New representations of the Taylor-Stratonovich expansion
Consider a regular functional \(R(x_s, s)\) of some \(\mathbb{R}^d\)-valued diffusion process \((x_s)\) which solves some well-behaved stochastic differential equation. A new variant of the Taylor expansion of \(R(x_s,s)\) near \(s= t\) is given, in terms of the multiple Stratonovich integrals \[ \int^s_t(t- \tau_1)^{l_k}\cdots \int^{\tau_{k-1}}_t (t- \tau_k)^{l_1}\circ dw^{i_1}_{\tau_k}\circ\cdots\circ dw^{i_k}_{\tau_1}, \] or alternatively \[ \int^s_t(s- \tau_1)^{l_k}\cdots \int^{\tau_{k-1}}_t (s- \tau_k)^{l_1}\circ dw^{i_1}_{\tau_k}\circ\cdots\circ dw^{i_k}_{\tau_1}. \] These formulas are intended to be useful for numerical solving of Itô stochastic differential equations.
scientific article; zbMATH DE number 2182729