A note on $L_2$-estimates for stable integrals with drift
HTML articles powered by AMS MathViewer
- by Vladimir Kurenok PDF
- Trans. Amer. Math. Soc. 360 (2008), 925-938 Request permission
Abstract:
Let $X$ be of the form $X_t=\int _0^tb_sdZ_s+\int _0^ta_sds, t\ge 0,$ where $Z$ is a symmetric stable process of index $\alpha \in (1,2)$ with $Z_0=0$. We obtain various $L_2$-estimates for the process $X$. In particular, for $m\in \mathbb N, t\ge 0,$ and any measurable, nonnegative function $f$ we derive the inequality \[ {\mathbf E}\int _0^{t\land \tau _m(X)}|b_s|^{\alpha }f(X_s)ds\le N\|f\|_{2,m}.\] As an application of the obtained estimates, we prove the existence of solutions for the stochastic equation $dX_t=b(X_{t-})dZ_t+a(X_t)dt$ for any initial value $x_0\in \mathbb R$.References
- David Aldous, Stopping times and tightness, Ann. Probability 6 (1978), no. 2, 335â340. MR 474446, DOI 10.1214/aop/1176995579
- S. Anulova and G. Pragarauskas, Weak Markov solutions of stochastic equations, Litovsk. Mat. Sb. 17 (1977), no. 2, 5â26, 219 (Russian, with Lithuanian and English summaries). MR 0651573
- Richard F. Bass, Stochastic differential equations driven by symmetric stable processes, SĂ©minaire de ProbabilitĂ©s, XXXVI, Lecture Notes in Math., vol. 1801, Springer, Berlin, 2003, pp. 302â313. MR 1971592, DOI 10.1007/978-3-540-36107-7_{1}1
- Claude Dellacherie and Paul-André Meyer, Probabilités et potentiel. Chapitres V à VIII, Revised edition, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1385, Hermann, Paris, 1980 (French). Théorie des martingales. [Martingale theory]. MR 566768
- H. J. Engelbert and W. Schmidt, On one-dimensional stochastic differential equations with generalized drift, Stochastic differential systems (Marseille-Luminy, 1984) Lect. Notes Control Inf. Sci., vol. 69, Springer, Berlin, 1985, pp. 143â155. MR 798317, DOI 10.1007/BFb0005069
- H. J. Engelbert and W. Schmidt, Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations. III, Math. Nachr. 151 (1991), 149â197. MR 1121203, DOI 10.1002/mana.19911510111
- Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, 2nd ed., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. MR 1011252
- Jean Jacod, Calcul stochastique et problĂšmes de martingales, Lecture Notes in Mathematics, vol. 714, Springer, Berlin, 1979 (French). MR 542115, DOI 10.1007/BFb0064907
- Olav Kallenberg, Foundations of modern probability, Probability and its Applications (New York), Springer-Verlag, New York, 1997. MR 1464694
- N. V. Krylov, Controlled diffusion processes, Applications of Mathematics, vol. 14, Springer-Verlag, New York-Berlin, 1980. Translated from the Russian by A. B. Aries. MR 601776, DOI 10.1007/978-1-4612-6051-6
- J.-P. Lepeltier and B. Marchal, ProblĂšme des martingales et Ă©quations diffĂ©rentielles stochastiques associĂ©es Ă un opĂ©rateur intĂ©gro-diffĂ©rentiel, Ann. Inst. H. PoincarĂ© Sect. B (N.S.) 12 (1976), no. 1, 43â103 (French). MR 0413288
- A. V. MelâČnikov, Stochastic equations and Krylovâs estimates for semimartingales, Stochastics 10 (1983), no. 2, 81â102. MR 716817, DOI 10.1080/17442508308833265
- N. I. Portenko, Some perturbations of drift-type for symmetric stable processes, Random Oper. Stochastic Equations 2 (1994), no. 3, 211â224. MR 1310558, DOI 10.1515/rose.1994.2.3.211
- H. Pragarauskas, On $L^p$-estimates of stochastic integrals, In: âProbab. Theory and Math. Statist.â, B.Grigelionis et al. (eds.), 579-588, VSP, Utrecht/TEV, Vilnius, 1999.
- G. Pragarauskas and P. A. Zanzotto, On one-dimensional stochastic differential equations with respect to stable processes, Liet. Mat. Rink. 40 (2000), no. 3, 361â385 (Russian, with English and Lithuanian summaries); English transl., Lithuanian Math. J. 40 (2000), no. 3, 277â295. MR 1803652, DOI 10.1007/BF02465137
- Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357, DOI 10.1007/978-3-662-06400-9
- J. RosiĆski and W. A. WoyczyĆski, On ItĂŽ stochastic integration with respect to $p$-stable motion: inner clock, integrability of sample paths, double and multiple integrals, Ann. Probab. 14 (1986), no. 1, 271â286. MR 815970, DOI 10.1214/aop/1176992627
- Hiroshi Tanaka, Masaaki Tsuchiya, and Shinzo Watanabe, Perturbation of drift-type for LĂ©vy processes, J. Math. Kyoto Univ. 14 (1974), 73â92. MR 368146, DOI 10.1215/kjm/1250523280
- P. A. Zanzotto, Representation of a class of semimartingales as stable integrals, Teor. Veroyatnost. i Primenen. 43 (1998), no. 4, 808â818 (English, with Russian summary); English transl., Theory Probab. Appl. 43 (1999), no. 4, 666â676. MR 1692452, DOI 10.1137/S0040585X97977252
- Pio Andrea Zanzotto, On stochastic differential equations driven by a Cauchy process and other stable LĂ©vy motions, Ann. Probab. 30 (2002), no. 2, 802â825. MR 1905857, DOI 10.1214/aop/1023481008
Additional Information
- Vladimir Kurenok
- Affiliation: Department of Natural and Applied Sciences, University of Wisconsin-Green Bay, 2420 Nicolet Drive, Green Bay, Wisconsin 54311-7001
- Email: kurenokv@uwgb.edu
- Received by editor(s): October 4, 2005
- Received by editor(s) in revised form: December 1, 2005
- Published electronically: September 25, 2007
- © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 360 (2008), 925-938
- MSC (2000): Primary 60H10, 60J60, 60J65, 60G44
- DOI: https://doi.org/10.1090/S0002-9947-07-04234-1
- MathSciNet review: 2346477