Lévy processes and quasi-shuffle algebras

We investigate the algebra of repeated integrals of semimartingales. We prove that a minimal family of semimartingales generates a quasi-shuffle algebra. In essence, to fulfil the minimality criterion, first, the family must be a minimal generator of the algebra of repeated integrals generated by its elements and by quadratic covariation processes recursively constructed from the elements of the family. Second, recursively constructed quadratic covariation processes may lie in the linear span of previously constructed quadratic covariation processes and of the family, but may not lie in the linear span of repeated integrals of these. We prove that a finite family of independent Lévy processes that have finite moments generates a minimal family. Key to the proof are the Teugels martingales and a strong orthogonalization of them. We conclude that a finite family of independent Lévy processes forms a quasi-shuffle algebra. We discuss important potential applications to constructing efficient numerical methods for the strong approximation of stochastic differential equations driven by Lévy processes.     

Square integrable martingales orthogonal to every stochastic integral

Examples of square integrable martingales adapted to processes with independent increments and orthogonal to all stochastic integrals are constructed. If every square integrable martingale adapted to a process with stationary independent increments is a stochastic integral it is shown that the process must be a Wiener process.     

Asymptotic theory for curve-crossing analysis

We consider asymptotic properties of curve-crossing counts of linear processes and nonlinear time series by curves. Central limit theorems are obtained for curve-crossing counts of short-range dependent processes. For the long-range dependence case, the asymptotic distributions are shown to be either multiple Wiener–Itô integrals or integrals with respect to stable Lévy processes, depending on the heaviness of tails of the underlying processes.     

Simulation of stochastic integrals with respect to Lévy processes of type G

We study the simulation of stochastic processes defined as stochastic integrals with respect to type G Lévy processes for the case where it is not possible to simulate the type G process exactly. The type G Lévy process as well as the stochastic integral can on compact intervals be represented as an infinite series. In a practical simulation we must truncate this representation. We examine the approximation of the remaining terms with a simpler process to get an approximation of the stochastic integral. We also show that a stochastic time change representation can be used to obtain an approximation of stochastic integrals with respect to type G Lévy processes provided that the integrator and the integrand are independent.     

Stochastic integrals of empirical-type processes with applications to censored regression

Motivated by the analysis of linear rank estimators and the Buckley-James nonparametric EM estimator in censored regression models, we study herein the asymptotic properties of stochastic integrals of certain two-parameter empirical processes. Applications of these results on empirical processes and their stochastic integrals to the asymptotic analysis of censored regression estimators are also given.     

Traces of harmonic functions,capacities, and traces of symmetric Markov processes

We derive explicit isomorphism formulas between weighted Dirichlet integrals for harmonic functions and boundary Dirichlet forms. Applications yield results on traces of Markov processes and convergence quasieverywhere of harmonic functions.     

Gel'fand三元组上Lévy过程的随机积分

本文研究了算子值过程关于Gel’fand三元组EHE*上Lévy过程的随机积分.利用再生核Hilbert空间上柱Lévy过程的随机积分,定义一类算子值过程关于E*-值Lévy过程的随机积分。     

A note on chaotic and predictable representations for Itô–Markov additive processes

In this article, we provide predictable and chaotic representations for Itô–Markov additive processes X. Such a process is governed by a finite-state continuous time Markov chain J which allows one to modify the parameters of the Itô-jump process (in so-called regime switching manner). In addition, the transition of J triggers the jump of X distributed depending on the states of J just prior to the transition. This family of processes includes Markov modulated Itô–Lévy processes and Markov additive processes. The derived chaotic representation of a square-integrable random variable is given as a sum of stochastic integrals with respect to some explicitly constructed orthogonal martingales. We identify the predictable representation of a square-integrable martingale as a sum of stochastic integrals of predictable processes with respect to Brownian motion and power-jumps martingales related to all the jumps appearing in the model. This result generalizes the seminal result of Jacod–Yor and is of importance in financial mathematics. The derived representation then allows one to enlarge the incomplete market by a series of power-jump assets and to price all market-derivatives.     

Modelling Credit Risk in the Jump Threshold Framework

ABSTRACT

The jump threshold framework for credit risk modelling developed by Garreau and Kercheval enjoys the advantages of both structural- and reduced-form models. In their article, the focus is on multidimensional default dependence, under the assumptions that stock prices follow an exponential Lévy process (i.i.d. log returns) and that interest rates and stock volatility are constant. Explicit formulas for default time distributions and basket credit default swap (CDS) prices are obtained when the default threshold is deterministic, but only in terms of expectations when the default threshold is stochastic. In this article, we restrict attention to the one-dimensional, single-name case in order to obtain explicit closed-form solutions for the default time distribution when the default threshold, interest rate and volatility are all stochastic. When the interest rate and volatility processes are affine diffusions and the stochastic default threshold is properly chosen, we provide explicit formulas for the default time distribution, prices of defaultable bonds and CDS premia. The main idea is to make use of the Duffie–Pan–Singleton method of evaluating expectations of exponential integrals of affine diffusions.     

