Bayesian Analysis of Linear Inverse Problems with Applications in Economics and Finance

In my PhD thesis I propose a Bayesian nonparametric estimation method for structural econometric models where the functional parameter of interest describes the economic agent's behavior. The structural parameter is characterized as the solution of a functional equation, or by using more technical words, as the solution of an inverse problem that can be either ill-posed or well-posed. From a Bayesian point of view, the parameter of interest is a random function and the solution to the inference problem is the posterior distribution of this parameter. A regular version of the posterior distribution in functional spaces is characterized. However, the infinite dimension of the considered spaces causes a problem of non continuity of the solution and then a problem of inconsistency, from a frequentist point of view, of the posterior distribution (i.e. problem of ill-posedness). The contribution of this essay is to propose new methods to deal with this problem of ill-posedness. The first one consists in adopting a Tikhonov regularization scheme in the construction of the posterior distribution so that I end up with a new object that I call regularized posterior distribution and that I guess it is solution of the inverse problem. The second approach consists in specifying a prior distribution on the parameter of interest of the g-prior type. Then, I detect a class of models for which the prior distribution is able to correct for the ill-posedness also in infinite dimensional problems. I study asymptotic properties of these proposed solutions and I prove that, under some regularity condition satisfied by the true value of the parameter of interest, they are consistent in a "frequentist" sense. Once I have set the general theory, I apply my bayesian nonparametric methodology to different estimation problems. First, I apply this estimator to deconvolution and to hazard rate, density and regression estimation. Then, I consider the estimation of an Instrumental Regression that is useful in micro-econometrics when we have to deal with problems of endogeneity. Finally, I develop an application in finance: I get the bayesian estimator for the equilibrium asset pricing functional by using the Euler equation defined in the Lucas'(1978) tree-type models.

Adaptive multiscale biological signal processing

Biological processes are very complex mechanisms, most of them being accompanied by or manifested as signals that reflect their essential characteristics and qualities. The development of diagnostic techniques based on signal and image acquisition from the human body is commonly retained as one of the propelling factors in the advancements in medicine and biosciences recorded in the recent past. It is a fact that the instruments used for biological signal and image recording, like any other acquisition system, are affected by non-idealities which, by different degrees, negatively impact on the accuracy of the recording. This work discusses how it is possible to attenuate, and ideally to remove, these effects, with a particular attention toward ultrasound imaging and extracellular recordings. Original algorithms developed during the Ph.D. research activity will be examined and compared to ones in literature tackling the same problems; results will be drawn on the base of comparative tests on both synthetic and in-vivo acquisitions, evaluating standard metrics in the respective field of application. All the developed algorithms share an adaptive approach to signal analysis, meaning that their behavior is not dependent only on designer choices, but driven by input signal characteristics too. Performance comparisons following the state of the art concerning image quality assessment, contrast gain estimation and resolution gain quantification as well as visual inspection highlighted very good results featured by the proposed ultrasound image deconvolution and restoring algorithms: axial resolution up to 5 times better than algorithms in literature are possible. Concerning extracellular recordings, the results of the proposed denoising technique compared to other signal processing algorithms pointed out an improvement of the state of the art of almost 4 dB.

Statistical methods for biomedical signal analysis and processing

Statistical modelling and statistical learning theory are two powerful analytical frameworks for analyzing signals and developing efficient processing and classification algorithms. In this thesis, these frameworks are applied for modelling and processing biomedical signals in two different contexts: ultrasound medical imaging systems and primate neural activity analysis and modelling. In the context of ultrasound medical imaging, two main applications are explored: deconvolution of signals measured from a ultrasonic transducer and automatic image segmentation and classification of prostate ultrasound scans. In the former application a stochastic model of the radio frequency signal measured from a ultrasonic transducer is derived. This model is then employed for developing in a statistical framework a regularized deconvolution procedure, for enhancing signal resolution. In the latter application, different statistical models are used to characterize images of prostate tissues, extracting different features. These features are then uses to segment the images in region of interests by means of an automatic procedure based on a statistical model of the extracted features. Finally, machine learning techniques are used for automatic classification of the different region of interests. In the context of neural activity signals, an example of bio-inspired dynamical network was developed to help in studies of motor-related processes in the brain of primate monkeys. The presented model aims to mimic the abstract functionality of a cell population in 7a parietal region of primate monkeys, during the execution of learned behavioural tasks.

