Aureli Alabert (Universitat Autònoma de Barcelona)
Title: Uniqueness for some non-Lipchitz SDE.
Abstract: We study the uniqueness in the “path-by-path” sense of solutions to stochastic differential equations with additive noise and non-Lipschitz autonomous drift. The notion of path-by-path solution involves considering a collection of ordinary differential equations and is, in principle, weaker than that of a strong solution, since no adaptability condition is required. We use results and ideas from the classical theory of ODE’s, together with probabilistic tools like Girsanov’s theorem. This is a joint work with Jorge A. León.
Sergey Foss (Heriot-Watt University, Edinburgh)
Title: Limit theorems for random directed graphs.
Abstract: We consider a stochastic directed graph on the integers whereby a directed edge between i and a larger integer j exists with probability p that may depend on the distance j-i, and there is no edges from bigger to smaller integers. Edge lengths L(i,j) may be constants or i.i.d. random variables. We introduce also a complementary “infinite bin” model. We study the asymptotics for the maximal path length in a long chunk of the graph. Under certain assumptions, the model has a regenerative structure that depends on the infinite future, and the SLLN and the CLT follow. Otherwise, we obtain scaling laws and asymptotic distributions expressed in terms of a “continuous last-passage percolation” model on [0,1]. If time allows, we introduce various and, in particular, multi-dimensional extensions of the models. We will also link this topic to contact processes.
The talk is partly based on joint papers with T Konstantopoulos (2003, MPRF), D Denisov and T Konstantopoulos (2012, AnnAP), J Martin and Ph. Schmidt (2014, AnnAP) and S Zachary (2013, AdvAP).
Gabriel Lord (Heriot-Watt University, Edinburgh)
Title: Stochastic travelling waves and computation.
Abstract: We examine a new numerical method for solving Stratonovich SDEs. In particular we are interested in computing stochastic travelling waves. Travelling waves are often of physical interest and we have applications from models of neural tissue that are both SPDEs and large SDE systems. We introduce a technique where we move to a travelling wave frame and stop the wave from moving. This has some computational advantages as a small domain can be used but we will also discuss some potential pitfalls.
Eulàlia Nualart (Universitat Pompeu Fabra, Barcelona)
Title: A truncated two-scale realized variance estimator robust to price jumps and small noises.
Abstract: In this work we propose an estimator that combines the truncation method with the two-scale realized variance estimator to estimate the integrated variance of an asset return. The observed log price of the asset is assumed to be driven by a diffusion process with jumps, and we assume the presence of market microstructure noise, which is decreasing in the number of observations during a day. A concentration inequality is derived to show the precision of this estimator. In addition, we also prove that this estimator converges to the quadratic variation of the efficient price process, which is the integrated variance plus the sum of the squares of the price jumps. We finally perform some simulations which show that our estimator is more efficient than the bipower variation and the truncated realized variance, which are estimators on the integrated variance robust to the price jumps. This is a joint work with Christian Brownlees and Yucheng Sun from Universitat Pompeu Fabra.
Lluís Quer-Sardanyons (Universitat Autònoma de Barcelona)
Title: A fully discrete approximation of the one-dimensional stochastic wave equation.
Abstract: In this talk, a fully discrete approximation of a one-dimensional nonlinear stochastic wave equations driven by multiplicative noise is presented. More precisely, we use a standard finite difference approximation in space and a stochastic trigonometric method for the temporal approximation. This explicit time integrator allows us to obtain error bounds in $L^p(\Omega)$, uniformly in time and space, in such a way that the time discretization does not suffer from any kind of CFL-type stepsize restriction. Moreover, uniform almost sure convergence of the numerical solution is also proved. We will present some numerical experiments which confirm the theoretical results. The talk is based on joint work with David Cohen (University of Umea).
Sotirios Sabanis (The University of Edinburgh)
Title: Explicit numerical schemes for SDEs driven by Levy noise and for Stochastic Evolution Equations.
Abstract: The idea of ‘tamed’ Euler schemes, which was pioneered by Hutzenthaler, Jentzen and Kloeden [1] and Sabanis [2], led to the development of a new generation of explicit numerical schemes
– for SDEs driven by Levy noise with superlinear coefficients and,
– for stochastic evolutions equations with super-linearly growing operators appearing in the drift.
Moreover, high order schemes (such as Milstein) are established (with optimal rates of convergence) by the natural extension of the aforementioned ideas. Theoretical results on this topic along with relevant simulation outputs will be presented during this talk.
[1] M. Hutzenthaler, A. Jentzen, P.E. Kloeden, Strong convergence of an explicit numerical method for SDEs with non-globally Lipschitz continuous coefficients. Ann. Appl. Probab. 22 (2012) 1611–1641.
[2] S. Sabanis, A note on tamed Euler approximations, Electron. Commun. Probab. 18 (2013), no. 47, 1–-10.
Joint work with Istvan Gyongy, David Siska, Chaman Kumar and Konstantinos Dareiotis.
