Approximations of the Wong–Zakai type for stochastic differential equations in M-type 2 Banach spaces with applications to loop spaces

On Asymptotics of Banach Space-valued Itô Functionals of Brownian Rough Paths

“…In this subsection, we consider SDEs on Mtype 2 Banach spaces. For detailed explanations and further references, we refer the reader to see Brzeźniak-Carroll [5] and Brzeźniak-Elworthy [6]. Let (X, H, µ) be an abstract Wiener space and w = (w t ) t≥0 be the Xvalued Brownian motion.…”

“…See Theorem 2.26 in [6] and Theorem 2 in [5] for the detail. Moreover they have already established the Wong-Zakai approximation theorem (Theorem 3 in [5]) for the SDE (25). Then by the same argument as in the previous subsection, we can obtain…”

Set-valued stochastic differential equation in M-type 2 Banach space

“…There are several references that consider stochastic integrals and stochastic differential equations in abstract Banach spaces (see for example [2,4,3,7,8,9,16,17,23,22] etc. ).…”

“…At first, mainly based on the work [4,3], we study precisely stochastic integrals of B(X , X )-valued random functions with respect to an X -valued Brownian motion, where B(X , X ) is the set of all bounded linear operators from X to X . Let L 2 ((X , µ), X ) denote a separable Banach space of all Borel measurable functions f : (X , B(X ), µ) → (X , B(X )) such that the norm of f is defined by…”

“…In a separable M-type 2 Banach space, Brzezniak [2] studied the stochastic integral with respect to a finite dimensional Brownian motion and the corresponding stochastic partial differential equations. Stochastic integration of operator-valued functions with respect to the Banach space-valued Wiener process is described briefly in [4,3].…”