Shrinkage drift parameter estimation for multi‐factor Ornstein–Uhlenbeck processes

We consider some inference problems concerning the drift parameters of multi‐factors Vasicek model (or multivariate Ornstein–Uhlebeck process). For example, in modeling for interest rates, the Vasicek model asserts that the term structure of interest rate is not just a single process, but rather a superposition of several analogous processes. This motivates us to develop an improved estimation theory for the drift parameters when homogeneity of several parameters may hold. However, the information regarding the equality of these parameters may be imprecise. In this context, we consider Stein‐rule (or shrinkage) estimators that allow us to improve on the performance of the classical maximum likelihood estimator (MLE). Under an asymptotic distributional quadratic risk criterion, their relative dominance is explored and assessed. We illustrate the suggested methods by analyzing interbank interest rates of three European countries. Further, a simulation study illustrates the behavior of the suggested method for observation periods of small and moderate lengths of time. Our analytical and simulation results demonstrate that shrinkage estimators (SEs) provide excellent estimation accuracy and outperform the MLE uniformly. An over‐ridding theme of this paper is that the SEs provide powerful extensions of their classical counterparts. Copyright © 2009 John Wiley & Sons, Ltd.     

Lévy driven moving averages and semimartingales

The aim of the present paper is to study the semimartingale property of continuous time moving averages driven by Lévy processes. We provide necessary and sufficient conditions on the kernel for the moving average to be a semimartingale in the natural filtration of the Lévy process, and when this is the case we also provide a useful representation. Assuming that the driving Lévy process is of unbounded variation, we show that the moving average is a semimartingale if and only if the kernel is absolutely continuous with a density satisfying an integrability condition.     

Transition probabilities of Lévy‐type processes: Parametrix construction

We present an existence result for Lévy‐type processes which requires only weak regularity assumptions on the symbol with respect to the space variable x. Applications range from existence and uniqueness results for Lévy‐driven SDEs with Hölder continuous coefficients to existence results for stable‐like processes and Lévy‐type processes with symbols of variable order. Moreover, we obtain heat kernel estimates for a class of Lévy and Lévy‐type processes. The paper includes an extensive list of Lévy(‐type) processes satisfying the assumptions of our results.     

Small-time expansions for the transition distributions of Lévy processes

Let X=(X_t)_t≥0 be a Lévy process with absolutely continuous Lévy measure ν. Small-time expansions of arbitrary polynomial order in t are obtained for the tails , y>0, of the process, assuming smoothness conditions on the Lévy density away from the origin. By imposing additional regularity conditions on the transition density p_t of X_t, an explicit expression for the remainder of the approximation is also given. As a byproduct, polynomial expansions of order n in t are derived for the transition densities of the process. The conditions imposed on p_t require that, away from the origin, its derivatives remain uniformly bounded as t→0. Such conditions are then shown to be satisfied for symmetric stable Lévy processes as well as some tempered stable Lévy processes such as the CGMY one. The expansions seem to correct the asymptotics previously reported in the literature.     

Ergodicity of stochastic 2D Navier–Stokes equation with Lévy noise

In this paper we deal with the 2D Navier-Stokes equation perturbed by a Lévy noise force whose white noise part is non-degenerate and that the intensity measure of Poisson measure is σ-finite. Existence and uniqueness of invariant measure for this equation is obtained, two main properties of the Markov semigroup associated with this equation are proved. In other words, strong Feller property and irreducibility hold in the same space.     

Infinite dimensional Ornstein–Uhlenbeck processes with unbounded diffusion – Approximation,quadratic variation,and Itô formula

The paper studies a class of Ornstein–Uhlenbeck processes on the classical Wiener space. These processes are associated with a diffusion type Dirichlet form whose corresponding diffusion operator is unbounded in the Cameron–Martin space. It is shown that the distributions of certain finite dimensional Ornstein–Uhlenbeck processes converge weakly to the distribution of such an infinite dimensional Ornstein–Uhlenbeck process. For the infinite dimensional processes, the ordinary scalar quadratic variation is calculated. Moreover, relative to the stochastic calculus via regularization, the scalar as well as the tensor quadratic variation are derived. A related Itô formula is presented.     

