Abstract
The stochastic equation dX t =dS t +a(t,X t )dt, t0, is considered where S is a one-dimensional Levy process with the characteristic exponent ψ(ξ),ξ â. We prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial value X 0=x 0 â when (e∈ ψ(ξ))-1=o(|ξ|-1) as |ξ|→∞. These conditions coincide with those found by Tanaka, Tsuchiya and Watanabe (J. Math. Kyoto Univ. 14(1), 73-92, 1974) 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.
Original language | English |
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Pages (from-to) | 859-869 |
Number of pages | 11 |
Journal | Journal of Theoretical Probability |
Volume | 20 |
Issue number | 4 |
DOIs | |
State | Published - Dec 2007 |
Keywords
- Krylov's estimates
- One-dimensional Levy processes
- Time-dependent drift
- Weak convergence