Studies of the atmospheric muon flux with the ANTARES detector

The thesis main topic is the determination of the vertical component of the atmospheric muon flux as a function of the sea depth at the ANTARES site. ANTARES is a Cherenkov neutrino telescope placed at 2500m depth in the Mediterranean Sea at 40 km from the southern cost of France. In order to retrieve back the physical flux from the experimental data a deconvolution algorithm has been perform which takes into consideration the trigger inefficiensies and the reconstruction errors on the zenith angle. The obtained results are in good agreement with other ANTARES indipendent analysis.

Analysis and modeling techniques for ultrasonic tissue characterization

This thesis introduces new processing techniques for computer-aided interpretation of ultrasound images with the purpose of supporting medical diagnostic. In terms of practical application, the goal of this work is the improvement of current prostate biopsy protocols by providing physicians with a visual map overlaid over ultrasound images marking regions potentially affected by disease. As far as analysis techniques are concerned, the main contributions of this work to the state-of-the-art is the introduction of deconvolution as a pre-processing step in the standard ultrasonic tissue characterization procedure to improve the diagnostic significance of ultrasonic features. This thesis also includes some innovations in ultrasound modeling, in particular the employment of a continuous-time autoregressive moving-average (CARMA) model for ultrasound signals, a new maximum-likelihood CARMA estimator based on exponential splines and the definition of CARMA parameters as new ultrasonic features able to capture scatterers concentration. Finally, concerning the clinical usefulness of the developed techniques, the main contribution of this research is showing, through a study based on medical ground truth, that a reduction in the number of sampled cores in standard prostate biopsy is possible, preserving the same diagnostic power of the current clinical protocol.

Statistical methods for analysis and processing of medical ultrasound: applications to segmentation and restoration

In this thesis two major topics inherent with medical ultrasound images are addressed: deconvolution and segmentation. In the first case a deconvolution algorithm is described allowing statistically consistent maximum a posteriori estimates of the tissue reflectivity to be restored. These estimates are proven to provide a reliable source of information for achieving an accurate characterization of biological tissues through the ultrasound echo. The second topic involves the definition of a semi automatic algorithm for myocardium segmentation in 2D echocardiographic images. The results show that the proposed method can reduce inter- and intra observer variability in myocardial contours delineation and is feasible and accurate even on clinical data.

Refined Estimation of Earthquake Source Parameters: Methods, Applications and Scaling Relationships

The objective of this work of thesis is the refined estimations of source parameters. To such a purpose we used two different approaches, one in the frequency domain and the other in the time domain. In frequency domain, we analyzed the P- and S-wave displacement spectra to estimate spectral parameters, that is corner frequencies and low frequency spectral amplitudes. We used a parametric modeling approach which is combined with a multi-step, non-linear inversion strategy and includes the correction for attenuation and site effects. The iterative multi-step procedure was applied to about 700 microearthquakes in the moment range 1011-1014 N•m and recorded at the dense, wide-dynamic range, seismic networks operating in Southern Apennines (Italy). The analysis of the source parameters is often complicated when we are not able to model the propagation accurately. In this case the empirical Green function approach is a very useful tool to study the seismic source properties. In fact the Empirical Green Functions (EGFs) consent to represent the contribution of propagation and site effects to signal without using approximate velocity models. An EGF is a recorded three-component set of time-histories of a small earthquake whose source mechanism and propagation path are similar to those of the master event. Thus, in time domain, the deconvolution method of Vallée (2004) was applied to calculate the source time functions (RSTFs) and to accurately estimate source size and rupture velocity. This technique was applied to 1) large event, that is Mw=6.3 2009 L’Aquila mainshock (Central Italy), 2) moderate events, that is cluster of earthquakes of 2009 L’Aquila sequence with moment magnitude ranging between 3 and 5.6, 3) small event, i.e. Mw=2.9 Laviano mainshock (Southern Italy).

