Remarks on Taylor series expansions and conditional expectations for Stratonovich SDEs with complete V-commutativity

TL;DR: In this article, an infinite series expansion of the conditional expectation was obtained with respect to the concept of mean square convergence under the assumption of "infinite smoothness" of drift a(t, x) and diffusion coefficients b j (t, X) and with finite initial second moments.

Abstract: It may happen that there is not a finite maximum order bound for numerical approximations of stochastic processes X = (X t : 0 ≤ t ≤ T) satisfying Stratonovich stochastic differential equations (SDEs) with some commutative structure along an appropriate functional V(t, X t ). This statement can be proven with respect to the concept of mean square convergence under the assumption of “infinite smoothness” of drift a(t, x) and diffusion coefficients b j (t, x) and with finite initial second moments. As a result, we obtain an infinite series expansion of the conditional expectation 𝔼[V(t, X t )|ℱ t N ] on any fixed finite time interval [0, T], provided that the information is collected by discretized σ‐field ℱ T N = σ{W t 0 , W t 1 , …, W t N−1 , W T } at N + 1 given time instants t i ∈ [0, T] with t 0 ≤ t 1 ≤ ··· ≤ t N−1 ≤ t N = T.