We investigate here the central limit theorem of the increment ratio statistic of a multifractional Brownian motion, leading to a CLT for the time varying Hurst index. The proofs are quite simple relying on Breuer–Major theorems and an original freezing of time strategy. A simulation study shows the goodness of fit of this estimator.
We obtain a local limit theorem for the laws of a class of Brownian additive functionals and we apply this result to a penalisation problem. We study precisely the case of the additive functional: . On the other hand, we describe Feynman-Kac type penalisation results for long Brownian bridges thus completing some similar previous study for standard Brownian motion (see [B. Roynette, P. Vallois and M. Yor, Studia Sci. Math. Hung.43 (2006) 171–246]).
A general theory is developed for the projection of martingale related processes onto smaller filtrations, to which they are not even adapted. Martingales, supermartingales, and semimartingales retain their nature, but the case of local martingales is more delicate, as illustrated by an explicit case study for the inverse Bessel process. This has implications for the concept of No Free Lunch with Vanishing Risk, in Finance.
A general theory is developed for the projection of martingale related processes onto smaller filtrations, to which they are not even adapted. Martingales, supermartingales, and semimartingales retain their nature, but the case of local martingales is more delicate, as illustrated by an explicit case study for the inverse Bessel process. This has implications for the concept of No Free Lunch with Vanishing Risk, in Finance.
Random interlacements at level is a one parameter family of connected random subsets of , (Ann. Math.171(2010) 2039–2087). Its complement, the vacant set at level , exhibits a non-trivial percolation phase transition in (Comm. Pure Appl. Math.62 (2009) 831–858; Ann. Math.171 (2010) 2039–2087), and the infinite connected component, when it exists, is almost surely unique (Ann. Appl. Probab.19(2009) 454–466). In this paper we study local percolative properties of the vacant set of random interlacements...
One of the earliest concepts for hedging and pricing in incomplete financial markets has been the quadratic criterion of local risk-minimization. However, definitions and theory have so far been established only for the case of a single (one-dimensional) risky asset. We extend the approach to a general multidimensional setting and prove that the basic martingale characterization result for locally risk-minimizing strategies still holds true. In comparison with existing literature, the self-contained...
We study the existence and the regularity of the local time of filtered white noises . We will also give Chung’s form of the law of iterated logarithm for , this shows that the result on the Hölder regularity, with respect to time, of the local time is sharp.
We study logarithmic estimates for a class of Fourier multipliers which arise from a nonsymmetric modulation of jumps of Lévy processes. In particular, this leads to corresponding tight bounds for second-order Riesz transforms on .