arXiv subjects

D. Belomestny

Publications and source records attributed to D. Belomestny.

7 recordsLinked to original sources

arXiv2022

Spectral bootstrap confidence bands for Lévy-driven moving average processes

In this paper we study the problem of constructing bootstrap confidence intervals for the Lévy density of the driving Lévy process based on high-frequency observations of a Lévy-driven moving average processes. Using a spectral estimator of the Lévy density, we propose a novel implementations of multiplier and empirical bootstraps to construct confidence bands on a compact set away from the origin. We also provide conditions under which the confidence bands are asymptotically valid.

math.ST↗

arXiv2021

Variance reduction for additive functional of Markov chains via martingale representations

In this paper we propose an efficient variance reduction approach for additive functionals of Markov chains relying on a novel discrete time martingale representation. Our approach is fully non-asymptotic and does not require the knowledge of the stationary distribution (and even any type of ergodicity) or specific structure of the underlying density. By rigorously analyzing the convergence properties of the proposed algorithm, we show that its cost-to-variance product is indeed smaller than one of the naive algorithm. The numerical performance of the new method is illustrated for the Langevin-type Markov Chain Monte Carlo (MCMC) methods.

stat.CO↗

arXiv2021

Empirical Variance Minimization with Applications in Variance Reduction and Optimal Control

We study the problem of empirical minimization for variance-type functionals over functional classes. Sharp non-asymptotic bounds for the excess variance are derived under mild conditions. In particular, it is shown that under some restrictions imposed on the functional class fast convergence rates can be achieved including the optimal non-parametric rates for expressive classes in the non-Donsker regime under some additional assumptions. Our main applications include variance reduction and optimal control.

math.NA↗

arXiv2020

Variance reduction for dependent sequences with applications to Stochastic Gradient MCMC

In this paper we propose a novel and practical variance reduction approach for additive functionals of dependent sequences. Our approach combines the use of control variates with the minimisation of an empirical variance estimate. We analyse finite sample properties of the proposed method and derive finite-time bounds of the excess asymptotic variance to zero. We apply our methodology to Stochastic Gradient MCMC (SGMCMC) methods for Bayesian inference on large data sets and combine it with existing variance reduction methods for SGMCMC. We present empirical results carried out on a number of benchmark examples showing that our variance reduction method achieves significant improvement as compared to state-of-the-art methods at the expense of a moderate increase of computational overhead.

math.ST↗

arXiv2020

Variance reduction for Markov chains with application to MCMC

In this paper we propose a novel variance reduction approach for additive functionals of Markov chains based on minimization of an estimate for the asymptotic variance of these functionals over suitable classes of control variates. A distinctive feature of the proposed approach is its ability to significantly reduce the overall finite sample variance. This feature is theoretically demonstrated by means of a deep non asymptotic analysis of a variance reduced functional as well as by a thorough simulation study. In particular we apply our method to various MCMC Bayesian estimation problems where it favourably compares to the existing variance reduction approaches.

math.ST↗

arXiv2019

Semi-tractability of optimal stopping problems via a weighted stochastic mesh algorithm

In this article we propose a Weighted Stochastic Mesh (WSM) Algorithm for approximating the value of a discrete and continuous time optimal stopping problem. We prove that in the discrete case the WSM algorithm leads to semi-tractability of the corresponding optimal problems in the sense that its complexity is bounded in order by $\varepsilon^{-4}\log^{d+2}(1/\varepsilon)$ with $d$ being the dimension of the underlying Markov chain. Furthermore we study the WSM approach in the context of continuous time optimal stopping problems and derive the corresponding complexity bounds. Although we can not prove semi-tractability in this case, our bounds turn out to be the tightest ones among the bounds known for the existing algorithms in the literature. We illustrate our theoretical findings by a numerical example.

q-fin.CP↗

arXiv2008

Holomorphic transforms with application to affine processes

In a rather general setting of Itô-Lévy processes we study a class of transforms (Fourier for example) of the state variable of a process which are holomorphic in some disc around time zero in the complex plane. We show that such transforms are related to a system of analytic vectors for the generator of the process, and we state conditions which allow for holomorphic extension of these transforms into a strip which contains the positive real axis. Based on these extensions we develop a functional series expansion of these transforms in terms of the constituents of the generator. As application, we show that for multidimensional affine Itô-Lévy processes with state dependent jump part the Fourier transform is holomorphic in a time strip under some stationarity conditions, and give log-affine series representations for the transform.

math.FA↗