Asymptotic normality for discretely observed Markov jump processes with an absorbing state

For a continuous-time Markov process, occasionally, only discrete-time observations are available. For a simple sample of homogeneous Markov jump processes with an absorbing state, observed each on a stochastic grid of time points, we establish asymptotic normality of the maximum likelihood estimator and close the gap in Kremer and Weißbach (2013). By showing that the solution of the Kolmogorov backward equation system is continuous differentiable, we can apply results for M-estimators.     

Testing long memory based on a discretely observed process

In this paper we consider the problem of testing long memory for a continuous time process based on high frequency data. We provide two test statistics to distinguish between a semimartingale and a fractional integral process with jumps, where the integral is driven by a fractional Brownian motion with long memory. The small–sample performances of the statistics are evidenced by means of simulation studies. The real data analysis shows that the fractional integral process with jumps can capture the long memory of some financial data.     

State estimation for partially observed jump processes

In this paper partially observed jump processes are considered and optimal filtering equations are given for the conditional expectation of a functional on the past of the process.Rudemo [6] derived filtering equations for a partially observed jump Markov process. Snyder [3] gives equations for the conditional characteristic function of a jump process. Segall et al. [2] discuss filtering for processes with counting observations. Their work carries over to processes with counting observations the martingale methods that Fujisaki et al. [1] had used to derive nonlinear filtering equations for processes governed by Ito equations. Many further references to filtering for processes with discrete state measurements are given in the references cited.The objective of this paper is to show that by making use of the concept of a representation of a functional the idea of Rudemo's proof of [6, pp. 595–599] can be carried over to jump processes. The author feels that this is a very interesting proof because of its simplicity. It involves only calculations with conditional expectations and the rule for differentiation of a quotient.     

Empirical likelihood estimation of discretely sampled processes of OU type

This paper presents an empirical likelihood estimation procedure for parameters of the discretely sampled process of Ornstein-Uhlenbeck type. The proposed procedure is based on the condi- tional characteristic function, and the maximum empirical likelihood estimator is proved to be consistent and asymptotically normal. Moreover, this estimator is shown to be asymptotically efficient under some mild conditions. When the background driving Lévy process is of type A or B, we show that the intensity parameter c...     

Maximum likelihood estimation for a branching diffusion process

This paper investigates the properties of the maximum likelihood estimators of the drift and diffusion coefficients under three sampling schemes for a branching diffusion process in which the branching process is a linear birth process and the diffusion is in accordance with the Brownian motion with drift.     

Parametric inference for discretely observed subordinate diffusions

Statistical Inference for Stochastic Processes - Subordinate diffusions are constructed by time changing diffusion processes with an independent Lévy subordinator. This is a rich family of...     

Non-parametric adaptive estimation of the drift for a jump diffusion process

In this article, we consider a jump diffusion process _(Xt)t≥0${\left(X_t\right)}_{t\ge 0}$ observed at discrete times t=0,Δ,…,nΔ$t=0,\Delta ,\dots ,n\Delta$. The sampling interval Δ$\Delta$ tends to 0 and nΔ$n\Delta$ tends to infinity. We assume that _(Xt)t≥0${\left(X_t\right)}_{t\ge 0}$ is ergodic, strictly stationary and exponentially β$\beta$-mixing. We use a penalised least-square approach to compute two adaptive estimators of the drift function b$b$. We provide bounds for the risks of the two estimators.     

Maximum likelihood estimation for birth and multidimensional Brownian process

The individuals of a population reproduce according to a linear birth process and simultaneously and independently diffuse inp-dimensional Euclidean space according to a Brownian process with dependent coordinates. The properties of the maximum likelihood estimators of the drift vector and the matrix of diffusion coefficients are investigated under two sampling schemes involving incomplete observations.     

The discontinuous Galerkin method for discretely observed Asian options

Asian options represent an important subclass of the path-dependent contracts that are identified by payoff depending on the average of the underlying asset prices over the prespecified period of option lifetime. Commonly, this average is observed at discrete dates, and also, early exercise features can be admitted. As a result, analytical pricing formulae are not always available. Therefore, some form of a numerical approximation is essential for efficient option valuation. In this paper, we study a PDE model for pricing discretely observed arithmetic Asian options with fixed as well as floating strike for both European and American exercise features. The pricing equation for such options is similar to the Black-Scholes equation with 1 underlying asset, and the corresponding average appears only in the jump conditions across the sampling dates. The objective of the paper is to present the comprehensive methodological concept that forms and improves the valuation process. We employ a robust numerical procedure based on the discontinuous Galerkin approach arising from the piecewise polynomial generally discontinuous approximations. This technique enables a simple treatment of discrete sampling by incorporation of jump conditions at each monitoring date. Moreover, an American early exercise constraint is directly handled as an additional nonlinear source term in the pricing equation. The proposed solving procedure is accompanied by an empirical study with practical results compared to reference values.     