A moment symbolic representation of Lévy processes with applications

By using a symbolic method, known in the literature as the classical umbral calculus, a symbolic representation of Lévy processes is given and a new family of time-space harmonic polynomials with respect to such processes, which includes and generalizes the exponential complete Bell polynomials, is introduced. The usefulness of time-space harmonic polynomials with respect to Lévy processes is that it is a martingale the stochastic process obtained by replacing the indeterminate x of the polynomials with a Lévy process, whereas the Lévy process does not necessarily have this property. Therefore to find such polynomials could be particularly meaningful for applications. This new family includes Hermite polynomials, time-space harmonic with respect to Brownian motion, Poisson-Charlier polynomials with respect to Poisson processes, Laguerre and actuarial polynomials with respect to Gamma processes , Meixner polynomials of the first kind with respect to Pascal processes, Euler, Bernoulli, Krawtchuk, and pseudo-Narumi polynomials with respect to suitable random walks. The role played by cumulants is stressed and brought to the light, either in the symbolic representation of Lévy processes and their infinite divisibility property, either in the generalization, via umbral Kailath-Segall formula, of the well-known formulae giving elementary symmetric polynomials in terms of power sum symmetric polynomials. The expression of the family of time-space harmonic polynomials here introduced has some connections with the so-called moment representation of various families of multivariate polynomials. Such moment representation has been studied here for the first time in connection with the time-space harmonic property with respect to suitable symbolic multivariate Lévy processes. In particular, multivariate Hermite polynomials and their properties have been studied in connection with a symbolic version of the multivariate Brownian motion, while multivariate Bernoulli and Euler polynomials are represented as powers of multivariate polynomials which are time-space harmonic with respect to suitable multivariate Lévy processes.

Inferences on earthquake kinematic properties from data inversion: two different approaches

During my PhD, starting from the original formulations proposed by Bertrand et al., 2000 and Emolo & Zollo 2005, I developed inversion methods and applied then at different earthquakes. In particular large efforts have been devoted to the study of the model resolution and to the estimation of the model parameter errors. To study the source kinematic characteristics of the Christchurch earthquake we performed a joint inversion of strong-motion, GPS and InSAR data using a non-linear inversion method. Considering the complexity highlighted by superficial deformation data, we adopted a fault model consisting of two partially overlapping segments, with dimensions 15x11 and 7x7 km2, having different faulting styles. This two-fault model allows to better reconstruct the complex shape of the superficial deformation data. The total seismic moment resulting from the joint inversion is 3.0x1025 dyne.cm (Mw = 6.2) with an average rupture velocity of 2.0 km/s. Errors associated with the kinematic model have been estimated of around 20-30 %. The 2009 Aquila sequence was characterized by an intense aftershocks sequence that lasted several months. In this study we applied an inversion method that assumes as data the apparent Source Time Functions (aSTFs), to a Mw 4.0 aftershock of the Aquila sequence. The estimation of aSTFs was obtained using the deconvolution method proposed by Vallée et al., 2004. The inversion results show a heterogeneous slip distribution, characterized by two main slip patches located NW of the hypocenter, and a variable rupture velocity distribution (mean value of 2.5 km/s), showing a rupture front acceleration in between the two high slip zones. Errors of about 20% characterize the final estimated parameters.

