A Series Approach to Stochastic Differential Equations with Infinite Dimensional Noise

Abstract

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.

Cylindrical Wiener Processes

“…Cylindrical Wiener processes in Banach or Hilbert spaces and their integral are treated for example in the monographs Da Prato and Zabcyzk [3], Kallianpur [5] and Metivier and Pellaumail [6]. In van Gaans [4] the series representation of the cylindrical Wiener process is used to define a stochastic integral in Hilbert spaces and in Berman and Root [1] an approach similar to ours is introduced. The fundamental observation in this work that not every Gaussian cylindrical measure has a nice covariance operator was pointed out to me the first time by Dave Applebaum.…”

Differential equations driven by Lévy white noise in spaces of Hilbert space-valued stochastic distributions

“…And as has been shown in Ref. [28] for the case of Itô integration with respect to a weak Wiener process, L 2 (H, U) is the appropriate space for integrands to take values. The same makes sense for integration with respect to a weak Lévy process, as we will see.…”

Existence of Lévy term structure models

“…Remark. If the driving process X is a (possibly infinite dimensional) Brownian motion, the equivalence of the van Gaans integral with the usual stochastic integral (see Da Prato and Zabczyk [13] for the infinite dimensional case) is provided in [26,Sec. 3].…”

“…, X n ), also taking into account the occurrence of jumps. If X is a Lévy process, this leads to df (t, T ) = α HJM (t, T )dt + We are grateful to Bohdan Maslowski, Barbara Rüdiger, Josef Teichmann and Jerzy Zabczyk for their helpful remarks and discussions, and an anonymous referee for bringing our attention to the results of van Gaans [26,27].…”

“…The ground for this result, Theorem C.1, is prepared by two works of van Gaans [26,27]. In addition to his result [27, Thm.…”