Stochastic equations with time-dependent drift driven by Lévy processes. (English) Zbl 1145.60033

Author’s summary: The stochastic equation \(dX_{t}= dS_{t}+ a (t, X_{t}) dt, t \geq 0\), is considered where \(S\) is a one-dimensional Levy process with the characteristic exponent \(\psi (\xi)\), \(\xi \in \mathbb R\). We prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial value \(X_{0}= x_{0}\in \mathbb R\) when \((e \psi (\xi))^{- 1}= o (|\xi |^{- 1})\) as \(| \xi |\rightarrow \infty\). These conditions coincide with those found by H. Tanaka, M. Tsuchiya and S. Watanabe [J. Math. Kyoto Univ. 14, 73–92 (1974; Zbl 0281.60064)] in the case of a \((t, x)= a(x)\). Our approach is based on Krylov’s estimates for Levy processes with time-dependent drift. Some variants of those estimates are derived in this note.


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60B10 Convergence of probability measures
60G52 Stable stochastic processes
60G99 Stochastic processes


Zbl 0281.60064
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