Curs 2019-2020: Abstracts i Slides

UPCOMING SEMINARS

06/11/2019: Carsten Chong, EPFL.

High-frequency analysis of parabolic stochastic PDEs with multiplicative noise

We consider the stochastic heat equation driven by a multiplicative Gaussian noise that is white in time and spatially homogeneous in space. Assuming that the spatial correlation function is given by a Riesz kernel of order $\alpha \in (0,1)$, we prove a central limit theorem for the power variations of the solution. At the same time, we show that the same central limit theorem fails in general if $\alpha = 1$ (in dimension $d \geq 2$) or if the noise is a space-time white noise (in dimension $d = 1$).

PREVIOUS SEMINARS

2/10/2019: Verónica Miró Pina, UNAM.

El coalescente simétrico

Trabajo conjunto con: Arno Siri-Jégousse y Adrián González Casanova

09/09/2019: Archil Gulisashvilli, Ohio University.

Gaussian Stochastic Volatility Models: Scaling Regimes, Large Deviations and Moment Explosions

In a Gaussian stochastic volatility model, the evolution of volatility is described by a stochastic process that can be represented as a positive continuous function (the volatility function) of a continuous Gaussian process (the volatility process). If the volatility process exhibits fractional features, then the model is called a Gaussian fractional stochastic volatility model. Important examples of fractional volatility processes are fractional Brownian motion, the Riemann-Liouville fractional Brownian motion, and the fractional Ornstein-Uhlenbeck process. If the volatility process admits a Volterra type representation, then the model is called a Volterra type stochastic volatility model. Forde and Zhang established a large deviation principle for the log-price process in a Volterra type model under the assumptions that the volatility function is globally Hölder  continuous and the volatility process is fractional Brownian motion. We prove a similar small-noise large deviation principle under significantly weaker restrictions. More precisely, we assume that the volatility function satisfies a mild local regularity condition, while the volatility process is any Volterra type Gaussian process. Moreover, we establish a sample path large  deviation principle for the log-price process in a Volterra type model, and a sample path moderate deviation principle for general Gaussian models. In addition, applications are given to the study of the asymptotic behavior of exit probabilities, call pricing functions, and the implied volatility in various mixed scaling regimes.

Another problem addressed in our work concerns moment explosions for asset price processes. We prove that for such a process in an uncorrelated Gaussian stochastic volatility model, all the moments of order greater than one explode provided that the volatility function grows faster than linearly. Partial results are also obtained for correlated models.

16/09/2019: Jorge León, CINVESTAV.

Dos esquemas numéricos para ecuaciones diferenciales estocásticas gobernadas por un movimiento browniano fraccionario

En esta charla estudiaremos métodos numéricos para ecuaciones diferenciales estocásticas governadas por un movimiento browniano fraccionario en los dos siguientes casos:

a) El parámetro de Hurst 1/4<H<1/2 y la integral estocástica es en el sentido de Stratonovich.
b) Aquí H>1/2 y la integral estocástica es la extensión de la integral de Young dada por M. Zähle.

En él primer (resp. segundo) caso usamos una aproximación de Taylor de primer orden de la representación de Doss-Sussman de la solución (resp. de los coeficientes) para obtener nuestros resultados . En el segundo caso la aproximación es la solución explícita de una ecuación lineal a trozos.

Trabajo conjunto con H. Araya y S. Torres.

22/10/2019: Jozsef Lorinczi, University of Loughborough.

The location of extrema of solutions for a class of non-local equations

I will consider a class of non-local equations, assuming that their solutions have a global maximum, and discuss estimates on their position. Such points are of special interest as they describe where hot-spots settle on the long run or, in a probabilistic description, the region around which paths typically concentrate. I will address, on the one hand, the interplay of competing parameters whose compromise the resulting location is, and the impact of geometric constraints, on the other.