The Calculus of Differentials for the Weak Stratonovich Integral

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


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.


Stochastic 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.


  1. 1.
    Burdzy, K., Swanson, J.: A change of variable formula with Itô correction term. Ann. Probab. 38(5), 1817–1869 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Cheridito, P., Nualart, D.: Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter \(H \in (0, \frac{1} {2})\). Ann. Inst. H. Poincaré Probab. Statist. 41(6), 1049–1081 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 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. 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
  5. 5.
    Nourdin, I., Réveillac, A.: Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: the critical case H = 1 ∕ 4. Ann. Probab. 37(6), 2200–2230 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Nourdin, I., Réveillac, A., Swanson, J.: The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1 ∕ 6. Electron. J. Probab. 15, 2087–2116 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Nualart, D., Ortiz-Latorre, S.: Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stoch. Process Appl. 118(4), 614–628 (2008)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.University of Central FloridaOrlandoUSA

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