Martingale Noise Research Articles

A stochastic Gronwall lemma and well-posedness of path-dependent SDEs driven by martingale noise

We show existence and uniqueness of solutions of stochastic path-dependent differential equations driven by cadlag martingale noise under joint local monotonicity and coercivity assumptions on the coefficients with a bound in terms of the supremum norm. In this set-up, the usual proof using the ordinary Gronwall lemma together with the Burkholder-Davis-Gundy inequality seems impossible. In order to solve this problem, we prove a new and quite general stochastic Gronwall lemma for cadlag martingales using Lenglart's inequality.

Oracle inequalities for the stochastic differential equations

This paper is a survey of recent results on the adaptive robust non parametric methods for the continuous time regression model with the semi - martingale noises with jumps. The noises are modeled by the L\'evy processes, the Ornstein -- Uhlenbeck processes and semi-Markov processes. We represent the general model selection method and the sharp oracle inequalities methods which provide the robust efficient estimation in the adaptive setting. Moreover, we present the recent results on the improved model selection methods for the nonparametric estimation problems.

Fractional Calculus And Pathwise Integration for Volterra Processes Driven by Lévy and Martingale Noise

Abstract We introduce a pathwise integration for Volterra processes driven by Lévy noise or martingale noise. These processes are widely used in applications to turbulence, signal processes, biology, and in environmental finance. Indeed they constitute a very flexible class of models, which include fractional Brownian and Lévy motions and it is part of the so-called ambit fields. A pathwise integration with respect of such Volterra processes aims at producing a framework where modelling is easily understandable from an information perspective. The techniques used are based on fractional calculus and in this there is a bridging of the stochastic and deterministic techniques. The present paper aims at setting the basis for a framework in which further computational rules can be devised. Our results are general in the choice of driving noise. Additionally we propose some further details in the relevant context subordinated Wiener processes.

Robust model selection for a semimartingale continuous time regression from discrete data

The paper considers the problem of estimating a periodic function in a continuous time regression model observed under a general semimartingale noise with an unknown distribution in the case when continuous observation cannot be provided and only discrete time measurements are available. Two specific types of noises are studied in detail: a non-Gaussian Ornstein–Uhlenbeck process and a time-varying linear combination of a Brownian motion and compound Poisson process. We develop new analytical tools to treat the adaptive estimation problems from discrete data. A lower bound for the frequency sampling, needed for the efficiency of the procedure constructed by discrete observations, has been found. Sharp non-asymptotic oracle inequalities for the robust quadratic risk have been derived. New convergence rates for the efficient procedures have been obtained. An example of the regression with a martingale noise exhibits that the minimax robust convergence rate may be both higher or lower as compared with the minimax rate for the “white noise” model. The results of Monte-Carlo simulations are given.

A Dependent Hidden Markov Model of Credit Quality

We propose a dependent hidden Markov model of credit quality. We suppose that the "true" credit quality is not observed directly but only through noisy observations given by posted credit ratings. The model is formulated in discrete time with a Markov chain observed in martingale noise, where "noise" terms of the state and observation processes are possibly dependent. The model provides estimates for the state of the Markov chain governing the evolution of the credit rating process and the parameters of the model, where the latter are estimated using the EM algorithm. The dependent dynamics allow for the so-called "rating momentum" discussed in the credit literature and also provide a convenient test of independence between the state and observation dynamics.

Milstein Approximation for Advection-Diffusion Equations Driven by Multiplicative Noncontinuous Martingale Noises

In this paper, the strong approximation of a stochastic partial differential equation, whose differential operator is of advection-diffusion type and which is driven by a multiplicative, infinite dimensional, càdlàg, square integrable martingale, is presented. A finite dimensional projection of the infinite dimensional equation, for example a Galerkin projection, with nonequidistant time stepping is used. Error estimates for the discretized equation are derived in L 2 and almost sure senses. Besides space and time discretizations, noise approximations are also provided, where the Milstein double stochastic integral is approximated in such a way that the overall complexity is not increased compared to an Euler–Maruyama approximation. Finally, simulations complete the paper.

Stochastic approximation with long range dependent and heavy tailed noise

Stability and convergence properties of stochastic approximation algorithms are analyzed when the noise includes a long range dependent component (modeled by a fractional Brownian motion) and a heavy tailed component (modeled by a symmetric stable process), in addition to the usual `martingale noise'. This is motivated by the emergent applications in communications. The proofs are based on comparing suitably interpolated iterates with a limiting ordinary differential equation. Related issues such as asynchronous implementations, Markov noise, etc. are briefly discussed.

