An efficient control variate method for pricing variance derivatives

This paper studies the pricing of variance swap derivatives with stochastic volatility by the control variate method. A closed form solution is derived for the approximate model with deterministic volatility, which plays the key role in the paper, and an efficient control variate technique is therefore proposed when the volatility obeys the log-normal process. By the analysis of moments for the underlying processes, the optimal volatility function in the approximate model is constructed. The numerical results show the high efficiency of our method; the results coincide with the theoretical results. The idea in the paper is also applicable for the valuation of other types of variance swap, options with stochastic volatility and other financial derivatives with multi-factor models.     

Forward and reverse representations for Markov chains

In this paper we carry over the concept of reverse probabilistic representations developed in Milstein, Schoenmakers, Spokoiny [G.N. Milstein, J.G.M. Schoenmakers, V. Spokoiny, Transition density estimation for stochastic differential equations via forward–reverse representations, Bernoulli 10 (2) (2004) 281–312] for diffusion processes, to discrete time Markov chains. We outline the construction of reverse chains in several situations and apply this to processes which are connected with jump–diffusion models and finite state Markov chains. By combining forward and reverse representations we then construct transition density estimators for chains which have root-N$N$ accuracy in any dimension and consider some applications.     

Estimation for stochastic damping hamiltonian systems under partial observation—I. Invariant density

In this paper, we study the non-parametric estimation of the invariant density of some ergodic hamiltonian systems, using kernel estimators. The main result is a central limit theorem for such estimators under partial observation (only the positions are observed). The main tools are mixing estimates and refined covariance inequalities, the main difficulty being the strong degeneracy of such processes. This is the first paper of a series of at least two, devoted to the estimation of the characteristics of such processes: invariant density, drift term, volatility.     

On the Monte Carlo simulation of BSDEs: An improvement on the Malliavin weights

We propose a generic framework for the analysis of Monte Carlo simulation schemes of backward SDEs. The general results are used to re-visit the convergence of the algorithm suggested by Bouchard and Touzi (2004) [6]. By keeping the higher order terms in the expansion of the Skorohod integrals resulting from the Malliavin integration by parts in [6], we introduce a variant of the latter algorithm which allows for a significant reduction of the numerical complexity. We prove the convergence of this improved Malliavin-based algorithm, and derive a bound on the induced error. In particular, we show that the price to pay for our simplification is to use a more accurate localizing function.     

Nonlinear weighted least squares estimation of a three-parameter Weibull density with a nonparametric start

This paper is concerned with the parameter estimation problem for the three-parameter Weibull density which is widely employed as a model in reliability and lifetime studies. Our approach is a combination of nonparametric and parametric methods. The basic idea is to start with an initial nonparametric density estimate which needs to be as good as possible, and then apply the nonlinear least squares method to estimate the unknown parameters. As a main result, a theorem on the existence of the least squares estimate is obtained. Some simulations are given to show that our approach is satisfactory if the initial density is of good enough quality.     

Application of fast spherical Fourier transform to density estimation

This paper is on density estimation on the 2-sphere, S², using the orthogonal series estimator corresponding to spherical harmonics. In the standard approach of truncating the Fourier series of the empirical density, the Fourier transform is replaced with a version of the discrete fast spherical Fourier transform, as developed by Driscoll and Healy. The fast transform only applies to quantitative data on a regular grid. We will apply a kernel operator to the empirical density, to produce a function whose values at the vertices of such a grid will be the basis for the density estimation. The proposed estimation procedure also contains a deconvolution step, in order to reduce the bias introduced by the initial kernel operator. The main issue is to find necessary conditions on the involved discretization and the bandwidth of the kernel operator, to preserve the rate of convergence that can be achieved by the usual computationally intensive Fourier transform. Density estimation is considered in L²(S²) and more generally in Sobolev spaces H_v(S²), any v?0, with the regularity assumption that the probability density to be estimated belongs to H_s(S²) for some s>v. The proposed technique to estimate the Fourier transform of an unknown density keeps computing cost down to order O(n), where n denotes the sample size.     

On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions

We introduce a sequence of stopping times that allow us to study an analogue of a life-cycle decomposition for a continuous time Markov process, which is an extension of the well-known splitting technique of Nummelin to the continuous time case. As a consequence, we are able to give deterministic equivalents of additive functionals of the process and to state a generalisation of Chen’s inequality. We apply our results to the problem of non-parametric kernel estimation of the drift of multi-dimensional recurrent, but not necessarily ergodic, diffusion processes.     

