n-covariation, generalized Dirichlet processes and calculus with respect to finite cubic variation processes

In this paper, we introduce first a natural generalization of the concept of Dirichlet process, providing significant examples. The second important tool concept is the n-covariation and the related n-variation. The n-variation of a continuous process and the n-covariation of a vector of continuous processes, are defined through a regularization procedure. We calculate explicitly the n-variation process, when it exists, of a martingale convolution. For processes having finite cubic variation, a basic stochastic calculus is developed. We prove an Ito formula and we study existence and uniqueness of the solution of a stochastic differential equation, in a symmetric-Stratonovich sense, with respect to those processes.

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