Ricerc@Sapienza

We combine two existing estimators of the local Hurst exponent to improve both the goodness of fit and the computational speed of the algorithm. An application with simulated time series is implemented, and a Monte Carlo simulation is performed to provide evidence of the improvement. The estimation of the Hurst exponent of a time series is a recurring problem of great interest in many fields: finance, biology, hydrology, ecology, and signal processing, to quote a few.

The Pandora's Box Problem, originally formalized by Weitzman in 1979, models selection from a set of options each with stochastic parameters, when evaluation (i.e. sampling) is costly. This includes, for example, the problem of hiring a skilled worker, where only one hire can be made, but the evaluation of each candidate is an expensive procedure.

We propose a probabilistic construction for the solution of a general class of fractional high-order heat-type equations in the one-dimensional case, by using a sequence of random walks in the complex plane with a suitable scaling. A time change governed by a class of subordinated processes allows to handle the fractional part of the derivative in space. We first consider evolution equations with space fractional derivatives of any order, and later we show the extension to equations with time fractional derivative (in the sense of Caputo derivative) of order α∈ (0 , 1).