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Denis Belomestny

Publications and source records attributed to Denis Belomestny.

At least 73 records · Page 4Linked to original sources
arXiv2018

Solving linear parabolic rough partial differential equations

We study linear rough partial differential equations in the setting of [Friz and Hairer, Springer, 2014, Chapter 12]. More precisely, we consider a linear parabolic partial differential equation driven by a deterministic rough path $\mathbf{W}$ of Hölder regularity $α$ with $1/3 < α \le 1/2$. Based on a stochastic representation of the solution of the rough partial differential equation, we propose a regression Monte Carlo algorithm for spatio-temporal approximation of the solution. We provide a full convergence analysis of the proposed approximation method which essentially relies on the new bounds for the higher order derivatives of the solution in space. Finally, a comprehensive simulation study showing the applicability of the proposed algorithm is presented.

math.PR↗
arXiv2017

Minimax theorems for American options in incomplete markets without time-consistency

In this paper we give sufficient conditions guaranteeing the validity of the well-known minimax theorem for the lower Snell envelope with respect to a family of absolutely continuous probability measures. Such minimax results play an important role in the characterisation of arbitrage-free prices of American contingent claims in incomplete markets. Our conditions do not rely on the notions of stability under pasting or time-consistency and reveal some unexpected connection between the minimax result and the path properties of the corresponding density process.

math.PR↗
arXiv2017

Variance reduction for discretised diffusions via regression

In this paper we present a novel approach towards variance reduction for discretised diffusion processes. The proposed approach involves specially constructed control variates and allows for a significant reduction in the variance for the terminal functionals. In this way the complexity order of the standard Monte Carlo algorithm ($\varepsilon^{-3}$ in the case of a first order scheme and $\varepsilon^{-2.5}$ in the case of a second order scheme) can be reduced down to $\varepsilon^{-2+δ}$ for any $δ\in [0,0.25)$ with $\varepsilon$ being the precision to be achieved. These theoretical results are illustrated by several numerical examples.

math.PR↗
arXiv2017

Semiparametric estimation in the normal variance-mean mixture model

In this paper we study the problem of statistical inference on the parameters of the semiparametric variance-mean mixtures. This class of mixtures has recently become rather popular in statistical and financial modelling. We design a semiparametric estimation procedure that first estimates the mean of the underlying normal distribution and then recovers nonparametrically the density of the corresponding mixing distribution. We illustrate the performance of our procedure on simulated and real data.

stat.OT↗
arXiv2017

Truncated control variates for weak approximation schemes

In this paper we present an enhancement of the regression-based variance reduction approaches recently proposed in Belomestny et al. This enhancement is based on a truncation of the control variate and allows for a significant reduction of the computing time, while the complexity stays of the same order. The performances of the proposed truncated algorithms are illustrated by a numerical example.

math.PR↗
arXiv2017

Low-rank diffusion matrix estimation for high-dimensional time-changed Lévy processes

The estimation of the diffusion matrix $Σ$ of a high-dimensional, possibly time-changed Lévy process is studied, based on discrete observations of the process with a fixed distance. A low-rank condition is imposed on $Σ$. Applying a spectral approach, we construct a weighted least-squares estimator with nuclear-norm-penalisation. We prove oracle inequalities and derive convergence rates for the diffusion matrix estimator. The convergence rates show a surprising dependency on the rank of $Σ$ and are optimal in the minimax sense for fixed dimensions. Theoretical results are illustrated by a simulation study.

math.ST↗
arXiv2017

Variance reduced multilevel path simulation: going beyond the complexity $\varepsilon^{-2}$

In this paper a novel modification of the multilevel Monte Carlo approach, allowing for further significant complexity reduction, is proposed. The idea of the modification is to use the method of control variates to reduce variance at level zero. We show that, under a proper choice of control variates, one can reduce the complexity order of the modified MLMC algorithm down to $\varepsilon^{-2+δ}$ for any $δ\in [0,1)$ with $\varepsilon$ being the precision to be achieved. These theoretical results are illustrated by several numerical examples.

q-fin.CP↗
arXiv2017

Regression-based variance reduction approach for strong approximation schemes

In this paper we present a novel approach towards variance reduction for discretised diffusion processes. The proposed approach involves specially constructed control variates and allows for a significant reduction in the variance for the terminal functionals. In this way the complexity order of the standard Monte Carlo algorithm ($\varepsilon^{-3}$) can be reduced down to $\varepsilon^{-2}\sqrt{\left|\log(\varepsilon)\right|}$ in case of the Euler scheme with $\varepsilon$ being the precision to be achieved. These theoretical results are illustrated by several numerical examples.

math.PR↗
arXiv2016

Low frequency estimation of continuous-time moving average Lévy processes

In this paper we study the problem of statistical inference for a continuous-time moving average Lévy process of the form $$Z_{t} = \int_{\mathbb{R}}\mathcal{K}(t-s)\, dL_{s},\quad t\in\mathbb{R}$$ with a deterministic kernel (\K\) and a L{é}vy process (L\). Especially the estimation of the Lévy measure (ν\) of $L$ from low-frequency observations of the process $Z$ is considered. We construct a consistent estimator, derive its convergence rates and illustrate its performance by a numerical example. On the technical level, the main challenge is to establish a kind of exponential mixing for continuous-time moving average Lévy processes.

