Estimation of jump-diffusion processes via empirical characteristic functions

Preface The starting point for this work and eventually the subject of the whole thesis was the question: how to estimate parameters of the affine stochastic volatility jump-diffusion models. These models are very important for contingent claim pricing. Their major advantage, availability T of analytical solutions for characteristic functions, made them the models of choice for many theoretical constructions and practical applications. At the same time, estimation of parameters of stochastic volatility jump-diffusion models is not a straightforward task. The problem is coming from the variance process, which is non-observable. There are several estimation methodologies that deal with estimation problems of latent variables. One appeared to be particularly interesting. It proposes the estimator that in contrast to the other methods requires neither discretization nor simulation of the process: the Continuous Empirical Characteristic function estimator (EGF) based on the unconditional characteristic function. However, the procedure was derived only for the stochastic volatility models without jumps. Thus, it has become the subject of my research. This thesis consists of three parts. Each one is written as independent and self contained article. At the same time, questions that are answered by the second and third parts of this Work arise naturally from the issues investigated and results obtained in the first one. The first chapter is the theoretical foundation of the thesis. It proposes an estimation procedure for the stochastic volatility models with jumps both in the asset price and variance processes. The estimation procedure is based on the joint unconditional characteristic function for the stochastic process. The major analytical result of this part as well as of the whole thesis is the closed form expression for the joint unconditional characteristic function for the stochastic volatility jump-diffusion models. The empirical part of the chapter suggests that besides a stochastic volatility, jumps both in the mean and the volatility equation are relevant for modelling returns of the S&P500 index, which has been chosen as a general representative of the stock asset class. Hence, the next question is: what jump process to use to model returns of the S&P500. The decision about the jump process in the framework of the affine jump- diffusion models boils down to defining the intensity of the compound Poisson process, a constant or some function of state variables, and to choosing the distribution of the jump size. While the jump in the variance process is usually assumed to be exponential, there are at least three distributions of the jump size which are currently used for the asset log-prices: normal, exponential and double exponential. The second part of this thesis shows that normal jumps in the asset log-returns should be used if we are to model S&P500 index by a stochastic volatility jump-diffusion model. This is a surprising result. Exponential distribution has fatter tails and for this reason either exponential or double exponential jump size was expected to provide the best it of the stochastic volatility jump-diffusion models to the data. The idea of testing the efficiency of the Continuous ECF estimator on the simulated data has already appeared when the first estimation results of the first chapter were obtained. In the absence of a benchmark or any ground for comparison it is unreasonable to be sure that our parameter estimates and the true parameters of the models coincide. The conclusion of the second chapter provides one more reason to do that kind of test. Thus, the third part of this thesis concentrates on the estimation of parameters of stochastic volatility jump- diffusion models on the basis of the asset price time-series simulated from various "true" parameter sets. The goal is to show that the Continuous ECF estimator based on the joint unconditional characteristic function is capable of finding the true parameters. And, the third chapter proves that our estimator indeed has the ability to do so. Once it is clear that the Continuous ECF estimator based on the unconditional characteristic function is working, the next question does not wait to appear. The question is whether the computation effort can be reduced without affecting the efficiency of the estimator, or whether the efficiency of the estimator can be improved without dramatically increasing the computational burden. The efficiency of the Continuous ECF estimator depends on the number of dimensions of the joint unconditional characteristic function which is used for its construction. Theoretically, the more dimensions there are, the more efficient is the estimation procedure. In practice, however, this relationship is not so straightforward due to the increasing computational difficulties. The second chapter, for example, in addition to the choice of the jump process, discusses the possibility of using the marginal, i.e. one-dimensional, unconditional characteristic function in the estimation instead of the joint, bi-dimensional, unconditional characteristic function. As result, the preference for one or the other depends on the model to be estimated. Thus, the computational effort can be reduced in some cases without affecting the efficiency of the estimator. The improvement of the estimator s efficiency by increasing its dimensionality faces more difficulties. The third chapter of this thesis, in addition to what was discussed above, compares the performance of the estimators with bi- and three-dimensional unconditional characteristic functions on the simulated data. It shows that the theoretical efficiency of the Continuous ECF estimator based on the three-dimensional unconditional characteristic function is not attainable in practice, at least for the moment, due to the limitations on the computer power and optimization toolboxes available to the general public. Thus, the Continuous ECF estimator based on the joint, bi-dimensional, unconditional characteristic function has all the reasons to exist and to be used for the estimation of parameters of the stochastic volatility jump-diffusion models.

