Bo Martin Bibby

Martingale estimation functions for discretely observed diffusion processes

We consider three different martingale estimating functions based on discrete-time observations of a diffusion process. One is the discretized continuous-time score function adjusted by its compensator. The other two emerge naturally when optimality properties of the first are considered. Subject to natural regularity conditions, we show that all three martingale estimating functions result in consistent and asymptotically normally distributed estimators when the underlying diffusion is ergodic. Practical problems with implementing the estimation procedures are discussed through simulation studies of three specific examples. These studies also show that our estimators have good properties even for moderate sample sizes and that they are a considerable improvement compared with the estimator based on the unadjusted discretized continuous-time likelihood function, which can be seriously biased.

Hyperbolic Processes in Finance

Abstract Distributions that have tails heavier than the normal distribution are ubiquitous in finance. For purposes such as risk management and derivative pricing it is important to use relatively simple models that can capture the heavy tails and other relevant features of financial data. A class of distributions that is very often able to Iit the distributions of financial data is the class of generalized hyperbolic distributions. This has been established in numerous investigations, see, e.g., Eberlein ad Keller (1995), Bibby and Sorensen (1997), Hurst (1997), Eberlein, Keller and Prause (1998), Rydberg (1999), Kuchler et al. (1999), Jiang (2000), and Bamdorff-Nielsen and Shephard (2001c). The class of generalized hyperbolic distributions includes the standard hyperbolic distributions, the normal inverse Gaussian distributions, the scaled t-distributions and the variance-gamma distributions. The use of scaled t-distributions in finance was studied by Praetz (1972) and Blattberg and Gonedes (1974), while Madan and Seneta (1990) introduced the variance-gamma distributions in the financial literature. The normal distribution appears as a limit of generalized hyperbolic distributions. The tail behaviour of the generalized hyperbolic distributions thus span a range from Gaussian tails via exponential tails to the power tails of the t-distributions. In Section 1 we present the generalized hyperbolic distributions and their most important properties. We also discuss the generalized inverse Gaussian distributions which play an important role in the theory of generalized hyperbolic distributions and processes. This class of distributions is also of interest in its own right as a model of positive quantities in finance. Its right-hand tail behaviour spans a range from exponential decrease to a Pareto tail. In the following sections we present a number of stochastic process models for which the marginal distributions or the distributions of increments (or both) are generalized hyperbolic. The models are increasingly complex. They are thus able to Iit an increasing number of the stylized features of financial data. The well established features of financial data are for instance reviewed inBamdorff-Nielsen (1998) and Rydberg (2000). In Section 2 we discuss Levy process models, while in Section 3 we discuss models defined by stochastic differential equations. These include classical diffusion models and Omstein–Uhlenbeck models driven by Levy processes as well as superpositions of such models. In the final Section 4 we present generalized hyperbolic stochastic volatility models.

A Hyperbolic Diffusion Model for Stock Prices

Abstract. In the present paper we consider a model for stock prices which is a generalization of the model behind the Black–Scholes formula for pricing European call options. We model the log-price as a deterministic linear trend plus a diffusion process with drift zero and with a diffusion coefficient (volatility) which depends in a particular way on the instantaneous stock price. It is shown that the model possesses a number of properties encountered in empirical studies of stock prices. In particular the distribution of the adjusted log-price is hyperbolic rather than normal. The model is rather successfully fitted to two different stock price data sets. Finally, the question of option pricing based on our model is discussed and comparison to the Black–Scholes formula is made. The paper also introduces a simple general way of constructing a zero-drift diffusion with a given marginal distribution, by which other models that are potentially useful in mathematical finance can be developed.

