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 Probability Surveys > Vol. 1 (2004) open journal systems 


Stochastic differential equations with jumps

Richard F. Bass, Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009


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

AMS 2000 subject classifications: Primary 60H10; secondary 60H30, 60J75.

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

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Bass, Richard F., Stochastic differential equations with jumps , Probability Surveys, 1, (2004), 1-19 (electronic).

References

[AS]    Albeverio, Sergio; Song, Shiqi. Closability and resolvent of Dirichlet forms perturbed by jumps. Potential Anal. 2 (1993), no. 2, 115–130. MR1246745

[AT]    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. MR1817587

[Ap]    Applebaum, David. Lévy processes and stochastic calculus, Cambridge Univ. Press, May 2004, to appear.

[Ba1]    Bass, Richard F. Uniqueness in law for pure jump Markov processes. Probab. Theory Related Fields 79 (1988), no. 2, 271–287. MR958291

[Ba2]    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

[Ba3]    Bass, Richard F. Probabilistic techniques in analysis. Probability and its Applications (New York). Springer-Verlag, New York, 1995. MR1329542

[Ba4]    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

[Ba5]    Bass, Richard F. Stochastic calculus for discontinuous processes, http://www.math.uconn.edu/~bass/scdp.pdf

[Ba6]    Bass, Richard F. General theory of processes, http://www.math.uconn.edu/~bass/gtp.pdf

[BBC]   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

[BK]    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

[BL1]   Bass, Richard F.; Levin, David A. Harnack inequalities for jump processes. Potential Anal. 17 (2002), no. 4, 375–388. MR1918242

[BL2]   Bass, Richard F.; Levin, David A. Transition probabilities for symmetric jump processes. Trans. Amer. Math. Soc. 354 (2002), no. 7, 2933–2953. MR1895210

[Ch]    Chen, Zhen-Qing. Multidimensional symmetric stable processes. Korean J. Comput. Appl. Math. 6 (1999), no. 2, 227–266. MR1687746

[CK]    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. MR2008600

[CJ]    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

[DM1]   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. MR488194

[DM2]   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

[Fu]    Fujiwara, Tsukasa. Stochastic differential equations of jump type on manifolds and Lévy flows. J. Math. Kyoto Univ. 31 (1991), no. 1, 99–119. MR1093330

[FK]    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

[FOT]   Fukushima, Masatoshi; ˉO   shima, Yˉo   ichi; Takeda, Masayoshi. Dirichlet forms and symmetric Markov processes. de Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 1994. MR1303354

[HWY]   He, Sheng Wu; Wang, Jia Gang; Yan, Jia An. Semimartingale theory and stochastic calculus. CRC Press, Boca Raton, FL, 1992. MR1219534

[Ho1]   Hoh, Walter. The martingale problem for a class of pseudo-differential operators. Math. Ann. 300 (1994), no. 1, 121–147. MR1289834

[Ho2]   Hoh, Walter. Pseudodifferential operators with negative definite symbols and the martingale problem. Stochastics Stochastics Rep. 55 (1995), no. 3-4, 225–252. MR1378858

[Ho3]   Hoh, Walter. Pseudo differential operators with negative definite symbols of variable order. Rev. Mat. Iberoamericana 16 (2000), no. 2, 219–241. MR1809340

[Ho4]   Hoh, Walter. Perturbations of pseudodifferential operators with negative definite symbol. Appl. Math. Optim. 45 (2002), no. 3, 269–281. MR1885821

[HJ1]   Hoh, Walter; Jacob, Niels. Some Dirichlet forms generated by pseudo differential operators. Bull. Sci. Math. 116 (1992), no. 3, 383–398. MR1177287

[HJ2]   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

[Ja1]    Jacob, Niels. Further pseudodifferential operators generating Feller semigroups and Dirichlet forms. Rev. Mat. Iberoamericana 9 (1993), no. 2, 373–407. MR1232848

[Ja2]    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

[Ja3]    Jacob, Niels. Pseudo-differential operators and Markov processes. Mathematical Research, 94. Akademie Verlag, Berlin, 1996. MR1409607

