Francesco Mori
Time Between The Maximum And The Minimum Of One-dimensional Brownian Motion.
Rel. Alessandro Pelizzola, Satya N. Majumdar, Paolo Biscari. Politecnico di Torino, Corso di laurea magistrale in Physics Of Complex Systems (Fisica Dei Sistemi Complessi), 2019
Abstract
The study of maxima and minima over a set of random variables is fundamental for the description of a plethora of phenomena, from climate studies to finance. Although the case of independent identically distributed random variables is well understood, in many relevant applications variables are strongly correlated and, without the assumption of independence, finding exact results becomes highly non trivial. An example of strongly correlated variables, for which it is possible to find exact solutions, are random walks. For instance, extrema of Brownian motion have been widely studied and analytical results are known, e.g. the probability distribution of the time $t_{max}$ at which a one-dimensional Brownian motion reaches its maximal value.
However, nothing is known about probability distributions of quantities that depend on both the time of the maximum $t_{max}$ and the time of the minimum $t_{min}$
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