The Time Change Method and SDEs with Nonnegative Drift

Kurenok, V. P.

Canadian mathematical bulletin, Tome 53 (2010) no. 3, pp. 503-515

Résumé

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.

Détail

DOI : 10.4153/CMB-2010-048-9

Mots-clés : 60H10, 60J60, 60J65, 60G44, One-dimensional SDEs, symmetric stable processes, nonnegative drift, time change, integral estimates, weak convergence