Quasi sure quadratic variation of local times of smooth semimartingales

In this paper, we prove that the process of the quadratic variation of local times of smooth semimartingales can be constructed as the quasi sure limit of the form ∑_Δn(L_t^a_i+1n−L_t^a_in)², where Δ_n=(a_iⁿ,a_i+1ⁿ) is a sequence of subdivisions of [a,b], a_iⁿ=i(b−a)/2ⁿ+a, i=0,1,…,2ⁿ.     

On the prediction of vector-valued random fields and the spectral distribution of their evanescent component

Let (X_m,n)_(m,n)∈Z² be a C^p-valued wide sense stationary process. We study the prediction theory of such processes according to different total orders on Z². In the case of a “rational order”, we give the spectral distribution of the resulting evanescent component and prove that for two different rational orders, the resulting evanescent components are mutually orthogonal.     

Characterization of dependence of multidimensional Lévy processes using Lévy copulas

This paper suggests Lévy copulas in order to characterize the dependence among components of multidimensional Lévy processes. This concept parallels the notion of a copula on the level of Lévy measures. As for random vectors, a version of Sklar's theorem states that the law of a general multivariate Lévy process is obtained by combining arbitrary univariate Lévy processes with an arbitrary Lévy copula. We construct parametric families of Lévy copulas and prove a limit theorem, which indicates how to obtain the Lévy copula of a multivariate Lévy process X from the ordinary copula of the random vector X_t for small t.     

Dynamic Markov bridges motivated by models of insider trading

Given a Markovian Brownian martingale Z, we build a process X which is a martingale in its own filtration and satisfies X₁=Z₁. We call X a dynamic bridge, because its terminal value Z₁ is not known in advance. We compute its semimartingale decomposition explicitly under both its own filtration F^X and the filtration F^X,Z jointly generated by X and Z. Our construction is heavily based on parabolic partial differential equations and filtering techniques. As an application, we explicitly solve an equilibrium model with insider trading that can be viewed as a non-Gaussian generalization of the model of Back and Pedersen (1998) [3], where the insider’s additional information evolves over time.     

Conditional large deviations for a sequence of words

Cut an i.i.d. sequence (_Xi)$\left(X_i\right)$ of ‘letters’ into ‘words’ according to an independent renewal process. Then one obtains an i.i.d. sequence of words, and thus the level 3 large deviation behaviour of this sequence of words is governed by the specific relative entropy. We consider the corresponding problem for the conditional   empirical process of words, where one conditions on a typical underlying (_Xi)$\left(X_i\right)$. We find that if the tails of the word lengths decay exponentially, the large deviations under the conditional distribution are almost surely again governed by the specific relative entropy, but the set of attainable limits is restricted.     

Exceedance probability of the integral of a stochastic process

Let X={X(s)}_s∈S be an almost sure continuous stochastic process (S compact subset of R^d) in the domain of attraction of some max-stable process, with index function constant over S. We study the tail distribution of ∫_SX(s)ds, which turns out to be of Generalized Pareto type with an extra ‘spatial’ parameter (the areal coefficient from Coles and Tawn (1996) [3]). Moreover, we discuss how to estimate the tail probability P(∫_SX(s)ds>x) for some high value x, based on independent and identically distributed copies of X. In the course we also give an estimator for the areal coefficient. We prove consistency of the proposed estimators. Our methods are applied to the total rainfall in the North Holland area; i.e. X represents in this case the rainfall over the region for which we have observations, and its integral amounts to total rainfall.The paper has two main purposes: first to formalize and justify the results of Coles and Tawn (1996) [3]; further we treat the problem in a non-parametric way as opposed to their fully parametric methods.     

Estimating the error distribution in nonparametric multiple regression with applications to model testing

In this paper we consider the estimation of the error distribution in a heteroscedastic nonparametric regression model with multivariate covariates. As estimator we consider the empirical distribution function of residuals, which are obtained from multivariate local polynomial fits of the regression and variance functions, respectively. Weak convergence of the empirical residual process to a Gaussian process is proved. We also consider various applications for testing model assumptions in nonparametric multiple regression. The model tests obtained are able to detect local alternatives that converge to zero at an n^−1/2-rate, independent of the covariate dimension. We consider in detail a test for additivity of the regression function.     

