The calculus of differentials for the weak Stratonovich integral
Abstract
The weak Stratonovich integral is defined as the limit, in law, of Stratonovichtype 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 Itôtype formula for this integral. As in the case where $g$ is the identity, the Itôtype formula has a correction term which is a classical Itô integral, and which is related to the socalled signed cubic variation of $g(B)$. Finally, we derive a surprising formula for calculating with differentials. We show that if $dM = X dN$, then $Z dM$ can be written as $ZX dN$ minus a stochastic correction term which is again related to the signed cubic variation.
 Publication:

arXiv eprints
 Pub Date:
 March 2011
 arXiv:
 arXiv:1103.0341
 Bibcode:
 2011arXiv1103.0341S
 Keywords:

 Mathematics  Probability
 EPrint:
 16 pages