The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6

Details The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6 The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6

2010-06-22

arxiv.org/abs/1006.4238v1

Mathematics

Probability

45 pages

Scientific paper

Let $B$ be a fractional Brownian motion with Hurst parameter $H=1/6$. It is known that the symmetric Stratonovich-style Riemann sums for $\int g(B(s))\,dB(s)$ do not, in general, converge in probability. We show, however, that they do converge in law in the Skorohod space of c\`adl\`ag functions. Moreover, we show that the resulting stochastic integral satisfies a change of variable formula with a correction term that is an ordinary It\^o integral with respect to a Brownian motion that is independent of $B$.

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