Martin Jacobsen

CHAPTER 4 – Estimating Functions for Discretely Sampled Diffusion-Type Models

Publisher Summary This chapter demonstrates that estimating functions can be found not only for ordinary diffusions but also for stochastic volatility models and diffusions with jumps. For stochastic volatility models, the estimating functions are constructed in such a way that asymptotic properties of the estimator can easily be established. The main advantage of the estimating functions discussed in this chapter is that they usually require less computation than the alternative methods. It is a particularly useful approach when quick estimators are needed. These simple estimators have a rather high efficiency when the estimating function is well chosen. The hallmark of the estimating functions approach is the use of a given collection of relations between observations at different time points to construct an optimal estimator, i.e., the most efficient estimator possible based on these relations. In a high-frequency sampling asymptotic scenario, optimal martingale estimating functions are, in fact, efficient for diffusion models.

Paroxysmal atrial fibrillation occurs often in cryptogenic ischaemic stroke. Final results from the SURPRISE study

Atrial fibrillation (AF) increases the risk of stroke fourfold and is associated with a poor clinical outcome. Despite work‐up in compliance with guidelines, up to one‐third of patients have cryptogenic stroke (CS). The prevalence of asymptomatic paroxysmal atrial fibrillation (PAF) in CS remains unknown. The SURPRISE project aimed at determining this rate using long‐term cardiac monitoring.

The time to ruin for a class of Markov additive risk process with two-sided jumps

We consider risk processes that locally behave like Brownian motion with some drift and variance, these both depending on an underlying Markov chain that is also used to generate the claims arrival process. Thus, claims arrive according to a renewal process with waiting times of phase type. Positive claims (downward jumps) are always possible but negative claims (upward jumps) are also allowed. The claims are assumed to form an independent, identically distributed sequence, independent of everything else. As main results, the joint Laplace transform of the time to ruin and the undershoot at ruin, as well as the probability of ruin, are explicitly determined under the assumption that the Laplace transform of the positive claims is a rational function. Both the joint Laplace transform and the ruin probability are decomposed according to the type of ruin: ruin by jump or ruin by continuity. The methods used involve finding certain martingales by first finding partial eigenfunctions for the generator of the Markov process composed of the risk process and the underlying Markov chain. We also use certain results from complex function theory as important tools.

Discretely Observed Diffusions: Classes of Estimating Functions and Small Δ‐optimality

Ergodic diffusions in several dimensions, depending on an unknown multivariate parameter are considered. For estimation, when the diffusion is observed only at finitely many equidistant time points, unbiased estimating functions leading to consistent and asymptotically Gaussian estimators are used. Different types of estimating functions are discussed and the concept of small Δ-optimality is introduced to help select good estimating functions. Explicit criteria for small Δ-optimality are given. Also some exact optimality conditions are presented as well as, for one-dimensional diffusions, methods for improving estimators using time reversibility.

Martingales and the distribution of the time to ruin

We determine the ultimate ruin probability and the Laplace transform of the distribution of the time to ruin in the classical risk model, where claims arrive according to a renewal process, with waiting times that are of phase-type, while the claims themselves follow a distribution with a Laplace transform that is a rational function. The main tools are martingales, the optional sampling theorem and results from the theory of piecewise deterministic Markov processes.

Weak convergence of marked point processes generated by crossings of multivariate jump processes. Applications to neural network modeling

Abstract We consider the multivariate point process determined by the crossing times of the components of a multivariate jump process through a multivariate boundary, assuming to reset each component to an initial value after its boundary crossing. We prove that this point process converges weakly to the point process determined by the crossing times of the limit process. This holds for both diffusion and deterministic limit processes. The almost sure convergence of the first passage times under the almost sure convergence of the processes is also proved. The particular case of a multivariate Stein process converging to a multivariate Ornstein–Uhlenbeck process is discussed as a guideline for applying diffusion limits for jump processes. We apply our theoretical findings to neural network modeling. The proposed model gives a mathematical foundation to the generalization of the class of Leaky Integrate-and-Fire models for single neural dynamics to the case of a firing network of neurons. This will help future study of dependent spike trains.

Multivariate Counting Processes

We shall define and construct multivariate counting processes in a manner similar to the one used in Chapter 1 for the one-dimensional case.

One-Dimensional Homogeneous Diffusions

When constructing a model defined by a stochastic differential equation (SDE) the basic problem is whether the equation has a solution and if so, when an initial condition is given, whether the solution is unique. Once the existence and uniqueness of the solution has been established so that the model is well-defined one may then proceed to study specific properties of the solution such as its long term behaviour, stationarity and the form of the invariant distribution, boundedness or positivity and whatever other properties are needed for the problem at hand. The solution to an SDE is a stochastic process, i.e, a randomly generated function of time so that formally the solution may be viewed as a typically huge collection of ordinary functions of time. It is this that makes SDEs much more difficult to deal with than ordinary differential equations where a unique solution is just one function of time.

One-Dimensional Counting Processes

Consider the half-line (0,∞] (0 excluded, ∞ included) equipped with the Borel σ-algebra β of subsets generated by the subintervals of (0,∞].

Birth times, death times and time substitutions in Markov chains

SummaryGiven a Markov chain (Xn)n≧0, random times τ are studied which are birth times or death times in the sense that the post-τ and pre-τ processes are independent given the present (Xτ−1, Xτ) at time τ and the conditional post-τ process (birth times) or the conditional pre-τ process (death times) is again Markovian. The main result for birth times characterizes all time substitutions through homogeneous random sets with the property that all points in the set are birth times. The main result for death times is the dual of this and appears as the birth time theorem with the direction of time reversed.