The Calculus of Differentials for the Weak Stratonovich Integral
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.
KeywordsStochastic integration Stratonovich integral Fractional Brownian motion Weak convergence
- 3.Errami, M., Russo, F.: n-covariation, generalized Dirichlet processes and calculus with respect to finite cubic variation processes. Stoch. Process Appl. 104(2), 259–299 (2003)Google Scholar
- 4.Gradinaru, M., Nourdin, I., Russo, F., Vallois, P.: m-order integrals and generalized Itô’s formula: the case of a fractional Brownian motion with any Hurst index. Ann. Inst. H. Poincaré Probab. Statist. 41(4), 781–806 (2005)Google Scholar