Generalization of Ito's formula for smooth nondegenerate martingales

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:91

Article

Moret, S ; Nualart, D

关键词: Ito's formula;    Malliavin calculus;    quadratic covariation;   

DOI  :  10.1016/S0304-4149(00)00058-2

来源: Elsevier

【 摘 要 】

In this paper we prove the existence of the quadratic covariation [(partial derivativeF/partial derivativex(k))(X), X-k] for all 1 less than or equal to k less than or equal to d, where F belongs locally to the Sobolev space W-1,W-p(R-d) for some p > d and X is a d-dimensional smooth nondegenerate martingale adapted to a d-dimensional Brownian motion. This result is based on some moment estimates for Riemann sums which are established by means of the techniques of the Malliavin calculus. As a consequence we obtain an extension of Ito's formula where the complementary term is one-half the sum of the quadratic covariations above. (C) 2001 Elsevier Science B.V. All rights reserved. MSG: 60H05; 60H07.

【 授权许可】

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