Diffusion approximation for nonparametric autoregression

A nonparametric statistical model of small diffusion type is compared with its discretization by a stochastic Euler difference scheme. It is shown that the discrete and continuous models are asymptotically equivalent in the sense of Le Cam's deficiency distance for statistical experiments, when the discretization step decreases with the noise intensity ε. Received: 12 April 1996 / Revised version: 29 October 1997     

On Gittins’ index theorem in continuous time

We give a new and comparably short proof of Gittins’ index theorem for dynamic allocation problems of the multi-armed bandit type in continuous time under minimal assumptions. This proof gives a complete characterization of optimal allocation strategies as those policies which follow the current leader among the Gittins indices while ensuring that a Gittins index is at an all-time low whenever the associated project is not worked on exclusively. The main tool is a representation property of Gittins index processes which allows us to show that these processes can be chosen to be pathwise lower semi-continuous from the right and quasi-lower semi-continuous from the left. Both regularity properties turn out to be crucial for our characterization and the construction of optimal allocation policies.     

Spatial autoregression and related spatio-temporal models

We propose a spatial autoregressive random field of order p on the spatial domain for p?2 in this paper, whose univariate margins are the continuous-time autoregression of order p on the real line, and introduce a class of semiparametric spatio-temporal covariance models stationary in space with the spatial autoregressive margin.     

Default swap games driven by spectrally negative Lévy processes

This paper studies game-type credit default swaps that allow the protection buyer and seller to raise or reduce their respective positions once prior to default. This leads to the study of an optimal stopping game subject to early default termination. Under a structural credit risk model based on spectrally negative Lévy processes, we apply the principles of smooth and continuous fit to identify the equilibrium exercise strategies for the buyer and the seller. We then rigorously prove the existence of the Nash equilibrium and compute the contract value at equilibrium. Numerical examples are provided to illustrate the impacts of default risk and other contractual features on the players’ exercise timing at equilibrium.     

An estimator for the quadratic covariation of asynchronously observed Itô processes with noise: Asymptotic distribution theory

The article is devoted to the nonparametric estimation of the quadratic covariation of non-synchronously observed Itô processes in an additive microstructure noise model. In a high-frequency setting, we aim at establishing an asymptotic distribution theory for a generalized multiscale estimator including a feasible central limit theorem with optimal convergence rate on convenient regularity assumptions. The inevitably remaining impact of asynchronous deterministic sampling schemes and noise corruption on the asymptotic distribution is precisely elucidated. A case study for various important examples, several generalizations of the model and an algorithm for the implementation warrant the utility of the estimation method in applications.     

Some theoretical properties of Silverman's method for Smoothed functional principal component analysis

Principal component analysis (PCA) is one of the key techniques in functional data analysis. One important feature of functional PCA is that there is a need for smoothing or regularizing of the estimated principal component curves. Silverman’s method for smoothed functional principal component analysis is an important approach in a situation where the sample curves are fully observed due to its theoretical and practical advantages. However, lack of knowledge about the theoretical properties of this method makes it difficult to generalize it to the situation where the sample curves are only observed at discrete time points. In this paper, we first establish the existence of the solutions of the successive optimization problems in this method. We then provide upper bounds for the bias parts of the estimation errors for both eigenvalues and eigenfunctions. We also prove functional central limit theorems for the variation parts of the estimation errors. As a corollary, we give the convergence rates of the estimations for eigenvalues and eigenfunctions, where these rates depend on both the sample size and the smoothing parameters. Under some conditions on the convergence rates of the smoothing parameters, we can prove the asymptotic normalities of the estimations.     

Information,no-arbitrage and completeness for asset price models with a change point

We consider a general class of continuous asset price models where the drift and the volatility functions, as well as the driving Brownian motions, change at a random time τ$\tau$. Under minimal assumptions on the random time and on the driving Brownian motions, we study the behavior of the model in all the filtrations which naturally arise in this setting, establishing martingale representation results and characterizing the validity of the NA1 and NFLVR no-arbitrage conditions.     

