Optimal discretization of stochastic integrals driven by general Brownian semimartingale

Emmanuel Gobet (1) , Uladzislau Stazhynski (2, 1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 DMA - Département de Mathématiques et Applications - ENS Paris

We study the optimal discretization error of stochastic integrals, driven by a multidimensional continuous Brownian semimartingale. In this setting we establish a pathwise lower bound for the renormalized quadratic variation of the error and we provide a sequence of discretiza- tion stopping times, which is asymptotically optimal. The latter is defined as hitting times of random ellipsoids by the semimartingale at hand. In comparison with previous available results, we allow a quite large class of semimartingales (relaxing in particular the non degeneracy conditions usually requested) and we prove that the asymptotic lower bound is attainable.

Keywords: discretization of stochastic integrals; hitting times; random ellipsoids; almost sure convergence

Domains: Probability [math.PR]

Emmanuel Gobet, Uladzislau Stazhynski. Optimal discretization of stochastic integrals driven by general Brownian semimartingale. 2017. ⟨hal-01241190v2⟩