Exact temporal evolution for some non-linear diffusion process

Exact solutions to FokkerPlanck equations with nonlinear drift are considered. Applications of these exact solutions for concrete models are studied. We arrive at the conclusion that for certain drifts we obtain divergent moments (and infinite relaxation time) if the diffusion process can be extended without any obstacle to the whole space. But if we introduce a potential barrier that limits the diffusion process, moments converge with a finite relaxation time.

Localization of HLA-A2.1-restricted T cell epitopes in the circumsporozoite protein of Plasmodium falciparum.

Localization of human MHC class I-restricted T cell epitopes in the circumsporozoite (CS) protein of the human parasite Plasmodium falciparum is an important objective in the development of antimalarial vaccines. To this purpose, we synthesized a series of overlapping synthetic 20-mer peptides, spanning the entire sequence of the 7G8 CS molecule except for the central repeat B cell domain. The P.f.CS peptides were first tested for their ability to bind to the human MHC class I HLA-A2.1 molecule on T2, a human cell line. Subsequently, the use of a series of shorter peptide analogues allowed us to determine the optimal A2.1 binding sequence present in several of the 20-mers. Binding P.f.CS peptides were further tested for their capacity to activate PBL from HLA-A2.1+ immune donors living in a malaria-endemic area. Specific IFN-gamma production was detected in the supernatant of cultures of PBL from exposed individuals. Cytotoxic T cell lines and clones were derived from the PBL of one responder, and their activity was shown to be HLA-A2.1-restricted and specific for the peptide 334-342 of the CS protein. In addition, double transgenic HLA-A2.1 x human beta 2-microglobulin mice were immunized with peptide 1-10 of the CS protein. T cells derived from immune lymph nodes displayed a peptide-specific HLA-A2.1-restricted cytolytic activity after one in vitro stimulation.

Diffusion Spectrum Imaging after stroke shows structural changes in the contra-lateral motor network correlating with functional recovery

There is growing interest in understanding the role of the non-injured contra-lateral hemisphere in stroke recovery. In the experimental field, histological evidence has been reported that structural changes occur in the contra-lateral connectivity and circuits during stroke recovery. In humans, some recent imaging studies indicated that contra-lateral sub-cortical pathways and functional and structural cortical networks are remodeling, after stroke. Structural changes in the contra-lateral networks, however, have never been correlated to clinical recovery in patients. To determine the importance of the contra-lateral structural changes in post-stroke recovery, we selected a population of patients with motor deficits after stroke affecting the motor cortex and/or sub-cortical motor white matter. We explored i) the presence of Generalized Fractional Anisotropy (GFA) changes indicating structural alterations in the motor network of patientsâeuro? contra-lateral hemisphere as well as their longitudinal evolution ii) the correlation of GFA changes with patientsâeuro? clinical scores, stroke size and demographics data iii) and a predictive model.

Expression hierarchy of T cell epitopes from melanoma differentiation antigens: unexpected high level presentation of tyrosinase-HLA-A2 Complexes revealed by peptide-specific, MHC-restricted, TCR-like antibodies.

