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The Time Change Method and SDEs with Nonnegative Drift

Published online by Cambridge University Press:  20 November 2018

V. P. Kurenok
V. P. Kurenok*
Affiliation:
Department of Natural and Applied Sciences, University of Wisconsin-Green Bay, 2420 Nicolet Drive, Green Bay, WI 54311-7001, USA e-mail: kurenokv@uwgb.edu
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Abstract

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Using the time change method we show how to construct a solution to the stochastic equation $d{{X}_{t}}\,=\,b({{X}_{t}}\_)d{{Z}_{t}}\,+\,a({{X}_{t}})dt$ with a nonnegative drift a provided there exists a solution to the auxililary equation $d{{L}_{t}}=[{{a}^{-1/\alpha }}b]({{L}_{t}}\_)d\overline{{{Z}_{t}}}+dt$ where $Z,\,\overline{Z}$ are two symmetric stable processes of the same index $\alpha \,\in \,(0,\,2]$. This approach allows us to prove the existence of solutions for both stochastic equations for the values $0\,<\,\alpha \,<\,1$ and only measurable coefficients $a$ and $b$ satisfying some conditions of boundedness. The existence proof for the auxililary equation uses the method of integral estimates in the sense of Krylov.

Keywords

60H1060J6060J6560G44One-dimensional SDEssymmetric stable processesnonnegative drifttime changeintegral estimatesweak convergence
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 3 , 01 September 2010 , pp. 503 - 515
Copyright
Copyright © Canadian Mathematical Society 2010

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