On driftless one-dimensional SDE’s with respect to stable Levy processes
- 39 Downloads
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.
Keywordsone-dimensional stochastic equations measurable coefficients symmetric stable processes time change equation monotone convergence
Unable to display preview. Download preview PDF.
- 1.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.Google Scholar
- 5.N. V. Krylov, Controlled Diffusion Processes, Springer, Berlin (1982).Google Scholar
- 6.I. P. Natanson, Theory of Functions of a Real Variable (in Russian), Gostekhizdat, Moscow (1954).Google Scholar