arXiv subjects

Denis Belomestny

Publications and source records attributed to Denis Belomestny.

82 records · Page 5Linked to original sources

arXiv2013

Statistical inference for exponential functionals of Lévy processes

In this paper, we consider the exponential functional $A_{\infty}=\int_0^\infty e^{-ξ_s}ds$ of a L{é}vy process $ξ_s$ and aim to estimate the characteristics of $ξ_{s}$ from the distribution of $A_{\infty}$. We present a new approach, which allows to statistically infer on the L{é}vy triplet of $ξ_{t}$, and study the theoretical properties of the proposed estimators. The suggested algorithms are illustrated with numerical simulations.

stat.OT↗

arXiv2013

Solving optimal stopping problems via empirical dual optimization

In this paper we consider a method of solving optimal stopping problems in discrete and continuous time based on their dual representation. A novel and generic simulation-based optimization algorithm not involving nested simulations is proposed and studied. The algorithm involves the optimization of a genuinely penalized dual objective functional over a class of adapted martingales. We prove the convergence of the proposed algorithm and demonstrate its efficiency for optimal stopping problems arising in option pricing.

math.PR↗

arXiv2013

Concentration inequalities for smooth random fields

In this note we derive a sharp concentration inequality for the supremum of a smooth random field over a finite dimensional set. It is shown that this supremum can be bounded with high probability by the value of the field at some deterministic point plus an intrinsic dimension of the optimisation problem. As an application we prove the exponential inequality for a function of the maximal eigenvalue of a random matrix is proved.

math.ST↗

arXiv2012

Statistical inference for time-changed Lévy processes via composite characteristic function estimation

In this article, the problem of semi-parametric inference on the parameters of a multidimensional Lévy process $L_t$ with independent components based on the low-frequency observations of the corresponding time-changed Lévy process $L_{\mathcal{T}(t)}$, where $\mathcal{T}$ is a nonnegative, nondecreasing real-valued process independent of $L_t$, is studied. We show that this problem is closely related to the problem of composite function estimation that has recently gotten much attention in statistical literature. Under suitable identifiability conditions, we propose a consistent estimate for the Lévy density of $L_t$ and derive the uniform as well as the pointwise convergence rates of the estimate proposed. Moreover, we prove that the rates obtained are optimal in a minimax sense over suitable classes of time-changed Lévy models. Finally, we present a simulation study showing the performance of our estimation algorithm in the case of time-changed Normal Inverse Gaussian (NIG) Lévy processes.

stat.ME↗

arXiv2011

Spectral estimation of the Lévy density in partially observed affine models

The problem of estimating the Lévy density of a partially observed multidimensional affine process from low-frequency and mixed-frequency data is considered. The estimation methodology is based on the log-affine representation of the conditional characteristic function of an affine process and local linear smoothing in time. We derive almost sure uniform rates of convergence for the estimated Lévy density both in mixed-frequency and low-frequency setups and prove that these rates are optimal in the minimax sense. Finally, the performance of the estimation algorithms is illustrated in the case of the Bates stochastic volatility model.

stat.ME↗

arXiv2010

Spectral estimation of the fractional order of a Lévy process

We consider the problem of estimating the fractional order of a Lévy process from low frequency historical and options data. An estimation methodology is developed which allows us to treat both estimation and calibration problems in a unified way. The corresponding procedure consists of two steps: the estimation of a conditional characteristic function and the weighted least squares estimation of the fractional order in spectral domain. While the second step is identical for both calibration and estimation, the first one depends on the problem at hand. Minimax rates of convergence for the fractional order estimate are derived, the asymptotic normality is proved and a data-driven algorithm based on aggregation is proposed. The performance of the estimator in both estimation and calibration setups is illustrated by a simulation study.

math.ST↗

arXiv2009

On the rates of convergence of simulation based optimization algorithms for optimal stopping problems

In this paper we study simulation based optimization algorithms for solving discrete time optimal stopping problems. This type of algorithms became popular among practioneers working in the area of quantitative finance. Using large deviation theory for the increments of empirical processes, we derive optimal convergence rates and show that they can not be improved in general. The rates derived provide a guide to the choice of the number of simulated paths needed in optimization step, which is crucial for the good performance of any simulation based optimization algorithm. Finally, we present a numerical example of solving optimal stopping problem arising in option pricing that illustrates our theoretical findings.

math.OC↗

arXiv2009

Pricing Bermudan options using nonparametric regression: optimal rates of convergence for lower estimates

The problem of pricing Bermudan options using Monte Carlo and a nonparametric regression is considered. We derive optimal non-asymptotic bounds for a lower biased estimate based on the suboptimal stopping rule constructed using some estimates of continuation values. These estimates may be of different nature, they may be local or global, with the only requirement being that the deviations of these estimates from the true continuation values can be uniformly bounded in probability. As an illustration, we discuss a class of local polynomial estimates which, under some regularity conditions, yield continuation values estimates possessing this property.

q-fin.PR↗

arXiv2007

Spatial aggregation of local likelihood estimates with applications to classification

This paper presents a new method for spatially adaptive local (constant) likelihood estimation which applies to a broad class of nonparametric models, including the Gaussian, Poisson and binary response models. The main idea of the method is, given a sequence of local likelihood estimates (``weak'' estimates), to construct a new aggregated estimate whose pointwise risk is of order of the smallest risk among all ``weak'' estimates. We also propose a new approach toward selecting the parameters of the procedure by providing the prescribed behavior of the resulting estimate in the simple parametric situation. We establish a number of important theoretical results concerning the optimality of the aggregated estimate. In particular, our ``oracle'' result claims that its risk is, up to some logarithmic multiplier, equal to the smallest risk for the given family of estimates. The performance of the procedure is illustrated by application to the classification problem. A numerical study demonstrates its reasonable performance in simulated and real-life examples.

math.ST↗

arXiv2003

Rates of convergence for constrained deconvolution problem

Let $X$ and $Y$ be two independent identically distributed random variables with density $p(x)$ and $Z=αX+βY$ for some constants $α>0$ and $β>0$. We consider the problem of estimating $p(x)$ by means of the samples from the distribution of $Z$. Non-parametric estimator based on the sync kernel is constructed and asymptotic behaviour of the corresponding mean integrated square error is investigated.

math.ST↗