Semimartingale stochastic approximation procedure and recursive estimation

The semimartingale stochastic approximation procedure, precisely, the Robbins-Monro type SDE, is introduced, which naturally includes both generalized stochastic approximation algorithms with martingale noises and recursive parameter estimation procedures for statistical models associated with semimartingales. General results concerning the asymptotic behavior of the solution are presented. In particular, the conditions ensuring the convergence, the rate of convergence, and the asymptotic expansion are established. The results concerning the Polyak weighted averaging procedure are also presented.

On sequential estimation of parameters in semimartingale regression models with continuous time parameter

We consider the problem of parameter estimation for multidimensional continuous-time linear stochastic regression models with an arbitrary finite number of unknown parameters and with martingale noise. The main result of the paper claims that the unknown parameters can be estimated with prescribed mean-square precision in this general model providing a unified description of both discrete and continuous time process. Among the conditions on the regressors there is one bounding the growth of the maximal eigenvalue of the design matrix with respect to its minimal eigenvalue. This condition is slightly stronger as compared with the corresponding conditions usually imposed on the regressors in asymptotic investigations but still it enables one to consider models with different behavior of the eigenvalues. The construction makes use of a two-step procedure based on the modified least-squares estimators and special stopping rules.

Linear systems driven by martingale noise

A method is developed for calculating second moment properties and moments of order three and higher of the state X of a linear filter driven by martingale noise. The martingale noise is interpreted as the formal derivative of a square integrable martingale with continuous samples. The Gaussian white noise is an example of a martingale noise. It is shown that the differential equations of the mean and correlation functions of the state X developed in the paper resemble the corresponding equations of the classical linear random vibration and coincide with these equations if the input is a Gaussian white noise. The moment equations are derived by (1) the Itô formula for semimartingales and (2) the classical Itô formula applied to a diffusion process whose coordinates include X . An advantage of the second method is use of more familiar concepts. However, this method requires to calculate unnecessary moments and can be applied only for a class of martingale noise processes. Examples are presented to illustrate and evaluate the two methods for calculating moments of X and demonstrate the use of these methods in linear random vibration.

On two-parameter non-degenerate brownian martingales

In this note we give an estimate of the density of a certain class of Brownian martingales in the plane, and analyze its asymptotic behaviour near the axes. In a second part, we give a result of equi-absolute continuity for the occupation measure of the Brownian martingale, and deduce the existence and uniqueness of solution for a class of hyperbolic stochastic differential equations with additive martingale noise and nondecreasing coefficient.

The robbins-monro type stochastic differential equations. I. convergence of solutions

The Robbins-Monro (RM) type SDE is introduced, which naturally includes both generalized RM stochastic approximation algorithms with martingale noises and recursive estimationprocedures for general statistical models. The approach of the investigation of the a.s.convergence as of the strong solution of such type equations is proposed. The approach based on the new description of the semimartingales convergence sets andnonstandard representation of the process of bounded variation (in the decomposition of thespecial semimartingale in the form of difference of two increasing predictable processes

Convergence of least squares identifiers of time series with martingale difference and binary white Markov generating processes

Abstract This paper gives some results concerning the convergence of sequential least squares (SLS) identifiers of autoregressive (AR) time series models. Convergence depends on the second moment ergodicity (SME) of the generating (input) process to the time series model. Since SME conditions are difficult to prove for specific generating sequences, this paper proves this condition to be satisfied with respect to two concrete classes of generating sequences, namely, martingale difference sequences and binary white noise Markov sequences to give insight to the convergence of identifiers via binary white noise interrogation inputs. This extends the result of Mann and Wald of convergence for the case of independent and identically distributed (IID) input processes to concrete non-IID input classes.

Stochastic Differential Equations in Hilbert Space and their Application to Delay Systems

We consider infinite dimensional stochastic differential equations with state dependent martingale noise using semigroup approach. Several kinds of solutions to such equations are studied and the interrelationsbetween them investigated. We prove also some existence theorems for a general class of stochastic evolution equations. New results on equations with state dependent Gaussian white noise are obtained. The general theory is applied to stochastic delay equations with state dependent noise. A representation theorem as well as an existence and uniqueness theorem for such equations are proved.