The quintessential option pricing formula under Lévy processes

The basic digital method for option pricing developed in Ingersoll [J. Ingersoll, Digital contracts: Simple tools for pricing complex derivatives, Journal of Business 73 (1) (2000) 67–88] and Buchen and Skipper [P. Buchen, M. Skipper, The quintessential option pricing formula, School of Mathematics and Statistics, University of Sydney, 2003, pp. 1–31] is generalized to a Lévy environment. The approach is combined with the mathematical methodology of Boyarchenko and Levendorski [S.I. Boyarchenko, S.Z. Levendorski, Non-Gaussian Merton–Black–Scholes theory, World Scientific, 2002] that employs pseudo-differential operators whose symbol is expressed in terms of the characteristic exponent of the underlying Lévy process. Some new valuation formulas are obtained.     

Lévy noise with memory

In this article we propose a characteristic functional for Lévy stable noise with temporal correlations. The non-Markovian characteristic functional is inferred from a physical model that describes a particle interacting with an inhomogeneous environment.     

A risk model driven by Lévy processes

We present a general risk model where the aggregate claims, as well as the premium function, evolve by jumps. This is achieved by incorporating a Lévy process into the model. This seeks to account for the discrete nature of claims and asset prices. We give several explicit examples of Lévy processes that can be used to drive a risk model. This allows us to incorporate aggregate claims and premium fluctuations in the same process. We discuss important features of such processes and their relevance to risk modeling. We also extend classical results on ruin probabilities to this model. Copyright © 2003 John Wiley & Sons, Ltd.     

On a Dirichlet Problem Involving an Ornstein–Uhlenbeck Operator

We consider an elliptic Dirichlet problem which involves Ornstein–Uhlenbeck operators of special form in a half space of R ⁿ . We obtain necessary and sufficient conditions under which global Schauder estimates in spaces of Hölder continuous and bounded functions hold. For this purpose we use analytical tools, in particular semigroups and interpolation theory. Moreover we extend a theorem on the analiticity of subordinated semigroups (see Carasso and Kato; Trans. Amer. Math. Soc. 327 (1990, 867–877)) to a class of Markov type semigroups. We also provide explicit formulas for the Poisson kernels.     

Banach Space-Valued Ornstein–Uhlenbeck Processes with General Drift Coefficients

We construct Ornstein–Uhlenbeck processes with values in Banach space and with continuous paths. The drift coefficient must only generate a strongly continuous semigroup on the Hilbert space which determines the Brownian motion. We admit arbitrary starting points and consider also invariant measures for the process, generalizing earlier work in many directions. A price for the generality is that sometimes one has to enlarge the phase space but most previously known results are covered.The constructions are based on abstract Wiener space methods, more precisely on images of abstract Wiener spaces under suitable linear transformations of the Cameron–Martin space. The image abstract Wiener measures are then given by stochastic extensions. We present the basic spaces and operators and the most important results on image spaces and stochastic extensions in some detail.     

Computing the first passage time density of a time-dependent Ornstein–Uhlenbeck process to a moving boundary

In this paper we use the method of images to derive the closed-form formula for the first passage time density of a time-dependent Ornstein–Uhlenbeck process to a parametric class of moving boundaries. The results are then applied to develop a simple, efficient and systematic approximation scheme to compute tight upper and lower bounds of the first passage time density through a fixed boundary.     