Peaks-over-threshold stability of multivariate generalized Pareto distributions

It is well-known that the univariate generalized Pareto distributions (GPD) are characterized by their peaks-over-threshold (POT) stability. We extend this result to multivariate GPDs.It is also shown that this POT stability is asymptotically shared by distributions which are in a certain neighborhood of a multivariate GPD. A multivariate extreme value distribution is a typical example.The usefulness of the results is demonstrated by various applications. We immediately obtain, for example, that the excess distribution of a linear portfolio with positive weights a_i, i≤d, is independent of the weights, if (U₁,…,U_d) follows a multivariate GPD with identical univariate polynomial or Pareto margins, which was established by Macke [On the distribution of linear combinations of multivariate EVD and GPD distributed random vectors with an application to the expected shortfall of portfolios, Diploma Thesis, University of Würzburg, 2004, (in German)] and Falk and Michel [Testing for tail independence in extreme value models. Ann. Inst. Statist. Math. 58 (2006) 261-290]. This implies, for instance, that the expected shortfall as a measure of risk fails in this case.     

Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction

Summary For Gibbsian systems of particles inR ^d , we investigate large deviations of the translation invariant empirical fields in increasing boxes. The particle interaction is given by a superstable, regular pair potential. The large deviation principle is established for systems with free or periodic boundary conditions and, under a stronger stability hypothesis on the potential, for systems with tempered boundary conditions, and for tempered (infinite-volume) Gibbs measures. As a by-product we obtain the Gibbs variational formula for the pressure. We also prove the asymptotic equivalence of microcanonical and grand canonical Gibbs distributions and establish a variational expression for the thermodynamic entropy density.     

Truncated variation, upward truncated variation and downward truncated variation of Brownian motion with drift — Their characteristics and applications

In ?ochowski (2008) [9] we defined truncated variation of Brownian motion with drift, W_t=B_t+μt,t≥0, where (B_t) is a standard Brownian motion. Truncated variation differs from regular variation in neglecting jumps smaller than some fixed c>0. We prove that truncated variation is a random variable with finite moment-generating function for any complex argument.We also define two closely related quantities — upward truncated variation and downward truncated variation.The defined quantities may have interpretations in financial mathematics. The exponential moment of upward truncated variation may be interpreted as the maximal possible return from trading a financial asset in the presence of flat commission when the dynamics of the prices of the asset follows a geometric Brownian motion process.We calculate the Laplace transform with respect to the time parameter of the moment-generating functions of the upward and downward truncated variations.As an application of the formula obtained we give an exact formula for the expected values of upward and downward truncated variations. We also give exact (up to universal constants) estimates of the expected values of the quantities mentioned.     

Path continuity of fractional Dirichlet functionals

We prove that the quasi continuous version of a functional in E^p_r is continuous along the sample paths of the Dirichlet process provided that p>2, 0<r?1 and pr>2, without assuming the Meyer equivalence. Parallel results for multi-parameter processes are also obtained. Moreover, for 1<p<2, we prove that a n parameter Dirichlet process does not touch a set of (p,2n)-zero capacity. As an example, we also study the quasi-everywhere existence of the local times of martingales on path space.     

On a class of martingale inequalities

Let Z = {Z₀, Z₁, Z₂,…} be a martingale, with difference sequence X₀ = Z₀, X_i = Z_i ? Z_{i ? 1}, i ≥ 1. The principal purpose of this paper is to prove that the best constant in the inequality λP(sup_i |X_i| ≥ λ) ≤ C sup_iE |Z_i|, for λ > 0, is C = (log 2)^?1. If Z is finite of length n, it is proved that the best constant is $\text{C}{}_{\mathrm{n}}\text{= [n(2}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}\text{? 1)]}{}^{\mathrm{?1}}$. The analogous best constant C_n(z) when Z₀ ≡ z is also determined. For these finite cases, examples of martingales attaining equality are constructed. The results follow from an explicit determination of the quantity G_n(z, E) = sup_zP(max_i=1,…,n |X_i| ≥ 1), the supremum being taken over all martingales Z with Z₀ ≡ z and E|Z_n| = E. The expression for G_n(z,E) is derived by induction, using methods from the theory of moments.     

