Multiple stochastic integrals appearing in the stochastic Taylor expansions
Consider the solution (started from \(x)\) \(X_t\) to a Stratonovitch s.d.e. \(dX_t = \sum^n_{i = 0} V_i (X_t) \circ d B^i_t\), where the \(V_i\)'s are smooth vector fields and the \(B_t^i\)'s are continuous semimartingales. Denote by \(B_t^{i_1 \dots i_m}\) the multiple stochastic integral \(\int^t_0 \circ d B^{i_m}_{s_m} \dots \int^{s_2}_0 \circ dB^{i_1}_{s_1}\). The purpose here is the study of the formal series in the \(V_i\)'s: \[ S_t : = 1 + \sum_{m \geq 1} \sum_{0 \leq i_1, \dots, i_m \leq n} B_t^{i_1 \dots i_m} V_{i_1} \dots V_{i_m}, \] motivated by the fact that if \(S_t (x)\) converges, then it equals \(X_t\). The first result gives an explicit algebraic expression for the series \(\text{Log} S_t\), which is in turn a formal series only in the (linear combinations of the) multiple (ordinary) brackets of the \(V_i\)'s. Then another description of \(\text{Log} S_t\) is given, as the solution (started from 0) of an s.d.e. \(dU_t = \sum^n_{i = 0} R_i (V_t) \circ d B^i_t\), where the vector field \(R_i\) can be given by means of the Campbell-Hausdorff formula or by means of a series in the ``ad'' morphism. These formulas become effective when for example the \(V_i\)'s generate a nilpotent Lie algebra. This work refines and precises precedent works by Ben Arous, Fliess and Normand-Cyrot, Yamato, Castell, Hu.
scientific article; zbMATH DE number 844442