhen-Qing; Kumagai, Takashi. Heat kernel estimates for stablelike processes on d-sets. Stoch. Proc. & their Appl. 108 (2003), no. 1, 27-62. MR2008600
16. 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. MR647786
17. 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 Scienti fiques et Industrielles, No. 1372. Hermann, Paris, 1975. MR488194
18. 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. MR566768
19. Fujiwara, Tsukasa. Stochastic differential equations of jump type on manifolds and Lévy ows. J. Math. Kyoto Univ. 31 (1991), no. 1, 99-119. MR1093330
20. 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. MR777247
21. Fukushima, Masatoshi; Ōshima, Yōichi; Takeda, Masayoshi. Dirichlet forms and symmetric Markov processes. de Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 1994. MR1303354
22. He, Sheng Wu; Wang, Jia Gang; Yan, Jia An. Semimartingale theory and stochastic calculus. CRC Press, Boca Raton, FL, 1992. MR1219534
23. Hoh, Walter. The martingale problem for a class of pseudo-differential operators. Math. Ann. 300 (1994), no. 1, 121-147. MR1289834
24. Hoh, Walter. Pseudodifferential operators with negative definite symbols and the martingale problem. Stochastics Stochastics Rep. 55 (1995), no. 3-4, 225-252. MR1378858
25. Hoh,Walter. Pseudo differential operators with negative definite symbols of variable order. Rev. Mat. Iberoamericana 16 (2000), no. 2, 219-241. MR1809340
26. Hoh, Walter. Perturbations of pseudodifferential operators with negative definite symbol. Appl. Math. Optim. 45 (2002), no. 3, 269-281. MR1885821
27. Hoh, Walter; Jacob, Niels. Some Dirichlet forms generated by pseudo differential operators. Bull. Sci. Math. 116 (1992), no. 3, 383-398. MR1177287
28. 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. MR1268244
29. Jacob, Niels. Further pseudodifferential operators generating Feller semigroups and Dirichlet forms. Rev. Mat. Iberoamericana 9 (1993), no. 2, 373-407. MR1232848
30. Jacob, Niels. Non-local (semi-) Dirichlet forms generated by pseudodifferential operators. Dirichlet forms and stochastic processes (Beijing, 1993), 223-233, de Gruyter, Berlin, 1995. MR1366438
31. Jacob, Niels. Pseudo-differential operators and Markov processes. Mathematical Research, 94. Akademie Verlag, Berlin, 1996. MR1409607
32. Jacob, Niels. Pseudo differential operators and Markov processes. Vol. I. Fourier analysis and semigroups. Imperial College Press, London, 2001. MR1873235
33. Jacob, Niels. Pseudo differential operators & Markov processes. Vol. II. Generators and their potential theory. Imperial College Press, London, 2002. (CMP 1 917 230)
34. 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. MR1243995
35. Jacob, Niels; Schilling, René L. Lévy-type processes and pseudodifferential operators. Lévy processes, 139-168, Birkhäuser, Boston, MA, 2001. MR1833696
36. Jacod, Jean. Calcul stochastique et problèmes de martingales. Lecture Notes in Mathematics, 714. Springer, Berlin, 1979. MR542115
37. Janicki, A.; Michna, Z.; Weron, A. Approximation of stochastic differential equations driven by ff-stable Lévy motion. Appl. Math. (Warsaw) 24 (1996), no. 2, 149-168. MR1424549
38. Kolokoltsov, Vassili. Symmetric stable laws and stable-like jumpdi ffusions. Proc. London Math. Soc. (3) 80 (2000), no. 3, 725-768. MR1744782
39. Kolokoltsov, Vassili. On Markov processes with decomposable pseudodi fferential generators, preprint, http://www.ima.umn.edu/probpde/reprints-preprints/kolokoltsov/fel.pdf
40. Komatsu, Takashi. Markov processes associated with certain integrodi fferential operators. Osaka J. Math. 10 (1973), 271-303. MR359017
41. Komatsu, Takashi. On the pathwise uniqueness of solutions of onedimensional stochastic differential equations of jump type. Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), no. 8, 353-356. MR683262
42. 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. MR735994
43. Komatsu, Takashi. On the martingale problem for generators of stable processes with perturbations. Osaka J. Math. 21 (1984), no. 1, 113-132. MR736974
44. Komatsu, Takashi. On stable-like processes. Probability theory and mathematical statistics (Tokyo, 1995), 210-219, World Sci. Publishing, River Edge, NJ, 1996. MR1467941
45. Komatsu, Takashi. Uniform estimates for fundamental solutions associated with non-local Dirichlet forms. Osaka J. Math. 32 (1995), no. 4, 833-860. MR1380729
46. Kunita, Hiroshi. Stochastic differential equations with jumps and stochastic flows of diffeomorphisms. Itô's stochastic calculus and probability theory, 197-211, Springer, Tokyo, 1996. MR1439526
47. 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. MR770393
48. 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. MR413288
49. Lyons, Terry J. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998), no. 2, 215-310. MR1654527
50. Meyer, P.-A. Un cours sur les intégrales stochastiques. Séminaire de Probabilit és, 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. MR501332
51. Nakao, S. On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations. Osaka J. Math. 9 (1972), 513-518. MR326840
52. 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. MR1262797
53. 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. MR936008
54. 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 MR1803652
55. Protter, Philip. Stochastic integration and differential equations, 2nd ed. Applications of Mathematics, 21. Springer-Verlag, Berlin, 2004. MR2020294
56. Rong, Situ. On solutions of backward stochastic differential equations with jumps and applications. Stochastic Process. Appl. 66 (1997), no. 2, 209-236. MR1440399
57. 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 MR185620
58. Song, Renming; Vondracek, Zoran. Harnack inequality for some classes of Markov processes, Math. Z., to appear, http://www.math.uiuc/~rsong/harnack-rev.pdf MR2031452
59. Stroock, Daniel W. Diffusion processes associated with Lévy generators. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32 (1975), no. 3, 209- 244. MR433614
60. Tanaka, Hiroshi; Tsuchiya, Masaaki; Watanabe, Shinzo. Perturbation of drift-type for Lévy processes. J. Math. Kyoto Univ. 14 (1974), 73-92. MR368146
61. Tsuchiya, Masaaki. On a small drift of Cauchy process. J. Math. Kyoto Univ. 10 (1970), 475-492. MR307361
62. 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. MR443105
63. Tsuchiya, Masaaki. Lévy measure with generalized polar decomposition and the associated SDE with jumps. Stochastics Stochastics Rep. 38 (1992), no. 2, 95-117. MR1274897
64. Uemura, Toshihiro. On some path properties of symmetric stable-like processes for one dimension. Potential Anal. 16 (2002), no. 1, 79-91. MR1880349
65. von Weizsäcker, Heinrich; Winkler, Gerhard. Stochastic integrals. An introduction. Friedr. Vieweg & Sohn, Braunschweig, 1990. MR1062600
66. Williams, David R. E. Path-wise solutions of stochastic differential equations driven by Lévy processes. Rev. Mat. Iberoamericana 17 (2001), no. 2, 295-329. MR1891200
67. Yamada, Toshio; Watanabe, Shinzo. On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11 1971 155-167. MR278420
68. 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. MR1454833
69. 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. MR1905857