Variations and estimators for the selfsimilarity order through Malliavin calculus

Using multiple stochastic integrals and the Malliavin calculus, we analyze the asymptotic behavior of quadratic variations for a specific non-Gaussian self-similar process, the Rosenblatt process. We apply our results to the design of strongly consistent statistical estimators for the self-similarity parameter $H$. Although, in the case of the Rosenblatt process, our estimator has non-Gaussian asymptotics for all $H>1/2$, we show the remarkable fact that the process's data at time 1 can be used to construct a distinct, compensated estimator with Gaussian asymptotics for $H\in(1/2,2/3)$

Lagrangian for the Convection-Diffusion Equation

Using the asymmetric fractional calculus of variations, we derive a fractional Lagrangian variational formulation of the convection-diffusion equation in the special case of constant coefficients

Almost sure optimal hedging strategy

In this work, we study the optimal discretization error of stochastic integrals, in the context of the hedging error in a multidimensional It\^{o} model when the discrete rebalancing dates are stopping times. We investigate the convergence, in an almost sure sense, of the renormalized quadratic variation of the hedging error, for which we exhibit an asymptotic lower bound for a large class of stopping time strategies. Moreover, we make explicit a strategy which asymptotically attains this lower bound a.s. Remarkably, the results hold under great generality on the payoff and the model. Our analysis relies on new results enabling us to control a.s. processes, stochastic integrals and related increments.Comment: Published in at http://dx.doi.org/10.1214/13-AAP959 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org