Soluble CD163, a macrophage activation marker, is independently associated with fibrosis in patients with chronic viral hepatitis B and C

Macrophages are involved in inflammation and liver fibrosis and soluble (s)CD163 is a specific marker of activated macrophages. We investigated associations between sCD163 and biochemical and histological parameters of inflammatory activity and fibrosis in 551 patients with chronic hepatitis C virus (HCV) and 203 patients with chronic hepatitis B virus (HBV) before antiviral treatment. Scheuer histological scores of activity and fibrosis were obtained. Clinical, biochemical, and metabolic parameters were recorded. We measured sCD163 by enzyme‐linked immunosorbent assay (ELISA). Soluble CD163 was higher in patients with HCV compared to HBV (3.6 [interquartile range (IQR) 2.5‐5.4] versus 2.4 [IQR 1.8‐3.6] mg/L, P < 0.001). sCD163 was associated with fibrosis stages for both HCV (odds ratio [OR] 1.49, 95% confidence interval [CI]: 1.38‐1.61) and HBV (OR 1.32, 95% CI: 1.17‐1.49) patients, with highest levels in patients with advanced fibrosis and cirrhosis. sCD163 was a marker of fibrosis independent of other biochemical parameters and known risk factors. We created two novel sCD163‐based fibrosis scores, CD163‐HCV‐FS and CD163‐HBV‐FS, which showed areas under the receiver operating characteristics curve (AUROC) of 0.79 (95% CI: 0.74‐0.83) and 0.71 (95% CI: 0.62‐0.79), respectively, for significant fibrosis. Compared to existing fibrosis scores, CD163‐HCV‐FS was significantly superior to the aspartate aminotransferase (AST) to platelet ratio index (APRI) for all fibrosis stages and to FIB‐4 for significant fibrosis, but CD163‐HBV‐FS was not. Conclusion: sCD163 levels are increased in patients with chronic viral hepatitis, reflecting macrophage activation. Increased sCD163 is associated with the severity of disease and predicts fibrosis. A sCD163‐based fibrosis score, CD163‐HCV‐FS, is superior to APRI and FIB‐4 for the diagnosis of significant fibrosis in patients with HCV infection. (Hepatology 2014;60:521–530)

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.

Simplified Estimating Functions for Diffusion Models with a High-dimensional Parameter

We consider estimating functions for discretely observed diffusion processes of the following type: for one part of the parameter of interest we propose to use a simple and explicit estimating function of the type studied by Kessler (2000); for the remaining part of the parameter we use a martingale estimating function. Such an approach is particularly useful in practical applications when the parameter is high-dimensional. It is also often necessary to supplement a simple estimating function by another type of estimating function because only the part of the parameter on which the invariant measure depends can be estimated by a simple estimating function. Under regularity conditions the resulting estimators are consistent and asymptotically normal. Several examples are considered in order to demonstrate the idea of the estimating procedure. The method is applied to two data sets comprising wind velocities and stock prices. In one example we also propose a general method for constructing diffusion models with a prescribed marginal distribution which have a flexible dependence structure.

Measuring consciousness: Task accuracy and awareness as sigmoid functions of stimulus duration

When consciousness is examined using subjective ratings, the extent to which processing is conscious or unconscious is often estimated by calculating task performance at the subjective threshold or by calculating the correlation between accuracy and awareness. However, both these methods have certain limitations. In the present article, we propose describing task accuracy and awareness as functions of stimulus intensity (thus obtaining an accuracy and an awareness curve) as suggested by Koch and Preuschoff (2007). The estimated lag between the curves describes how much stimulus intensity must increase for awareness to change proportionally as much as accuracy and the slopes of the curves are used to assess how fast accuracy and awareness increases and whether awareness is dichotomous. The method is successfully employed to assess consciousness characteristics on data from four different awareness scales.

Exenatide Alters Myocardial Glucose Transport and Uptake Depending on Insulin Resistance and Increases Myocardial Blood Flow in Patients with Type 2 Diabetes

