We consider a stochastic delay differential equation driven by a general Lévy process. Both, the drift and the noise term may depend on the past, but only the drift term is assumed to be linear. We show that the segment process is eventually Feller, but in general not eventually strong Feller on the Skorokhod space. The existence of an invariant measure is shown by proving tightness of the segments using semimartingale characteristics and the Krylov-Bogoliubov method. A counterexample shows that the stationary solution in completely general situations may not be unique, but in more specific cases uniqueness is established.
A stochastic integral of Banach space valued deterministic functions with respect to Banach space valued Lévy processes is defined. There are no conditions on the Banach spaces nor on the Lévy processes. The integral is defined analogously to the Pettis integral. The integrability of a function is characterized by means of a radonifying property of an integral operator associated to the integrand. The integral is used to prove a Lévy-Itô decomposition for Banach space valued Lévy processes and to study existence and uniqueness of solutions of stochastic Cauchy problems driven by Lévy processes.
In a vector lattice, ideals and bands are well-investigated subjects. We study similar notions in a pre-Riesz space. The pre-Riesz spaces are exactly the order dense linear subspaces of vector lattices. Restriction and extension properties of ideals, solvex ideals and bands are investigated. Since every Archimedean directed partially ordered vector space is pre-Riesz, we establish properties of ideals and bands in such spaces.
Mathematics Subject Classification (2000). Primary 46A40; Secondary 06F20.
A generalization ofÉmery's inequality for stochastic integrals is shown for convolution integrals of the form (t 0 g(t − s)Y (s−) dZ(s)) t 0 , where Z is a semimartingale, Y an adapted càdlàg process, and g a deterministic function. The function g is assumed to be absolutely continuous with a derivative that is continuous or of bounded variation or a sum of such functions. The function g may also have jumps, as long as the jump sizes are absolutely summable. The inequality is used to prove existence and uniqueness of solutions of equations of variation-of-constants type. As a consequence, it is shown that the solution of a stochastic delay differential equation with linear drift, bounded functional Lipschitz diffusion coefficient, and noise driven by a general semimartingale satisfies a variation-of-constants formula. The proof of the inequality consists of an approximation argument and a reduction toÉmery's original inequality.
This paper considers semilinear stochastic differential equations in Hilbert spaces with Lipschitz nonlinearities and with the noise terms driven by sequences of independent scalar Wiener processes (Brownian motions). The interpretation of such equations requires a stochastic integral. By means of a series of Itô integrals, an elementary and direct construction of a Hilbert space valued stochastic integral with respect to a sequence of independent scalar Wiener processes is given. As an application, existence and strong and weak uniqueness for the stochastic differential equation are shown by exploiting the series construction of the integral.
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