Finite difference approximations and dynamics simulations for the Lévy Fractional Klein‐Kramers equation

The Klein‐Kramers equation describes position and velocity distribution of Langevin dynamics, the diffusion equation and Fokker‐Planck equation are its special cases for characterizing position distribution and velocity distribution, respectively. Incorporating the mechanisms of Lévy flights into the Klein‐Kramers formalism leads to the Lévy fractional Klein‐Kramers equation, which can effectively describe Lévy flights in the presence of an external force field in the phase space. For numerically solving the Lévy fractional Klein‐Kramers equation, this article presents the explicit and implicit finite difference schemes. The discrete maximum principle is generalized, using this result the detailed stability and convergence analyses of the schemes are given. And the extrapolation and some other possible techniques for improving the convergent rate or making the schemes efficient in more general cases are also discussed. The extensive numerical experiments are performed to confirm the effectiveness of the numerical schemes or simulate the superdiffusion processes. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

The distribution of tax payments in a Lévy insurance risk model with a surplus-dependent taxation structure

We study the distribution of tax payments in the model of Kyprianou and Zhou [Kyprianou, A.E., Zhou, X., 2009. General tax structures and the Lévy insurance risk model. J. Appl. Probab. (in press)], that is a Lévy insurance risk model with a surplus-dependent tax rate. More precisely, after a short discussion on the so-called tax identity, we derive a recursive formula for arbitrary moments of the discounted tax payments until ruin and we identify the distribution of the tax payments when there is no force of interest.     

Optimal control problems constrained by the stochastic Navier–Stokes equations with multiplicative Lévy noise

We consider the controlled stochastic Navier–Stokes equations in a bounded multidimensional domain, where the noise term allows jumps. In order to prove existence and uniqueness of an optimal control w.r.t. a given control problem, we first need to show the existence and uniqueness of a local mild solution of the considered controlled stochastic Navier–Stokes equations. We then discuss the control problem, where the related cost functional includes stopping times dependent on controls. Based on the continuity of the cost functional, we can apply existence and uniqueness results provided in [4], which enables us to show that a unique optimal control exists.     

Ergodicity for the 3D stochastic Navier–Stokes equations perturbed by Lévy noise

In this work we construct a Markov family of martingale solutions for 3D stochastic Navier–Stokes equations (SNSE) perturbed by Lévy noise with periodic boundary conditions. Using the Kolmogorov equations of integrodifferential type associated with the SNSE perturbed by Lévy noise, we construct a transition semigroup and establish the existence of a unique invariant measure. We also show that it is ergodic and strongly mixing.     

Hölder Exponent for a Two-Parameter Lévy Process

We study the regularity of a two-parameter Lévy process in the neighbourhood of a fixed point and then we compute the Hölder exponent of such a process.     

Quadratic Hermite–Padé polynomials associated with the exponential function

The asymptotic behavior of quadratic Hermite–Padé polynomials associated with the exponential function is studied for n→∞. These polynomials are defined by the relation

(*)

p_n(z)+q_n(z)e^z+r_n(z)e^2z=O(z³ⁿ⁺²) as z→0,

where O(·) denotes Landau's symbol. In the investigation analytic expressions are proved for the asymptotics of the polynomials, for the asymptotics of the remainder term in (*), and also for the arcs on which the zeros of the polynomials and of the remainder term cluster if the independent variable z is rescaled in an appropriate way. The asymptotic expressions are defined with the help of an algebraic function of third degree and its associated Riemann surface. Among other possible applications, the results form the basis for the investigation of the convergence of quadratic Hermite–Padé approximants, which will be done in a follow-up paper.     

The weak type inequality for the maximal operator of the Marcinkiewicz–Fejér means of the two-dimensional Walsh–Fourier series

The main aim of this paper is to prove that the maximal operator σ^* of the Marcinkiewicz–Fejér means of the two-dimensional Walsh–Fourier series is bounded from the Hardy space H_2/3 to the space weak-L_2/3.     

A remark on the definitions of viscosity solutions for the integro-differential equations with Lévy operators

We prove the equivalence of the three different definitions of the viscosity solution for the integro-differential equation with the Lévy operator. The two of the definitions are known in the preceding works of the author and the others, and the last one is new. A construction of a sequence of the approximating test functions to the subsolution (or the supersolution) is indispensable for the proof, and it is done explicitly in the paper.