Remarks on Taylor Series Expansions and Conditional Expectations for Stratonovich SDEs with Complete V‐Commutativity

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.