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 d M = X d N, then Z d M can be written as ZX d N minus a stochastic correction term which is again related to the signed cubic variation.
KeywordsStochastic integration Stratonovich integral Fractional Brownian motion Weak convergence
Thanks go to Tom Kurtz and Frederi Viens for stimulating and helpful comments, feedback, and discussions. Jason Swanson was supported in part by NSA grant H98230-09-1-0079.
- 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