Inference under functional proportional and common principal component models

In many situations, when dealing with several populations with different covariance operators, equality of the operators is assumed. Usually, if this assumption does not hold, one estimates the covariance operator of each group separately, which leads to a large number of parameters. As in the multivariate setting, this is not satisfactory since the covariance operators may exhibit some common structure. In this paper, we discuss the extension to the functional setting of the common principal component model that has been widely studied when dealing with multivariate observations. Moreover, we also consider a proportional model in which the covariance operators are assumed to be equal up to a multiplicative constant. For both models, we present estimators of the unknown parameters and we obtain their asymptotic distribution. A test for equality against proportionality is also considered.     

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.     

Regular variation of order 1 nonlinear AR-ARCH models

We prove both geometric ergodicity and regular variation of the stationary distribution for a class of nonlinear stochastic recursions that includes nonlinear AR-ARCH models of order 1. The Lyapounov exponent for the model, the index of regular variation and the spectral measure for the regular variation all are characterized by a simple two-state Markov chain.     

On location estimation for LARCH processes

We consider location estimation when the error process is a stationary LARCH process with long memory in the second moments. The asymptotic distribution of the sample mean and nonlinear M-estimators of the location parameter are derived. Essential assumptions for obtaining asymptotic normality with -rate of convergence are symmetry of the innovation distribution and skew-symmetry of the ψ-function.     

Sample path Large Deviations and optimal importance sampling for stochastic volatility models

Sample path Large Deviation Principles (LDP) of the Freidlin–Wentzell type are derived for a class of diffusions, which govern the price dynamics in common stochastic volatility models from Mathematical Finance. LDP are obtained by relaxing the non-degeneracy requirement on the diffusion matrix in the standard theory of Freidlin and Wentzell. As an application, a sample path LDP is proved for the price process in the Heston stochastic volatility model.     

Stochastic optimal multi-modes switching with a viscosity solution approach

We consider the problem of optimal multi-modes switching in finite horizon, when the state of the system, including the switching cost functions are arbitrary (_gij(t,x)≥0$g_{ij}(t,x)\ge 0$). We show existence of the optimal strategy, via a verification theorem. Finally, when the state of the system is a Markov process, we show that the vector of value functions of the optimal problem is the unique viscosity solution to the system of m$m$ variational partial differential inequalities with inter-connected obstacles.     

Corrigendum to “Prediction for some processes related to a fractional Brownian motion”

This note is a correction for the statement of the results presented in Proposition 2 of Duncan (2006).     

Splitting multidimensional BSDEs and finding local equilibria

We introduce a new notion of local solution of backward stochastic differential equations (BSDEs) and prove that multidimensional quadratic BSDEs are locally but not globally solvable. Applied in a financial context on optimal investment, our results show that there exist local but no global equilibria when agents take both the absolute and the relative performance compared to their peers into account.     

Tridiagonal implicit method to evaluate European and American options under infinite activity Lévy models

We propose an efficient implicit method to evaluate European and American options when the underlying asset follows an infinite activity Lévy model. Since the Lévy measure of the infinite activity model has the singularity at the origin, we approximate infinitely many small jumps by samples of a diffusion. The proposed methods to solve partial integro–differential equations for European options and linear complementarity problems for American options via an operator splitting method involve solving linear systems with tridiagonal matrices and so can significantly reduce the computations associated with the discrete integral operators. The numerical experiments verify that the proposed method has the second-order convergence rate under an infinite activity Lévy model.     

Optimal consumption, investment and insurance with insurable risk for an investor in a Lévy market

Numerous researchers have applied the martingale approach for models driven by Lévy processes to study optimal investment problems. The aim of this paper is to apply the martingale approach to obtain a closed form solution for the optimal investment, consumption and insurance strategies of an individual in the presence of an insurable risk when the insurable risk and risky asset returns are described by Lévy processes and the utility is a constant absolute risk aversion (CARA). The model developed in this paper can potentially be applied to absorb large insurable losses in the absence of insurance protection and to examine the level of diminishing current utility and consumption.     

On the distribution of the (un)bounded sum of random variables

We propose a general treatment of random variables aggregation accounting for the dependence among variables and bounded or unbounded support of their sum. The approach is based on the extension to the concept of convolution to dependent variables, involving copula functions. We show that some classes of copula functions (such as Marshall-Olkin and elliptical) cannot be used to represent the dependence structure of two variables whose sum is bounded, while Archimedean copulas can be applied only if the generator becomes linear beyond some point. As for the application, we study the problem of capital allocation between risks when the sum of losses is bounded.