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 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.
AMS: Primary 60H05; secondary 60F17, 60G22
Received 4/5/2011; Accepted 9/3/2011; Final 1/4/2012
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Acknowledgements
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.
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Swanson, J. (2013). The Calculus of Differentials for the Weak Stratonovich Integral. In: Viens, F., Feng, J., Hu, Y., Nualart , E. (eds) Malliavin Calculus and Stochastic Analysis. Springer Proceedings in Mathematics & Statistics, vol 34. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-5906-4_5
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DOI: https://doi.org/10.1007/978-1-4614-5906-4_5
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