Abstract
The mild Ito formula proposed in Theorem 1 in [Da Prato, G., Jentzen, A., \& R\"ockner, M., A mild Ito 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 It\^o 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.
Highlights
The standard Ito formula for finite dimensional Ito processes has been generalized in the literature to infinite dimensions so that it is applicable to Ito processes with values in infinite dimensional Hilbert or Banach spaces; see Theorem 2.4 in Brzezniak, Van Neerven, Veraar & Weiss [5]
This work is partly financed by the NWO-research programme VENI Vernieuwingsimpuls with project number 639.031.549. It is partly financed by SNSF-Research project 200021 156603 “Numerical approximations of nonlinear stochastic ordinary and partial differential equations”. ∗ Corresponding author: Sonja Cox
Assume the setting in Subsection 3.1, let X : [t0, T ] × Ω → V be a mild Ito process with evolution family S : ∠ → L(V, V ), mild drift Y : [t0, T ] × Ω → V, and mild diffusion Z : [t0, T ] × Ω → γ(U, V ), let X : [t0, T ] × Ω → Vbe a stochastic process with continuous sample paths which satisfies ∀ t ∈ [t0, T ) : P Xt = St,T Xt = 1, let φ ∈ C2(V, V), and let τ : Ω → [t0, T ] be an F-stopping time which satisfies that min E
Summary
The standard Ito formula for finite dimensional Ito processes has been generalized in the literature to infinite dimensions so that it is applicable to Ito processes with values in infinite dimensional Hilbert or Banach spaces; see Theorem 2.4 in Brzezniak, Van Neerven, Veraar & Weiss [5]. Note that Lemma 2.3 (with (Ω, F , ν) = (O, B(O), λO), E = Rk, E = L(m+1)(Rk, Rl), p = p(1 + ε)(1 + m), q = p(1 + 1/ε), φ = f (m+1), f0 = v, fj = v + rhj for r ∈ [0, 1], j ∈ N, v ∈ Lq(λO; Rk), (hj )j∈N ∈ {(uj )j∈N ⊆ Lp(1+ε)(1+m)(λO; Rk) : lim supj→∞ uj Lp(1+ε)(1+m)(λO;Rk) = 0}, ε ∈ (0, ∞) in the notation of Lemma 2.3), the fact that supx∈Rk f (m+1)(x) L(m+1)(Rk,Rl) < ∞, and the fact that f (m+1) is continuous ensure that for all r ∈ [0, 1], v ∈ Lq(λO; Rk), ε ∈ (0, ∞), (hj)j∈N ⊆ Lp(1+ε)(1+m)(λO; Rk) with lim supj→∞ hj Lp(1+ε)(1+m)(λO;Rk) = 0 it holds that lim sup j→∞.
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