Good rough path sequences and applications to anticipating stochastic calculus. (English) Zbl 1132.60053

This article is concerned with anticipative Stratonovich stochastic differential equations \[ d Y_t = V_0(Y_t) d t + \sum_{i=1}^d V_i(Y_t)\circ d B^i_t, \quad Y_0 = y_0 \] driven by some stochastic process which will be lifted to a rough path in a Lie group. Neither adaptedness of the initial point \(y_0\) and the \(C^1\) vector field \(V = (V_0,\dots, V_d)\) nor commuting conditions for the latter are assumed. Under simple conditions on the stochastic process, the authors prove that the unique solution of the above SDE understood in the rough path sense is actually a Stratonovich solution. It is shown that this condition is satisfied by Brownian motion.
As applications, the authors obtain rather flexible results for these solutions such as support theorems, large deviation principles and Wong-Zakai approximations for SDEs driven by Brownian motion along anticipating vectorfields. The authors suggest that this perspective may unify many results on anticipative SDEs.


60H99 Stochastic analysis
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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