On collision local time of two independent fractional ornstein-uhlenbeck processes

In this article, we study the existence of collision local time of two independent d-dimensional fractional Ornstein-Uhlenbeck processes X_t~(H_1)and _t~(H_2),with different parameters H_i∈(0, 1), i = 1, 2. Under the canonical framework of white noise analysis,we characterize the collision local time as a Hida distribution and obtain its' chaos expansion.     

Multivariate Markov-switching ARMA processes with regularly varying noise

The tail behaviour of stationary R^d-valued Markov-switching ARMA (MS-ARMA) processes driven by a regularly varying noise is analysed. It is shown that under appropriate summability conditions the MS-ARMA process is again regularly varying as a sequence. Moreover, it is established that these summability conditions are satisfied if the sum of the norms of the autoregressive parameters is less than one for all possible values of the parameter chain, which leads to feasible sufficient conditions.Our results complement in particular those of Saporta [Tail of the stationary solution of the stochastic equation Y_n+1=a_nY_n+b_n with Markovian coefficients, Stochastic Process. Appl. 115 (2005) 1954-1978.] where regularly varying tails of one-dimensional MS-AR(1) processes coming from consecutive large values of the parameter chain were studied.     

Sharp estimates for the CDF of quadratic forms of MPE random vectors

In this paper we develop an efficient analytical expansion of the cumulative distribution function (cdf) XBX^t where X=(X₁,…,X_n+1) with n≥2, follows a multivariate power exponential distribution (MPE). Our approach provides a sharp estimate of the cumulative distribution function of a quadratic form of MPE, together with explicit error estimates.     

Empirical likelihood confidence region for parameter in the errors-in-variables models

This paper proposes a constrained empirical likelihood confidence region for a parameter β₀ in the linear errors-in-variables model: Y_i=x_i^τβ₀+ε_i,X_i=x_i+u_i,(1?i?n), which is constructed by combining the score function corresponding to the squared orthogonal distance with a constrained region of β₀. It is shown that the coverage error of the confidence region is of order n⁻¹, and Bartlett corrections can reduce the coverage errors to n⁻². An empirical Bartlett correction is given for practical implementation. Simulations show that the proposed confidence region has satisfactory coverage not only for large samples, but also for small to medium samples.     

Extremes of conditioned elliptical random vectors

Let {X_n,n?1} be iid elliptical random vectors in R^d,d≥2 and let I,J be two non-empty disjoint index sets. Denote by X_n,I,X_n,J the subvectors of X_n with indices in I,J, respectively. For any a∈R^d such that a_J is in the support of X_1,J the conditional random sample X_n,I|X_n,J=a_J,n≥1 consists of elliptically distributed random vectors. In this paper we investigate the relation between the asymptotic behaviour of the multivariate extremes of the conditional sample and the unconditional one. We show that the asymptotic behaviour of the multivariate extremes of both samples is the same, provided that the associated random radius of X₁ has distribution function in the max-domain of attraction of a univariate extreme value distribution.     

Fluctuations of the empirical quantiles of independent Brownian motions

We consider iid Brownian motions, B_j(t), where B_j(0) has a rapidly decreasing, smooth density function f. The empirical quantiles, or pointwise order statistics, are denoted by B_j:n(t), and we consider a sequence Q_n(t)=B_j(n):n(t), where j(n)/n→α∈(0,1). This sequence converges in probability to q(t), the α-quantile of the law of B_j(t). We first show convergence in law in C[0,∞) of F_n=n^1/2(Q_n−q). We then investigate properties of the limit process F, including its local covariance structure, and Hölder-continuity and variations of its sample paths. In particular, we find that F has the same local properties as fBm with Hurst parameter H=1/4.