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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.

MSC:

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

Citations:

Zbl 0281.60064
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References:

[1] Aldous, D.: Stopping times and tightness. Ann. Probab. 6, 335–340 (1978) · Zbl 0391.60007 · doi:10.1214/aop/1176995579
[2] Anulova, S., Pragarauskas, H.: On strong Markov weak solutions of stochastic equations. Liet. Mat. Rink. XVII, 5–26 (1977) · Zbl 0381.60053
[3] Bertoin, J.: Levy Processes. Cambridge University Press (1996) · Zbl 0861.60003
[4] Dellacherie, C., Meyer, P.A.: Probabilities et potentiels B. Hermann, Paris (1980)
[5] Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. North-Holland, Tokyo (1989) · Zbl 0684.60040
[6] Krylov, N.V.: Controlled Diffusion Processes. Springer, New York (1980) · Zbl 0436.93055
[7] Kurenok, V.P.: A note on L 2-estimates for stable integrals with drift. Trans. Am. Math. Soc., to appear · Zbl 1137.60029
[8] Melnikov, A.V.: Stochastic equations and Krylov’s estimates for semimartingales. Stoch. Stoch. Rep. 10, 81–102 (1983) · Zbl 0539.60059
[9] Pragarauskas, H.: On L p -estimates of stochastic integrals. In: Grigelionis, B., et al. (eds.) Probab. Theory and Math. Stat., pp. 579–588. VSP, Utrecht/TEV, Vilnius (1999) · Zbl 0994.60059
[10] Tanaka, H., Tsuchiya, M., Watanabe, S.: Perturbation of drift-type for Levy processes. J. Math. Kyoto Univ. 14(1), 73–92 (1974) · Zbl 0281.60064
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