The Symbol Associated with the Solution of a Stochastic Differential Equation∗
نویسندگان
چکیده
Let (Zt)t3⁄40 be an Rn-valued Lévy process. We consider stochastic differential equations of the form dX x t = Φ(X x t−) dZt X x 0 = x , x ∈R d , where Φ : Rd → Rd×n is Lipschitz continuous. We show that the infinitesimal generator of the solution process (X x t )t3⁄40 is a pseudo-differential operator whose symbol p : R d ×Rd → C can be calculated by p(x ,ξ) :=− lim t↓0 Ex ei(X σ t −x) >ξ − 1 t ! . For a large class of Feller processes many properties of the sample paths can be derived by analysing the symbol. It turns out that the process (X x t )t3⁄40 is a Feller process if Φ is bounded and that the symbol is of the form p(x ,ξ) =ψ(Φ>(x)ξ), where ψ is the characteristic exponent of the driving Lévy process. ∗Acknowledgements: We would like to thank an anonymous referee for carefully reading the manuscript and offering useful suggestions which helped to improve the paper. †Institut für Mathematische Stochastik, Technische Universität Dresden, D-01062 Dresden, Germany, [email protected] ††Lehrstuhl IV, Fakultät für Mathematik, Technische Universität Dortmund, D-44227 Dortmund, Germany, [email protected]
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