Some data-analytic modifications to Bayes-Stein estimation

Summary The usual Bayes-Stein shrinkages of maximum likelihood estimates towards a common value may be refined by taking fuller account of the locations of the individual observations. Under a Bayesian formulation, the types of shrinkages depend critically upon the nature of the common distribution assumed for the parameters at the second stage of the prior model. In the present paper this distribution is estimated empirically from the data, permitting the data to determine the nature of the shrinkages. For example, when the observations are located in two or more clearly distinct groups, the maximum likelihood estimates are roughly speaking constrained towards common values within each group. The method also detects outliers; an extreme observation will either the regarded as an outlier and not substantially adjusted towards the other observations, or it will be rejected as an outlier, in which case a more radical adjustment takes place. The method is appropriate for a wide range of sampling distributions and may also be viewed as an alternative to standard multiple comparisons, cluster analysis, and nonparametric kernel methods.     

Nonparametric estimation of the location and scale parameters based on density estimation

Summary By representing the location and scale parameters of an absolutely continuous distribution as functionals of the usually unknown probability density function, it is possible to provide estimates of these parameters in terms of estimates of the unknown functionals. Using the properties of well-known methods of density estimates, it is shown that the proposed estimates possess nice large sample properties and it is indicated that they are also robust against dependence in the sample. The estimates perform well against other estimates of location and scale parameters.     

Local invariance principles and their application to density estimation

Summary Letx ₁,...,x _n be independent random variables with uniform distribution over [0, 1] ^d , andX( ⁿ ) be the centered and normalized empirical process associated tox ₁,...,x _n . Given a Vapnik-Chervonenkis classL of bounded functions from [0, 1] ^d intoR of bounded variation, we apply the one-dimensional dyadic scheme of Komlós, Major and Tusnády to get the best possible rate in Dudley's uniform central limit theorem for the empirical process {E ⁽ⁿ⁾(h):hL}. WhenL fulfills some extra condition, we prove there exists some sequenceB _n of Brownian bridges indexed byL such that whereK (L) denotes the maximal variation of the elements ofL. This result is then applied to maximal deviations distributions for kernel density estimators under minimal assumptions on the sequence of bandwith parameters. We also derive some results concerning strong approximations for empirical processes indexed by classes of sets with uniformly small perimeter. For example, it follows from Beck's paper that the above result is optimal, up to a possible factor , whenL is the class of Euclidean balls with radius less thanr.     

On non-parametric estimation of the Lévy kernel of Markov processes

We consider a recurrent Markov process which is an Itô semi-martingale. The Lévy kernel describes the law of its jumps. Based on observations _X0,_XΔ,…,_XnΔ$X_0,X_{\Delta },\dots ,X_{n\Delta }$, we construct an estimator for the Lévy kernel’s density. We prove its consistency (as nΔ→∞$n\Delta \to \infty$ and Δ→0$\Delta \to 0$) and a central limit theorem. In the positive recurrent case, our estimator is asymptotically normal; in the null recurrent case, it is asymptotically mixed normal. Our estimator’s rate of convergence equals the non-parametric minimax rate of smooth density estimation. The asymptotic bias and variance are analogous to those of the classical Nadaraya–Watson estimator for conditional densities. Asymptotic confidence intervals are provided.     

Efficient nonparametric estimation of distribution density in the basis of algebraic polynomials

A nonparametric estimatef ^* of an unknown distribution densityf W is called locally minimax iff it is minimax for all not too small neighborhoodsW _g ,g W, simultaneously, whereW is some dense subset ofW. Radaviius and Rudzkis proved the existence of such an estimate under some general conditions. However, the construction of the estimate is rather complicated. In this paper, a new estimate is proposed. This estimate is locally minimax under some additional assumptions which usually hold for orthobases of algebraic polynomial and is almost as simple as the linear projective estimate. Thus, it takes a form convenient for the construction of an adaptive estimator, which does not usea-priori information about the smoothness of the density. The adaptive estimation problem is briefly discussed and an unknown density fitting by Jacobi polynomials is investigated more explicitly.     