math.ST↗
arXiv2015

Generalized Post-Widder inversion formula with application to statistics

In this work we derive an inversion formula for the Laplace transform of a density observed on a curve in the complex domain, which generalizes the well known Post-Widder formula. We establish convergence of our inversion method and derive the corresponding convergence rates for the case of a Laplace transform of a smooth density. As an application we consider the problem of statistical inference for variance-mean mixture models. We construct a nonparametric estimator for the mixing density based on the generalized Post-Widder formula, derive bounds for its root mean square error and give a brief numerical example.

math.ST↗
arXiv2015

Optimal stopping under probability distortions and law invariant coherent risk measures

In this paper we study optimal stopping problems with respect to distorted expectations of the form \begin{eqnarray*} \mathcal{E}(X)=\int_{-\infty}^{\infty} x\,dG(F_X(x)), \end{eqnarray*} where $F_X$ is the distribution function of $X$ and $G$ is a convex distribution function on $[0,1].$ As a matter of fact, except for $G$ being the identity on $[0,1],$ dynamic versions of $\mathcal{E}(X)$ do not have the so-called time-consistency property necessary for the dynamic programming approach. So the standard approaches are not applicable to optimal stopping under $\mathcal{E}(X).$ In this paper, we prove a novel representation, which relates the solution of an optimal stopping problem under distorted expectation to the sequence of standard optimal stopping problems and hence makes the application of the standard dynamic programming-based approaches possible. Furthermore, by means of the well known Kusuoka representation, we extend our results to optimal stopping under general law invariant coherent risk measures. Finally, based on our novel representations, we develop several Monte Carlo approximation algorithms and illustrate their power for optimal stopping under Average Value at Risk and the absolute semideviation risk measures.

math.OC↗
arXiv2015

Statistical inference for generalized Ornstein-Uhlenbeck processes

In this paper, we consider the problem of statistical inference for generalized Ornstein-Uhlenbeck processes of the type \[ X_{t} = e^{-ξ_{t}} \left( X_{0} + \int_{0}^{t} e^{ξ_{u-}} d u \right), \] where \(ξ_s\) is a L{é}vy process. Our primal goal is to estimate the characteristics of the Lévy process \(ξ\) from the low-frequency observations of the process \(X\). We present a novel approach towards estimating the L{é}vy triplet of \(ξ,\) which is based on the Mellin transform technique. It is shown that the resulting estimates attain optimal minimax convergence rates. The suggested algorithms are illustrated by numerical simulations.

stat.ME↗
arXiv2014

Optimal stopping under model uncertainty: randomized stopping times approach

In this work we consider optimal stopping problems with conditional convex risk measures called optimised certainty equivalents. Without assuming any kind of time-consistency for the underlying family of risk measures, we derive a novel representation for the solution of the optimal stopping problem. In particular, we generalise the additive dual representation of Rogers (2002) to the case of optimal stopping under uncertainty. Finally, we develop several Monte Carlo algorithms and illustrate their power for optimal stopping under Average Value at Risk.

q-fin.MF↗
arXiv2014

Multilevel path simulation for weak approximation schemes

In this paper we discuss the possibility of using multilevel Monte Carlo (MLMC) methods for weak approximation schemes. It turns out that by means of a simple coupling between consecutive time discretisation levels, one can achieve the same complexity gain as under the presence of a strong convergence. We exemplify this general idea in the case of weak Euler scheme for Lévy driven stochastic differential equations, and show that, given a weak convergence of order $α\geq 1/2,$ the complexity of the corresponding "weak" MLMC estimate is of order $\varepsilon^{-2}\log ^{2}(\varepsilon).$ The numerical performance of the new "weak" MLMC method is illustrated by several numerical examples.

q-fin.CP↗
arXiv2014

Statistical Skorohod embedding problem and its generalizations

Given a Lévy process $L$, we consider the so-called statistical Skorohod embedding problem of recovering the distribution of an independent random time $T$ based on i.i.d. sample from $L_{T}.$ Our approach is based on the genuine use of the Mellin and Laplace transforms. We propose a consistent estimator for the density of $T,$ derive its convergence rates and prove their optimality. It turns out that the convergence rates heavily depend on the decay of the Mellin transform of $T.$ We also consider the application of our results to the problem of statistical inference for variance-mean mixture models and for time-changed Lévy processes.

math.ST↗
arXiv2013

Pricing American options via multi-level approximation methods

In this article we propose a novel approach to reduce the computational complexity of various approximation methods for pricing discrete time American options. Given a sequence of continuation values estimates corresponding to different levels of spatial approximation and time discretization, we propose a multi-level low biased estimate for the price of an American option. It turns out that the resulting complexity gain can be rather high and can even reach the order (\varepsilon^{-1}) with (\varepsilon) denoting the desired precision. The performance of the proposed multilevel algorithm is illustrated by a numerical example of pricing Bermudan max-call options.

q-fin.CP↗