Abstract
We prove that the theory of rough paths, which is used to define path-wise integrals and path-wise differential equations, can be used with continuous semi-martingales. We provide then an almost sure theorem of type Wong-Zakai. Moreover, we show that the conditions UT and UCV, used to prove that one can interchange limits and Ito or Stratonovich integrals, provide the same result when one uses the rough paths theory.
Highlights
The theory of rough paths allows to give a meaning to integrals like t zt = z0 + g(xs) dxs and controlled differential equations like t yt = y0 + f dxs when x is a continuous, irregular path in a Banach space, and g and f are differential forms and vector fields smooth enough (See [21, 19, 16])
For that, one needs to know the equivalent of the iterated integrals of x, that is 0≤s1≤···≤sk≤T dxs1 ⊗ · · · ⊗ dxsk, and to use the topology of p-variation, which is defined using the semi-norm k−1
For a large class of stochastic processes, it is possible to define the equivalent of the iterated integrals for the trajectories
Summary
The convergence under the topology of Skorokhod is kwnon since the papers of Memin and Slominski [22], and Jakubowski, Memin and Pages [11] This is fully coherent with the results concerning interchanging limits and stochastic integrals in the semi-martingales theory. These almost-sure Wong-Zakai results are not surprising We extend this result to a general continuous semi-martingales, and our proof includes recent techniques in the theory of rough path, that leads to some simplification of results. These proofs show the role played by Holder continuous paths among paths of finite p-variation, and that it is not a big deal to use Holder continuous path instead of path of finite p-variation using a time-change. One could think that this technique of using a time-change could be applied to generalize almost immediately to continuous semi-martingales some results given for the Brownian motion, as long as only the martingale property of the Brownian motion is involved in their proofs
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