Stochastic Equations with Time-Dependent Drift Driven by Levy Processes |
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Authors: | V P Kurenok |
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Institution: | (1) Department of Natural and Applied Sciences, University of Wisconsin-Green Bay, 2420 Nicolet Drive, Green Bay, WI 54311-7001, USA |
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Abstract: | The stochastic equation dX
t
=dS
t
+a(t,X
t
)dt, t≥0, is considered where S is a one-dimensional Levy process with the characteristic exponent ψ(ξ),ξ∈ℝ. We prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial value X
0=x
0∈ℝ when (ℛe
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. |
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Keywords: | One-dimensional Levy processes Time-dependent drift Krylov’ s estimates Weak convergence |
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