51 research outputs found
Explicit error bounds for lazy reversible Markov Chain Monte Carlo
We prove explicit, i.e., non-asymptotic, error bounds for Markov Chain Monte
Carlo methods, such as the Metropolis algorithm. The problem is to compute the
expectation (or integral) of f with respect to a measure which can be given by
a density with respect to another measure. A straight simulation of the desired
distribution by a random number generator is in general not possible. Thus it
is reasonable to use Markov chain sampling with a burn-in. We study such an
algorithm and extend the analysis of Lovasz and Simonovits (1993) to obtain an
explicit error bound
Limit Theorems for Individual-Based Models in Economics and Finance
There is a widespread recent interest in using ideas from statistical physics
to model certain types of problems in economics and finance. The main idea is
to derive the macroscopic behavior of the market from the random local
interactions between agents. Our purpose is to present a general framework that
encompasses a broad range of models, by proving a law of large numbers and a
central limit theorem for certain interacting particle systems with very
general state spaces. To do this we draw inspiration from some work done in
mathematical ecology and mathematical physics. The first result is proved for
the system seen as a measure-valued process, while to prove the second one we
will need to introduce a chain of embeddings of some abstract Banach and
Hilbert spaces of test functions and prove that the fluctuations converge to
the solution of a certain generalized Gaussian stochastic differential equation
taking values in the dual of one of these spaces.Comment: To appear in Stochastic Processes and their Application
Functional and Banach Space Stochastic Calculi: Path-Dependent Kolmogorov Equations Associated with the Frame of a Brownian Motion
First, we revisit basic theory of functional It\uf4/path-dependent calculus, using the formulation of calculus via regularization. Relations with the corresponding Banach space valued calculus are explored. The second part of the paper is devoted to the study of the Kolmogorov type equation associated with the so called window Brownian motion, called path-dependent heat equation, for which well-posedness at the level of strict solutions is established. Then, a notion of strong approximating solution, called strong-viscosity solution, is introduced which is supposed to be a substitution tool to the viscosity solution. For that kind of solution, we also prove existence and uniqueness
- …