On stochastic equations with measurable coefficients driven by symmetric stable processes
type: Item
instance of: scholarly article
title: On stochastic equations with measurable coefficients driven by symmetric stable processes
zbMATH Open document ID: 1241.60027
DOI: 10.1155/2012/258415
author: V. P. Kurenok
published in: International Journal of Stochastic Analysis
publication date: 2012-05-08T00:00:00Z
review text: Summary: We consider a one-dimensional stochastic equation \[ dX_t = b(t, X_{t-})dZ_t + a(t, X_t)dt,\;t \geq 0, \] with respect to a symmetric stable process \(Z\) of index \(0 < \alpha \leq 2\). It is shown that solving this equation is equivalent to solving of a two-dimensional stochastic equation \[ dL_t = B(L_{t-})dWt \] with respect to the semimartingale \(W = (Z, t)\) and the corresponding matrix \(B\). In the case of \(1 \leq \alpha < 2\), we provide new sufficient conditions for the existence of solutions of both equations with measurable coefficients. Existence proofs are established using the method of Krylov's estimates for processes satisfying the two-dimensional equation. Also, the Krylov estimates are based on some analytical facts of independent interest that are also proved.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60G52
zbMATH DE Number: 6031633
zbMATH Keywords: Krylov's estimates
zbMATH Keywords: existence of solutions
Wikidata QID: Q58689554
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2012/258415
OpenAlex ID: W2065690617
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description: scientific article; zbMATH DE number 6031633