Abstract
The time-dependent SDE dX t = b(t, X t−)dZ t with X 0 = x 0 ∈ ℝ, and a symmetric α-stable process Z, 1 < α ⩽ 2, is considered. We study the existence of nonexploding solutions of the given equation through the existence of solutions of the equation \(dA_t = \left| b \right|^\alpha (t,\bar Z \circ A_t )dt\) in class of time change processes, where \(\bar Z\) is a symmetric stable process of the same index α as Z. The approach is based on using the time change method, Krylov’s estimates for stable integrals, and properties of monotone convergence. The main existence result extends the results of Pragarauskas and Zanzotto (2000) for 1 < α < 2 and those of T. Senf (1993) for α = 2.
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References
H. J. Engelbert and V. P. Kurenok, On one-dimensional stochastic equations driven by symmetric stable processes, in: Stochastic Processes and Related Topics, R. Buckdahn, H. J. Engelbert, and M. Yor (Eds.), (2002), pp. 81–110.
H. J. Engelbert and W. Schmidt, Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations, III, Math. Nachr., 151, 149–197 (1991).
J. Jacod, Calcul Stochastique et Problèmes de Martingales, Lecture Notes Math., vol. 714, Springer, Berlin (1979).
O. Kallenberg, Foundations of Modern Probability, Springer, Berlin (1997).
N. V. Krylov, Controlled Diffusion Processes, Springer, Berlin (1982).
I. P. Natanson, Theory of Functions of a Real Variable (in Russian), Gostekhizdat, Moscow (1954).
H. Pragarauskas and P. A. Zanzotto, On one-dimensional stochastic differential equations driven by stable processes, Lith. Math. J., 40(1), 1–24 (2000).
P. Raupach, On driftless one-dimensional SDEs with time-dependent diffusion coefficients, Stochastics Stochastic Rep., 67, 207–230 (1998).
A. Rozkosz and L. Słomiński, On weak solutions of one-dimensional SDEs with time-dependent coefficients, Stochastics Stochastic Rep., 42, 199–208 (1993).
T. Senf, On one-dimensional stochastic differential equations without drift and with time-dependent diffusion coefficients, Stochastics Stochastic Rep., 43, 199–220 (1993).
P. A. Zanzotto, Representation of a class of semimartingales as stable integrals, Theory Probab. Appl., 43(4), 808–818 (1998).
P. A. Zanzotto, On stochastic differential equations driven by Cauchy process and the other stable Lévy motions, Ann. Probab., 30(2), 802–825 (2002).
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Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 517–531, October–December, 2007.
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Kurenok, V.P. On driftless one-dimensional SDE’s with respect to stable Levy processes. Lith Math J 47, 423–435 (2007). https://doi.org/10.1007/s10986-007-0030-x
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DOI: https://doi.org/10.1007/s10986-007-0030-x