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Transactions of the American Mathematical Society

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A note on $ L_2$-estimates for stable integrals with drift

Author: Vladimir Kurenok
Journal: Trans. Amer. Math. Soc. 360 (2008), 925-938
MSC (2000): Primary 60H10, 60J60, 60J65, 60G44
Published electronically: September 25, 2007
MathSciNet review: 2346477
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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

$\displaystyle {\mathbf E}\int_0^{t\land\tau_m(X)}\vert b_s\vert^{\alpha}f(X_s)ds\le N\Vert f\Vert _{2,m}.$

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}$.

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  • [A] D. Aldous, Stopping times and tightness, Ann. Prob., 6 (1978), 335-340. MR 0474446 (57:14086)
  • [AP] S. Anulova and H. Pragarauskas, On strong Markov weak solutions of stochastic equations, Liet. Math. Rinkinys, XVII (1977), 5-26. MR 58:31397
  • [B] R. Bass, Stochastic differential equations driven by symmetric stable processes, Séminaire de Probabilités, XXXVI, Lecture Notes in Math., 1801, 302-313, Springer, Berlin, 2003. MR 1971592 (2004b:60143)
  • [DM] C. Dellacherie and P.A. Meyer, Probabilities et Potentiels B, Hermann, Paris, 1980. MR 566768 (82b:60001)
  • [ES1] H.J. Engelbert and W. Schmidt, On one-dimensional stochastic differential equations with generalized drift, Lect. Notes Control Inf. Sci., 69, 143-155, Springer, Berlin, 1985. MR 798317 (86m:60144)
  • [ES2] -, Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations III, Math. Nachr., 151 (1991), 149-197. MR 1121203 (92m:60044)
  • [IW] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publ., Tokyo, 1989. MR 1011252 (90m:60069)
  • [J] J. Jacod, Calcul Stochastique et Problèmes de Martingales. Lect. Notes Math., Vol. 714, Springer, Berlin, 1979. MR 542115 (81e:60053)
  • [K] O. Kallenberg, Foundations of Modern Probability, Springer, Berlin, 1997. MR 1464694 (99e:60001)
  • [KR] N.V. Krylov, Controlled Diffusion Processes, Springer, New York, 1980. MR 601776 (82a:60062)
  • [LM] J.P. Lepeltier and B. Marchal, Probléme des martingales et équations différentielles stochastiques associées á un opérateur intégro-différentiel, Annales IHP, Vol. 12, No. 1 (1976), 43-103. MR 0413288 (54:1403)
  • [M] A.V. Melnikov, Stochastic equations and Krylov's estimates for semimartingales, Stochastics and Stoch. Rep., 10 (1983), 81-102. MR 716817 (85j:60113)
  • [PO] N.I. Portenko, Some perturbations of drift-type for symmetric stable processes, Random Oper. and Stoch. Equ., Vol. 2, No. 3 (1994), 211-224. MR 1310558 (95k:60144)
  • [PR] H. Pragarauskas, On $ L^p$-estimates of stochastic integrals, In: ``Probab. Theory and Math. Statist.'', B.Grigelionis et al. (eds.), 579-588, VSP, Utrecht/TEV, Vilnius, 1999.
  • [PZ] H. Pragarauskas and P.A. Zanzotto, On one-dimensional stochastic differential equations driven by stable processes, Liet. Mat. Rink., 40 (2000), 1-24. MR 2003a:60091
  • [RY] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, Berlin, 1999. MR 1725357 (2000h:60050)
  • [RW] J. Rosinski and W. Woyczynski, On Itô stochastic integration with respect to p-stable motion: inner clock, integrability of sample paths, double and multiple integrals, Ann. Probab., 14 (1986), 271-286. MR 815970 (87h:60109)
  • [TTW] H. Tanaka, M. Tsuchiya, S. Watanabe, Perturbation of drift-type for Levy processes, Journal Math. Kyoto University, 14, No. 1 (1974), 73-92. MR 0368146 (51:4388)
  • [Z1] P.A. Zanzotto, Representation of a class of semimartingales as stable integrals, Theory Probab. Appl., Vol. 43, No. 4 (1998), 808-818. MR 1692452 (2000a:60107)
  • [Z2] P.A. Zanzotto, On stochastic differential equations driven by Cauchy process and the other stable Lévy motions, Ann. Probab., Vol. 30, No. 2 (2002), 802-825. MR 1905857 (2003d:60120)

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Additional Information

Vladimir Kurenok
Affiliation: Department of Natural and Applied Sciences, University of Wisconsin-Green Bay, 2420 Nicolet Drive, Green Bay, Wisconsin 54311-7001

Keywords: One-dimensional stochastic equations, bounded drift, Krylov's estimates, weak convergence, symmetric stable processes.
Received by editor(s): October 4, 2005
Received by editor(s) in revised form: December 1, 2005
Published electronically: September 25, 2007
Article copyright: © Copyright 2007 American Mathematical Society

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