# A Stratonovich-Skorohod integral formula for Gaussian rough paths

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Title: | A Stratonovich-Skorohod integral formula for Gaussian rough paths |

Author(s): | Cass, T Lim, N |

Item Type: | Journal Article |

Abstract: | Given a Gaussian process $X$, its canonical geometric rough path lift $\mathbf{X}$, and a solution $Y$ to the rough differential equation (RDE) $\mathrm{d}Y_{t} = V\left (Y_{t}\right ) \circ \mathrm{d} \mathbf{X}_t$, we present a closed-form correction formula for $\int Y \circ \mathrm{d} \mathbf{X} - \int Y \, \mathrm{d} X$, i.e. the difference between the rough and Skorohod integrals of $Y$ with respect to $X$. When $X$ is standard Brownian motion, we recover the classical Stratonovich-to-It{\^o} conversion formula, which we generalize to Gaussian rough paths with finite $p$-variation, $p < 3$, and satisfying an additional natural condition. This encompasses many familiar examples, including fractional Brownian motion with $H > \frac{1}{3}$. To prove the formula, we first show that the Riemann-sum approximants of the Skorohod integral converge in $L^2(\Omega)$ by using a novel characterization of the Cameron-Martin norm in terms of higher-dimensional Young-Stieltjes integrals. Next, we append the approximants of the Skorohod integral with a suitable compensation term without altering the limit, and the formula is finally obtained after a re-balancing of terms. |

Date of Acceptance: | 8-Jan-2018 |

URI: | http://hdl.handle.net/10044/1/56747 |

ISSN: | 0091-1798 |

Publisher: | Institute of Mathematical Statistics |

Journal / Book Title: | Annals of Probability |

Copyright Statement: | This article is under embargo until publication |

Sponsor/Funder: | Engineering & Physical Science Research Council (EPSRC) |

Funder's Grant Number: | EP/M00516X/1 |

Keywords: | math.PR math.PR math.PR math.PR 0104 Statistics Statistics & Probability |

Embargo Date: | publication subject to indefinite embargo |

Appears in Collections: | Financial Mathematics Mathematics Faculty of Natural Sciences |