Limit theorems in the Fourier transform method for the estimation of multivariate volatility

In this paper, we prove some limit theorems for the Fourier estimator of multivariate volatility proposed by Malliavin and Mancino (2002, 2009) [14] and [15]. In a general framework of discrete time observations we establish the convergence of the estimator and some associated central limit theorems with explicit asymptotic variance. In particular, our results show that this estimator is consistent for synchronous data, but possibly biased for non-synchronous observations. Moreover, from our general central limit theorem, we deduce that the estimator can be efficient in the case of a synchronous regular sampling. In the non-synchronous sampling case, the expression of the asymptotic variance is in general less tractable. We study this case more precisely through the example of an alternate sampling.     

Quantile regression for longitudinal data

The penalized least squares interpretation of the classical random effects estimator suggests a possible way forward for quantile regression models with a large number of “fixed effects”. The introduction of a large number of individual fixed effects can significantly inflate the variability of estimates of other covariate effects. Regularization, or shrinkage of these individual effects toward a common value can help to modify this inflation effect. A general approach to estimating quantile regression models for longitudinal data is proposed employing ?₁ regularization methods. Sparse linear algebra and interior point methods for solving large linear programs are essential computational tools.     

On directional multiple-output quantile regression

This paper sheds some new light on projection quantiles. Contrary to the sophisticated set analysis used in Kong and Mizera (2008) [13], we adopt a more parametric approach and study the subgradient conditions associated with these quantiles. In this setup, we introduce Lagrange multipliers which can be interpreted in various interesting ways, in particular in a portfolio optimization context. The corresponding projection quantile regions were already shown to coincide with the halfspace depth ones in Kong and Mizera (2008) [13], but we provide here an alternative proof (completely based on projection quantiles) that has the advantage of leading to an exact computation of halfspace depth regions from projection quantiles. Above all, we systematically consider the regression case, which was barely touched in Kong and Mizera (2008) [13]. We show in particular that the regression quantile regions introduced in Hallin, Paindaveine, and Šiman (2010) [6] and [7] can also be obtained from projection (regression) quantiles, which may lead to a faster computation of those regions in some particular cases.     

Uniform central limit theorems for kernel density estimators

Let be the classical kernel density estimator based on a kernel K and n independent random vectors X _i each distributed according to an absolutely continuous law on . It is shown that the processes , , converge in law in the Banach space , for many interesting classes of functions or sets, some -Donsker, some just -pregaussian. The conditions allow for the classical bandwidths h _n that simultaneously ensure optimal rates of convergence of the kernel density estimator in mean integrated squared error, thus showing that, subject to some natural conditions, kernel density estimators are ‘plug-in’ estimators in the sense of Bickel and Ritov (Ann Statist 31:1033–1053, 2003). Some new results on the uniform central limit theorem for smoothed empirical processes, needed in the proofs, are also included.      

Asymptotic efficient estimation in semiparametric nonlinear regression models

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On estimation of a density and its derivatives

Summary Letf _n ^(p) be a recursive kernel estimate off ^(p) thepth order derivative of the probability density functionf, based on a random sample of sizen. In this paper, we provide bounds for the moments of and show that the rate of almost sure convergence of to zero isO(n ^−α), α<(r−p)/(2r+1), iff ^(r),r>p≧0, is a continuousL ₂(−∞, ∞) function. Similar rate-factor is also obtained for the almost sure convergence of to zero under different conditions onf. This work was supported in part by the Research Foundation of SUNY.     

The law of the Euler scheme for stochastic differential equations

Summary We study the approximation problem ofE f(X _T ) byE f(X _T ⁿ ), where (X _t ) is the solution of a stochastic differential equation, (X _T ⁿ ) is defined by the Euler discretization scheme with stepT/n, andf is a given function. For smoothf's, Talay and Tubaro have shown that the errorE f(X _T ) –f(X _T ⁿ ) can be expanded in powers of 1/n, which permits to construct Romberg extrapolation precedures to accelerate the convergence rate. Here, we prove that the expansion exists also whenf is only supposed measurable and bounded, under an additional nondegeneracy condition of Hörmander type for the infinitesimal generator of (X _t ): to obtain this result, we use the stochastic variations calculus. In the second part of this work, we will consider the density of the law ofX _T ⁿ and compare it to the density of the law ofX _T .