Peptide Ags presented by class I MHC molecules on human melanomas and that are recognized by CD8(+) T cells are the subjects of many studies of antitumor immunity and represent attractive candidates for therapeutic approaches. However, no direct quantitative measurements exist to reveal their expression hierarchy on the cell surface. Using novel recombinant Abs which bind these Ags with a peptide-specific, MHC-restricted manner, we demonstrate a defined pattern of expression hierarchy of peptide-HLA-A2 complexes derived from three major differentiation Ags: gp100, Melan-A/Mart-1, and tyrosinase. Studying melanoma cell lines derived from multiple patients, we reveal a surprisingly high level of presentation of tyrosinase-derived complexes and moderate to very low expression of complexes derived from other Ags. No correlation between Ag presentation and mRNA expression was found; however, protein stability may play a major role. These results provide new insights into the characteristics of Ag presentation and are particularly important when such targets are being considered for immunotherapy. These results may shed new light on relationships between Ag presentation and immune response to cancer Ags.

Diffusion in spatially and temporarily inhomogeneous media: Effects of turbulent mixing

We consider diffusion of a passive substance C in a phase-separating nonmiscible binary alloy under turbulent mixing. The substance is assumed to have different diffusion coefficients in the pure phases A and B, leading to a spatially and temporarily dependent diffusion ¿coefficient¿ in the diffusion equation plus convective term. In this paper we consider especially the effects of a turbulent flow field coupled to both the Cahn-Hilliard type evolution equation of the medium and the diffusion equation (both, therefore, supplemented by a convective term). It is shown that the formerly observed prolonged anomalous diffusion [H. Lehr, F. Sagués, and J.M. Sancho, Phys. Rev. E 54, 5028 (1996)] is no longer seen if a flow of sufficient intensity is supplied.

Diffusion in spatially and temporarily inhomogeneous media

In this paper we consider diffusion of a passive substance C in a temporarily and spatially inhomogeneous two-dimensional medium. As a realization for the latter we choose a phase-separating medium consisting of two substances A and B, whose dynamics is determined by the Cahn-Hilliard equation. Assuming different diffusion coefficients of C in A and B, we find that the variance of the distribution function of the said substance grows less than linearly in time. We derive a simple identity for the variance using a probabilistic ansatz and are then able to identify the interface between A and B as the main cause for this nonlinear dependence. We argue that, finally, for very large times the here temporarily dependent diffusion "constant" goes like t-1/3 to a constant asymptotic value D¿. The latter is calculated approximately by employing the effective-medium approximation and by fitting the simulation data to the said time dependence.

Reaction-diffusion fronts under stochastic advection

We study front propagation in stirred media using a simplified modelization of the turbulent flow. Computer simulations reveal the existence of the two limiting propagation modes observed in recent experiments with liquid phase isothermal reactions. These two modes respectively correspond to a wrinkled although sharp propagating interface and to a broadened one. Specific laws relative to the enhancement of the front velocity in each regime are confirmed by our simulations.

Morphological changes in convection-diffusion-limited deposition

The effect of hydrodynamic flow upon diffusion-limited deposition on a line is investigated using a Monte Carlo model. The growth process is governed by the convection and diffusion field. The convective diffusion field is simulated by the biased-random walker resulting from a superimposed drift that represents the convective flow. The development of distinct morphologies is found with varying direction and strength of drift. By introducing a horizontal drift parallel to the deposition plate, the diffusion-limited deposit changes into a single needle inclined to the plate. The width of the needle decreases with increasing strength of drift. The angle between the needle and the plate is about 45° at high flow rate. In the presence of an inclined drift to the plate, the convection-diffusion-limited deposit leads to the formation of a characteristic columnar morphology. In the limiting case where the convection dominates, the deposition process is equivalent to ballistic deposition onto an inclined surface.

Diffusion of passive scalars under stochastic convection

The diffusion of passive scalars convected by turbulent flows is addressed here. A practical procedure to obtain stochastic velocity fields with well¿defined energy spectrum functions is also presented. Analytical results are derived, based on the use of stochastic differential equations, where the basic hypothesis involved refers to a rapidly decaying turbulence. These predictions are favorable compared with direct computer simulations of stochastic differential equations containing multiplicative space¿time correlated noise.

