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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
DOI: https://doi.org/10.1090/S0002-9947-07-04234-1
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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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

DOI: https://doi.org/10.1090/S0002-9947-07-04234-1
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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