CONTEXT Glucagon-like peptide-1 (GLP-1) and GLP-1 receptor agonists provide beneficial cardiovascular effects by protecting against ischemia and reperfusion injury. Type 2 diabetes mellitus patients have reduced glycolysis in the heart. OBJECTIVE We hypothesized that cardioprotection by GLP-1 is achieved through increased glucose availability and utilization and aimed to assess the effect of exenatide, a synthetic GLP-1 receptor agonist, on myocardial glucose uptake (MGU), myocardial glucose transport, and myocardial blood flow (MBF). DESIGN AND METHODS We conducted a randomized, double-blinded, placebo-controlled crossover study in eight male, insulin-naive, type 2 diabetes mellitus patients without coronary artery disease. Positron emission tomography was used to determine the effect of exenatide on MGU and MBF during a pituitary-pancreatic hyperglycemic clamp with (18)F-fluorodeoxyglucose and (13)N-ammonia as tracers. RESULTS Overall, exenatide did not alter MGU. However, regression analysis revealed that exenatide altered initial clearance of glucose over the membrane of cardiomyocytes and MGU, depending on the level of insulin resistance (P = 0.017 and 0.010, respectively). Exenatide increased MBF from 0.73 ± 0.094 to 0.85 ± 0.091 ml/g · min (P = 0.0056). Except for an increase in C-peptide levels, no differences in circulating hormones or metabolites were found. CONCLUSIONS The action of exenatide as an activator or inhibitor of the glucose transport and glucose uptake in cardiomyocytes is dependent on baseline activity of glucose transport and insulin resistance. Exenatide increases MBF without changing MGU.

Global Left Ventricular Longitudinal Systolic Strain for Early Risk Assessment in Patients with Acute Myocardial Infarction Treated with Primary Percutaneous Intervention

BACKGROUND Left ventricular systolic function is a key determinant of outcome after ST-segment elevation myocardial infarction (STEMI). The aim of this study was to study speckle-tracking global longitudinal strain (GLS) for early risk evaluation in STEMI and compare it with left ventricular ejection fraction (LVEF), wall motion score index (WMSI), and end-systolic volume index (ESVI). METHODS Five-hundred seventy-six patients underwent echocardiography ≤24 hours after primary percutaneous coronary intervention for STEMI. The end point was the composite of death, hospitalization with reinfarction, congestive heart failure, or stroke. Associations with outcome were assessed by multivariate Cox regression with adjustment for clinical parameters. Hazard ratios (HRs) for events within the first year are reported per absolute percentage GLS increase. RESULTS During a median follow-up period of 24 months, 162 patients experienced at least one event. GLS was associated with the composite end point (adjusted HR, 1.20; 95% confidence interval [CI], 1.12-1.29) and also when controlling for LVEF (adjusted HR, 1.17; 95% CI, 1.07-1.29) and ESVI (adjusted HR, 1.18; 95% CI, 1.08-1.28). Although WMSI was significantly associated with outcome beyond any association accounted for by GLS, a borderline significant association was found after controlling for WMSI (adjusted HR for GLS, 1.10; 95% CI, 1.00-1.21). When GLS or WMSI was known, there was no significant association between LVEF or ESVI and outcome. CONCLUSIONS In a large population of patients with STEMI, GLS and WMSI were comparable and both superior for early risk assessment compared with volume-based left ventricular function indicators such as LVEF and ESVI. Compared with WMSI, the advantage of GLS is the provision of a semiautomated quantitative measure.

Glucagon-like peptide-1 decreases intracerebral glucose content by activating hexokinase and changing glucose clearance during hyperglycemia

Type 2 diabetes and hyperglycemia with the resulting increase of glucose concentrations in the brain impair the outcome of ischemic stroke, and may increase the risk of developing Alzheimers disease (AD). Reports indicate that glucagon-like peptide-1 (GLP-1) may be neuroprotective in models of AD and stroke: Although the mechanism is unclear, glucose homeostasis appears to be important. We conducted a randomized, double-blinded, placebo-controlled crossover study in nine healthy males. Positron emission tomography was used to determine the effect of GLP-1 on cerebral glucose transport and metabolism during a hyperglycemic clamp with 18fluoro-deoxy-glucose as tracer. Glucagon-like peptide-1 lowered brain glucose (P = 0.023) in all regions. The cerebral metabolic rate for glucose was increased everywhere (P = 0.039) but not to the same extent in all regions (P = 0.022). The unidirectional glucose transfer across the blood-brain barrier remained unchanged (P = 0.099) in all regions, while the unidirectional clearance and the phosphorylation rate increased (P = 0.013 and 0.017), leading to increased net clearance of the glucose tracer (P = 0.006). We show that GLP-1 plays a role in a regulatory mechanism involved in the actions of GLUT1 and glucose metabolism: GLP-1 ensures less fluctuation of brain glucose levels in response to alterations in plasma glucose, which may prove to be neuroprotective during hyperglycemia.