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# The Time Change Method and SDEs with Nonnegative Drift

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Published:2010-05-11
Printed: Sep 2010
• V. P. Kurenok,
Department of Natural and Applied Sciences, University of Wisconsin-Green Bay, Green Bay, WI, USA
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## Abstract

Using the time change method we show how to construct a solution to the stochastic equation $dX_t=b(X_{t-})dZ_t+a(X_t)dt$ with a nonnegative drift $a$ provided there exists a solution to the auxililary equation $dL_t=[a^{-1/\alpha}b](L_{t-})d\bar Z_t+dt$ where $Z, \bar Z$ 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: One-dimensional SDEs, symmetric stable processes, nonnegative drift, time change, integral estimates, weak convergence
 MSC Classifications: 60H10 - Stochastic ordinary differential equations [See also 34F05] 60J60 - Diffusion processes [See also 58J65] 60J65 - Brownian motion [See also 58J65] 60G44 - Martingales with continuous parameter

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