Maximum likelihood estimation in a partially observed stratified regression model with censored data

The stratified proportional intensity model generalizes Cox’s proportional intensity model by allowing different groups of the population under study to have distinct baseline intensity functions. In this article, we consider the problem of estimation in this model when the variable indicating the stratum is unobserved for some individuals in the studied sample. In this setting, we construct nonparametric maximum likelihood estimators for the parameters of the stratified model and we establish their consistency and asymptotic normality. Consistent estimators for the limiting variances are also obtained.     

On maximum likelihood estimation of a Pareto mixture

In this paper we deal with maximum likelihood estimation (MLE) of the parameters of a Pareto mixture. Standard MLE procedures are difficult to apply in this setup, because the distributions of the observations do not have common support. We study the properties of the estimators under different hypotheses; in particular, we show that, when all the parameters are unknown, the estimators can be found maximizing the profile likelihood function. Then we turn to the computational aspects of the problem, and develop three alternative procedures: an EM-type algorithm, a Simulated Annealing and an algorithm based on Cross-Entropy minimization. The work is motivated by an application in the operational risk measurement field: we fit a Pareto mixture to operational losses recorded by a bank in two different business lines. Under the assumption that each population follows a Pareto distribution, the appropriate model is a mixture of Pareto distributions where all the parameters have to be estimated.     

AIC type statistics for discretely observed ergodic diffusion processes

We consider the model selection problem for ergodic diffusion processes based on sampled data. The adaptive estimators for parameters of drift and diffusion coefficients are used in order to construct Akaike’s information criterion (AIC) type model selection statistics. Asymptotic properties of our proposed criteria are given for three kinds of the adaptive estimators.     

Test for parameter change in discretely observed diffusion processes

In this paper, we consider the problem of testing for the parameter change in discretely observed diffusion processes. For a test, we perform the cusum test based on the estimator proposed by Kessler (J Stat 24:211–229, 1997). It is shown that the test statistic weakly converges to the sup of the square of a Browian bridge. Simulation results are illustrated for the Ornstein-Uhlenbeck and Cox-Ingersoll-Ross processes.     

Extreme value estimation for discretely sampled continuous processes

In environmental applications of extreme value statistics, the underlying stochastic process is often modeled either as a max-stable process in continuous time/space or as a process in the domain of attraction of such a max-stable process. In practice, however, the processes are typically only observed at discrete points and one has to resort to interpolation to fill in the gaps. We discuss the influence of such an interpolation on estimators of marginal parameters as well as estimators of the exponent measure. In particular, natural conditions on the fineness of the observational scheme are developed which ensure that asymptotically the interpolated estimators behave in the same way as the estimators which use fully observed continuous processes.     

Estimation for discretely observed continuous state branching processes with immigration

We study the problem of parameter estimation for the continuous state branching processes with immigration, observed at discrete time points. The weighted conditional least square estimators (WCLSEs) are used for the drift parameters. Under the proper moment conditions, asymptotic distributions of the WCLSEs are obtained in the supercritical, sub- or critical cases.     

Penalized maximum likelihood estimation for a function of the intensity of a Poisson point process

Let be an unknown 2 times differentiable function and consider M to be an α- homogeneous Poisson process on Graf(f). The goal is to estimate f having a sample of the inhomogeneous Poisson process N constructed by dislocating each point of M perpendicularly to Graf(f) by a normal random variable with zero mean and constant variance σ². The exact formulas for the mean measure and the intensity function of N are obtained. Then, the function f is estimated directly using a hybrid spline approach to penalized maximum likelihood. Simulation results indicate the procedure to be consistent as and .      

Estimation of parameters for discretely observed diffusion processes with a variety of rates for information

A specific form of stochastic differential equation with unknown parameters are considered. We do not necessarily assume ergodicity or recurrency, and any moment conditions for the true process, but some tightness conditions for an information-like quantity. The interest is to estimate the parameters from discrete observations the step size of which tends to zero. Consistency and the rate of convergence of proposed estimators are presented. The rate is deduced naturally from the rate for the information-like quantities.     

A note on maximum likelihood estimation for the complex-valued first-order autoregressive process

The maximum likelihood estimation of the parameters of a complex-valued zero-mean normal stationary first-order autoregressive process is investigated. It is shown that the likelihood function corresponding to independent replicated series is uniquely maximized at a point in the interior of the parameter space. A closed-form expression is given for the estimator.