Morphology and segregation in two-component diffusion-limited aggregation

A diffusion-limited-aggregation (DLA) model with two components (A and B species) is presented to investigate the structure of the composite deposits. The sticking probability PAB (=PBA) between the different species is introduced into the original DLA model. By using computer simulation it is shown that various patterns are produced with varying the sticking probabilities PAB (=PBA) and PAA (= PBB), where PAA (=PBB) is the sticking probability between the same species. Segregated patterns can be analyzed under the condition PAB < PAA, assumed throughout the paper. With decreasing sticking probability PAB, a clustering of the same species occurs. With sufficiently small values of both sticking probabilities PAB and PAA, the deposit becomes dense and the segregated patterns of the composite deposit show a striped structure. The effect of the concentration on the pattern morphology is also shown.

Hepatic hemangiomas: Factors associated with T2 shine-through effect on diffusion-weighted MR sequences.

PURPOSE: To determine the frequency and factors associated with the presence of T2 shine-through effect in hepatic hemangiomas on diffusion-weighted (DW) magnetic resonance (MR) sequences. MATERIALS AND METHODS: This retrospective study was approved by institutional review board with waiver of informed consent. One hundred forty-nine consecutive patients with 388 hepatic hemangiomas who underwent a liver MR between January 2010 and November 2011 were included. MR analysis evaluated the lesion characteristics (signal intensities and enhancement patterns (classical, rapidly filling, delayed filling)), the presence of T2 shine-through effect on DW sequences (b values of 0, 150, and 600s/mm(2)), and apparent diffusion coefficient (ADC) values. Multivariate analysis was performed to study the factors associated with the T2 shine-through effect. RESULTS: T2 shine-through effect was observed in 204/388 (52.6%) of hepatic hemangiomas and in 100 (67.1%) patients. Mean ADC value of hemangiomas with T2 shine-through effect was significantly lower than hemangiomas without (2.0±0.48 vs 2.38±0.45, P<.0001). On multivariate analysis, high signal intensity on fat-suppressed T2-weighted fast spin-echo images, hemangiomas with classical or delayed enhancement, and the ADC of the liver were the only significant factors associated with T2 shine-through effect. CONCLUSION: T2 shine-through effect is commonly observed in hepatic hemangiomas and is related to hemangiomas characteristics. Radiologists should be aware of this phenomenon which could lead to misdiagnosis. Its presence should not question the diagnosis of hemangiomas when typical MR findings are found.

Restricted transgene expression in the brain with cell-type specific neuronal promoters.

Tissue-targeted expression is of major interest for studying the contribution of cellular subpopulations to neurodegenerative diseases. However, in vivo methods to investigate this issue are limited. Here, we report an analysis of the cell specificity of expression of fluorescent reporter genes driven by six neuronal promoters, with the ubiquitous phosphoglycerate kinase 1 (PGK) promoter used as a reference. Quantitative analysis of AcGFPnuc expression in the striatum and hippocampus of rodents showed that all lentiviral vectors (LV) exhibited a neuronal tropism; however, there was substantial diversity of transcriptional activity and cell-type specificity of expression. The promoters with the highest activity were those of the 67 kDa glutamic acid decarboxylase (GAD67), homeobox Dlx5/6, glutamate receptor 1 (GluR1), and preprotachykinin 1 (Tac1) genes. Neuron-specific enolase (NSE) and dopaminergic receptor 1 (Drd1a) promoters showed weak activity, but the integration of an amplification system into the LV overcame this limitation. In the striatum, the expression profiles of Tac1 and Drd1a were not limited to the striatonigral pathway, whereas in the hippocampus, Drd1a and Dlx5/6 showed the expected restricted pattern of expression. Regulation of the Dlx5/6 promoter was observed in a disease condition, whereas Tac1 activity was unaffected. These vectors provide safe tools that are more selective than others available, for the administration of therapeutic molecules in the central nervous system (CNS). Nevertheless, additional characterization of regulatory elements in neuronal promoters is still required.