The Finite Product Theorem for Certain Epireflections

We discuss a number of topics relating to multiple stochastic integration, where notions and ideas from point process theory seem particularly useful. Thus we give conditions for summability of certain multiple random series in terms of associated Poisson integrals, prove a decoupling result for divergence in probability to infinity, and give conditions for the existence of certain multiple integrals with respect to compensated POISSON and asymmetric LÉVY processes. In particular, the existence criteria for multiple p-stable integrals are shown to be independent of the skewness parameter.     

A functional non-central limit theorem for multiple-stable processes with long-range dependence

A functional limit theorem is established for the partial-sum process of a class of stationary sequences which exhibit both heavy tails and long-range dependence. The stationary sequence is constructed using multiple stochastic integrals with heavy-tailed marginal distribution. Furthermore, the multiple stochastic integrals are built upon a large family of dynamical systems that are ergodic and conservative, leading to the long-range dependence phenomenon of the model. The limits constitute a new class of self-similar processes with stationary increments. They are represented by multiple stable integrals, where the integrands involve the local times of intersections of independent stationary stable regenerative sets.     

Itô formula for stochastic integrals w.r.t. compensated Poisson random measures on separable Banach spaces

We prove the Ito formula (1.3) for Banach valued functions acting on stochastic processes with jumps, the martingale part given by stochastic integrals of time dependent Banach valued random functions w.r.t. compensated Poisson random measures. Such stochastic integrals have been discussed by Mandrekar and Rüdiger, Stochastics and Stochastic Reports 78(4), 189–212 (2006) and Rüdiger (2004), Stochastics and Stochastic Reports, 76, pp. 213–242.     

有界可料过程关于集值平方可积鞅的集值随机积分

本文定义了一类有界可料过程关于集值平方可积鞅的集值随机积分,并研究了集植随机积分的性质。此为建立集值随机分析的理论奠定了基础。     

Extremal ranks and transformation of variables for extremes of functions of multivariate Gaussian processes

The exit rate from a ‘safe region’ plays an important role in dynamic reliability theory with multivariate random loads. For Gaussian processes the exit rate is simply calculated only for spherical or linear boundaries. However, many smooth boundaries, not of any of these types, are asymptotically spherical in variables of lower dimension, having a greater curvature in the remaining variables. As is shown in this paper, the asymptotic exit rate is then simply expressed as the exit rate from a sphere for a process of the lower dimensions, corrected by an explicit factor.The procedure circumvents the need to calculate complicated exit rate integrals for general boundaries, reducing the problem to a Gaussian probability integral for independent variables.A result of independent interest relates the tail distribution for a sum of a noncentral χ²-variable and a weighted sum of squares of noncentral normal variables, to the tail distribution of the χ²-variable only.     

On the ergodicity and mixing of max-stable processes

Max-stable processes arise in the limit of component-wise maxima of independent processes, under appropriate centering and normalization. In this paper, we establish necessary and sufficient conditions for the ergodicity and mixing of stationary max-stable processes. We do so in terms of their spectral representations by using extremal integrals.     

Absolute Continuity in Infinite Dimensions and Anticipating Stochastic Calculus

We generalize the change of variables formula for infinite dimensional integrals with respect to the Gaussian and exponential densities to the case of the uniform measure. The presentation of the result and its interpretation in terms of stochastic processes and anticipating stochastic calculus is unified. The expression of the Radon–Nykodim density function uses a Carleman–Fredholm determinant and a divergence operator.     

扩散过程在Holder范数下的局部 Strassen重对数律

在该文中,作者应用扩散过程在Holder范数下的大偏差得到了扩散过程在Holder范数下的局部Strassen重对数律. 并且还得到了重Ito积分的泛函重对数律.     

A Brascamp-Lieb-Luttinger-type inequality and applications to symmetric stable processes

We derive an inequality for multiple integrals from which we conclude various generalized isoperimetric inequalities for Brownian motion and symmetric stable processes in convex domains of fixed inradius. Our multiple integral inequality is a replacement for the classical inequality of H. J. Brascamp, E. H. Lieb and J. M. Luttinger, where instead of fixing the volume of the domain one fixes its inradius.

     

Approximation of optimal stopping problems

We consider optimal stopping of independent sequences. Assuming that the corresponding imbedded planar point processes converge to a Poisson process we introduce some additional conditions which allow to approximate the optimal stopping problem of the discrete time sequence by the optimal stopping of the limiting Poisson process. The optimal stopping of the involved Poisson processes is reduced to a differential equation for the critical curve which can be solved in several examples. We apply this method to obtain approximations for the stopping of iid sequences in the domain of max-stable laws with observation costs and with discount factors.     

A Series Approach to Stochastic Differential Equations with Infinite Dimensional Noise

This paper considers semilinear stochastic differential equations in Hilbert spaces with Lipschitz nonlinearities and with the noise terms driven by sequences of independent scalar Wiener processes (Brownian motions). The interpretation of such equations requires a stochastic integral. By means of a series of Itô integrals, an elementary and direct construction of a Hilbert space valued stochastic integral with respect to a sequence of independent scalar Wiener processes is given. As an application, existence and strong and weak uniqueness for the stochastic differential equation are shown by exploiting the series construction of the integral.