UNIVERSAL LIMIT THEOREM
UNIVERSAL LIMIT THEOREM
The key result presented in this book and the result which justifies the definition of a rough path is the universal limit theorem. The chapter proves in section 6.3 that a system of differential equations controlled by a p-rough path has a unique meaning for p-rough paths, providing the system of equations has a Lipschitz smoothness > p. In fact, the proof of this result also shows that the response of the Itô functional to the driving signal is continuous in the p-rough path metric. The proof is a difficult iterative process, but because the bounds are uniform, one sees that the iterations converge uniformly, and so the limit is continuous.
Keywords: Chen iterated integral, universal limit theorem, stochastic differential equation, p-rough path, Itô functional
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