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# Nonparametric Estimation for I.I.D. Paths of a Martingale Driven Model with Application to Non-Autonomous Fractional SDE

Abstract : This paper deals with a projection least square estimator of the function $J_0$ computed from multiple independent observations on $[0,T]$ of the process $Z$ defined by $dZ_t = J_0(t)d\langle M\rangle_t + dM_t$, where $M$ is a centered, continuous and square integrable martingale vanishing at $0$. Risk bounds are established on this estimator and on an associated adaptive estimator. An appropriate transformation allows to rewrite the differential equation $dX_t = V(X_t)(b_0(t)dt +\sigma(t)dB_t)$, where $B$ is a fractional Brownian motion of Hurst parameter $H\in (1/2,1)$, as a model of the previous type. So, the second part of the paper deals with risk bounds on a nonparametric estimator of $b_0$ derived from the results on the projection least square estimator of $J_0$. In particular, our results apply to the estimation of the drift function in a non-autonomous extension of the fractional Black-Scholes model introduced in Hu et al. (2003).
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Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-03412261
Contributor : Nicolas Marie Connect in order to contact the contributor
Submitted on : Wednesday, November 3, 2021 - 2:07:19 AM
Last modification on : Saturday, November 6, 2021 - 3:42:00 AM
Long-term archiving on: : Friday, February 4, 2022 - 6:05:55 PM

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• HAL Id : hal-03412261, version 1

### Citation

Nicolas Marie. Nonparametric Estimation for I.I.D. Paths of a Martingale Driven Model with Application to Non-Autonomous Fractional SDE. 2021. ⟨hal-03412261⟩

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