[Ja4]    Jacob, Niels. Pseudo differential operators and Markov processes. Vol. I. Fourier analysis and semigroups. Imperial College Press, London, 2001. MR1873235

[Ja5]    Jacob, Niels. Pseudo differential operators & Markov processes. Vol. II. Generators and their potential theory. Imperial College Press, London, 2002. (CMP 1 917 230)

[JL]    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

[JS]    Jacob, Niels; Schilling, René L. Lévy-type processes and pseudodifferential operators. Lévy processes, 139–168, Birkhäuser, Boston, MA, 2001. MR1833696

[Jd]    Jacod, Jean. Calcul stochastique et problèmes de martingales. Lecture Notes in Mathematics, 714. Springer, Berlin, 1979. MR542115

[JMW]   Janicki, A.; Michna, Z.; Weron, A. Approximation of stochastic differential equations driven by α-stable Lévy motion. Appl. Math. (Warsaw) 24 (1996), no. 2, 149–168. MR1424549

[Kl1]    Kolokoltsov, Vassili. Symmetric stable laws and stable-like jump-diffusions. Proc. London Math. Soc. (3) 80 (2000), no. 3, 725–768. MR1744782

[Kl2]    Kolokoltsov, Vassili. On Markov processes with decomposable pseudo-differential generators, preprint, http://www.ima.umn.edu/prob-pde/reprints-preprints/kolokoltsov/fel.pdf

[Km1]   Komatsu, Takashi. Markov processes associated with certain integro-differential operators. Osaka J. Math. 10 (1973), 271–303. MR359017

[Km2]   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. MR683262

[Km3]   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

[Km4]   Komatsu, Takashi. On the martingale problem for generators of stable processes with perturbations. Osaka J. Math. 21 (1984), no. 1, 113–132. MR736974

[Km5]   Komatsu, Takashi. On stable-like processes. Probability theory and mathematical statistics (Tokyo, 1995), 210–219, World Sci. Publishing, River Edge, NJ, 1996. MR1467941

[Km6]   Komatsu, Takashi. Uniform estimates for fundamental solutions associated with non-local Dirichlet forms. Osaka J. Math. 32 (1995), no. 4, 833–860. MR1380729

[Ku]    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

[LG]    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

[LM]    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

[Ly]    Lyons, Terry J. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998), no. 2, 215–310. MR1654527

[Me]    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. MR501332

[Na]    Nakao, S. On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations. Osaka J. Math. 9 (1972), 513–518. MR326840

[Ne]    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

[NT]    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

[PZ]    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

[Pr]    Protter, Philip. Stochastic integration and differential equations, 2nd ed. Applications of Mathematics, 21. Springer-Verlag, Berlin, 2004. MR2020294

[Ro]    Rong, Situ. On solutions of backward stochastic differential equations with jumps and applications. Stochastic Process. Appl. 66 (1997), no. 2, 209–236. MR1440399

[Sk]    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

[SV]    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

[St]    Stroock, Daniel W. Diffusion processes associated with Lévy generators. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32 (1975), no. 3, 209–244. MR433614

[TTW]   Tanaka, Hiroshi; Tsuchiya, Masaaki; Watanabe, Shinzo. Perturbation of drift-type for Lévy processes. J. Math. Kyoto Univ. 14 (1974), 73–92. MR368146

[Ts1]    Tsuchiya, Masaaki. On a small drift of Cauchy process. J. Math. Kyoto Univ. 10 (1970), 475–492. MR307361

[Ts2]    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

[Ts3]    Tsuchiya, Masaaki. Lévy measure with generalized polar decomposition and the associated SDE with jumps. Stochastics Stochastics Rep. 38 (1992), no. 2, 95–117. MR1274897

[Ue]    Uemura, Toshihiro. On some path properties of symmetric stable-like processes for one dimension. Potential Anal. 16 (2002), no. 1, 79–91. MR1880349

[WW]   von Weizsäcker, Heinrich; Winkler, Gerhard. Stochastic integrals. An introduction. Friedr. Vieweg & Sohn, Braunschweig, 1990. MR1062600

[Wi]    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

[YW]   Yamada, Toshio; Watanabe, Shinzo. On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11 1971 155–167. MR278420

[Za1]    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

[Za2]    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




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