Noticias en español
First passage percolation and escape strategies
Abstract: Consider first passage percolation on Z^d with passage times given by i.i.d. random variables with common distribution F. Let t_\pi(u,v) be the time from u to v for a path \pi and t(u,v) the minimal time among all paths from u to v. We ask whether or not there exist points x,y \in Z^d and a semi-infinite path \pi=(y_0=y,y_1,\dots) such that t_\pi(y,y_{n+1})<t(x,y_n) for all n. Necessary and sufficient conditions on F are given for this to occur. When the support of F is unbounded, we also obtain results on the number of edges with large passage time used by...
Read MoreLímites de escala para paseos aleatorios en ambientes dinámicos (spanish only)
Resumen: Consideremos la siguiente dinámica. Comenzamos con un proceso de exclusión simple simétrico unidimensional (o el sistema de partículas conservativo favorito de la audiencia). Un paseo aleatorio se mueve con la siguiente regla: el paseo salta con tasa t; escoge saltar a la derecha si el proceso de exclusión tiene una partícula en su actual ubicación, y escoge saltar a la izquierda si el proceso de exclusión no tiene una partícula en la ubicación actual del paseo aleatorio. Mandando la tasa t a 0 con una velocidad adecuada, probaremos una ley de los grandes números para este paseo...
Read MoreQuenched local limit theorem for ergodic random conductance model
Abstract: We prove an almost sure local limit theorem for a symmetric random walk in an ergodic random environment. The proof is based on the parabolic Harnack inequality, which holds under some moment conditions on the conductances and the quenched invariance principle.
Read MorePricing Variance Swaps on Time-Changed Markov Processes
ABSTRACT We prove that the variance swap rate is just the price of a co-terminal European-style contract when the underlying is modeled as an exponential Markov process, time-changed by an arbitrary continuous stochastic clock, which has arbitrary correlation with the driving Markov process. The payoff function of the European contract that prices the variance swap satisfies an ordinary integro-differential equation, which depends only on the dynamics of the Markov process, not on the clock. We present examples of Markov processes whose payoff function can be computed...
Read MoreA stochastic model for speculative bubbles
Abstract: “One commonly says that the creation of speculative bubbles is the consequence of two phenomenons: the self-reinforcing effect of the investors and the tendency of these investors to follow the forecasting rule which consists in deciding that the price will increase if it has (strongly) increased in the past. Following these two rules, we build a model for speculative bubbles which is a Gaussian two-dimensional “turning” diffusion. Then, the main objective of the work is to obtain some sharp bounds for the time of return to a given price. In our main results,...
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