The Calculus of Differentials for the Weak Stratonovich Integral

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 34)

Abstract

The weak Stratonovich integral is defined as the limit, in law, of Stratonovich-type symmetric Riemann sums. We derive an explicit expression for the weak Stratonovich integral of f(B) with respect to g(B), where B is a fractional Brownian motion with Hurst parameter 1/6, and f and g are smooth functions. We use this expression to derive an Itô-type formula for this integral. As in the case where g is the identity, the Itô-type formula has a correction term which is a classical Itô integral and which is related to the so-called signed cubic variation of g(B). Finally, we derive a surprising formula for calculating with differentials. We show that if dM = XdN, then ZdM can be written as ZXdN minus a stochastic correction term which is again related to the signed cubic variation.

Keywords

Stochastic integration Stratonovich integral Fractional Brownian motion Weak convergence 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.University of Central FloridaOrlandoUSA

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