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The Time Change Method and SDEs with Nonnegative Drift

  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 One-dimensional SDEs, symmetric stable processes, nonnegative drift, time change, integral estimates, weak convergence
MSC Classifications: 60H10, 60J60, 60J65, 60G44 show english descriptions Stochastic ordinary differential equations [See also 34F05]
Diffusion processes [See also 58J65]
Brownian motion [See also 58J65]
Martingales with continuous parameter
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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