On the mild Itô formula in Banach spaces

Abstract

The mild Itô formula proposed in Theorem 1 in [Da Prato, G., Jentzen, A., & Röckner, M., A mild Itô formula for SPDEs, arXiv:1009.3526 (2012, To appear in the Trans. Amer. Math. Soc.] has turned out to be a useful instrument to study solutions and numerical approximations of stochastic partial differential equations (SPDEs) which are formulated as stochastic evolution equations (SEEs) on Hilbert spaces. In this article we generalize this mild Itô formula so that it is applicable to solutions and numerical approximations of SPDEs which are formulated as SEEs on UMD (unconditional martingale differences) Banach spaces. This generalization is especially useful for proving essentially sharp weak convergence rates for numerical approximations of SPDEs.

Weak convergence rates for numerical approximations of stochastic partial differential equations with nonlinear diffusion coefficients in UMD Banach spaces

“…Combining Proposition 2.6 in [15] (with k = k, l = k, d = d, n = n, p = q, q = p, O = (0, 1) d , f = f , F = G in the notation of Proposition 2.6 in [15]), (252), and the fact that I ∈ L(L q (λ (0,1) d ; R k ), V α ) hence establishes items (ii)-(vi). The proof of Corollary 9.1 is thus completed.…”

“…Throughout this article we frequently use the following elementary lemma (see, e.g., [11,Lemma 2.3] and [15,Lemma 2.2]).…”

“…It thus remains to prove (50). To do so, we apply the mild Itô formula in [15]. More formally, an application of Proposition 3.11 in [15] shows that for all (s, t) ∈ ∠ it holds that E ψ(e…”

“…It thus remains to provide a suitable estimate for the first summand on the right hand side of (53). For this we will employ the mild Itô formula in [15] again and this will allow us to obtain an appropriate upper bound for E ψ(e…”

“…It thus remains to prove (84). To do so, we make use of the mild Itô formula in [15]. For this let Fs,t : V × V × V → V, (s, t) ∈ ∠, be the functions which satisfy for all (s, t) ∈ ∠, u, v, w ∈ V that Fs,t (u, v, w) = ψ 1,0 e (t−s)A u, e (t−s)A v − ψ 1,0 e (t−s)A u, e (t−s)A u e (t−s)A F (w) + ψ 0,1 e (t−s)A u, e (t−s)A v e (t−⌊s⌋ h )A F (w) − ψ 0,1 e (t−s)A u, e (t−s)A u e (t−s)A F (w) (87) and let Bs,t : V ×V ×V → V, (s, t) ∈ ∠, be the functions which satisfy for all (s, t) ∈ ∠, u, v, w ∈ V that Bs,t (u, v, w)…”