Multilevel Domain Decomposition Algorithms for Monolithic Fluid-Structure Interaction Problems with Application to Haemodynamics

Finite element techniques for solving the problem of fluid-structure interaction of an elastic solid material in a laminar incompressible viscous flow are described. The mathematical problem consists of the Navier-Stokes equations in the Arbitrary Lagrangian-Eulerian formulation coupled with a non-linear structure model, considering the problem as one continuum. The coupling between the structure and the fluid is enforced inside a monolithic framework which computes simultaneously for the fluid and the structure unknowns within a unique solver. We used the well-known Crouzeix-Raviart finite element pair for discretization in space and the method of lines for discretization in time. A stability result using the Backward-Euler time-stepping scheme for both fluid and solid part and the finite element method for the space discretization has been proved. The resulting linear system has been solved by multilevel domain decomposition techniques. Our strategy is to solve several local subproblems over subdomain patches using the Schur-complement or GMRES smoother within a multigrid iterative solver. For validation and evaluation of the accuracy of the proposed methodology, we present corresponding results for a set of two FSI benchmark configurations which describe the self-induced elastic deformation of a beam attached to a cylinder in a laminar channel flow, allowing stationary as well as periodically oscillating deformations, and for a benchmark proposed by COMSOL multiphysics where a narrow vertical structure attached to the bottom wall of a channel bends under the force due to both viscous drag and pressure. Then, as an example of fluid-structure interaction in biomedical problems, we considered the academic numerical test which consists in simulating the pressure wave propagation through a straight compliant vessel. All the tests show the applicability and the numerical efficiency of our approach to both two-dimensional and three-dimensional problems.

Estimation of Quasi-Rational DSGE Models

Small-scale dynamic stochastic general equilibrium have been treated as the benchmark of much of the monetary policy literature, given their ability to explain the impact of monetary policy on output, inflation and financial markets. One cause of the empirical failure of New Keynesian models is partially due to the Rational Expectations (RE) paradigm, which entails a tight structure on the dynamics of the system. Under this hypothesis, the agents are assumed to know the data genereting process. In this paper, we propose the econometric analysis of New Keynesian DSGE models under an alternative expectations generating paradigm, which can be regarded as an intermediate position between rational expectations and learning, nameley an adapted version of the "Quasi-Rational" Expectatations (QRE) hypothesis. Given the agents' statistical model, we build a pseudo-structural form from the baseline system of Euler equations, imposing that the length of the reduced form is the same as in the `best' statistical model.

New methodologies in CT perfusion and MRI analysis to develop cancer imaging biomarkers

Quantitative imaging in oncology aims at developing imaging biomarkers for diagnosis and prediction of cancer aggressiveness and therapy response before any morphological change become visible. This Thesis exploits Computed Tomography perfusion (CTp) and multiparametric Magnetic Resonance Imaging (mpMRI) for investigating diverse cancer features on different organs. I developed a voxel-based image analysis methodology in CTp and extended its use to mpMRI, for performing precise and accurate analyses at single-voxel level. This is expected to improve reproducibility of measurements and cancer mechanisms’ comprehension and clinical interpretability. CTp has not entered the clinical routine yet, although its usefulness in the monitoring of cancer angiogenesis, due to different perfusion computing methods yielding unreproducible results. Instead, machine learning applications in mpMRI, useful to detect imaging features representative of cancer heterogeneity, are mostly limited to clinical research, because of results’ variability and difficult interpretability, which make clinicians not confident in clinical applications. In hepatic CTp, I investigated whether, and under what conditions, two widely adopted perfusion methods, Maximum Slope (MS) and Deconvolution (DV), could yield reproducible parameters. To this end, I developed signal processing methods to model the first pass kinetics and remove any numerical cause hampering the reproducibility. In mpMRI, I proposed a new approach to extract local first-order features, aiming at preserving spatial reference and making their interpretation easier. In CTp, I found out the cause of MS and DV non-reproducibility: MS and DV represent two different states of the system. Transport delays invalidate MS assumptions and, by correcting MS formulation, I have obtained the voxel-based equivalence of the two methods. In mpMRI, the developed predictive models allowed (i) detecting rectal cancers responding to neoadjuvant chemoradiation showing, at pre-therapy, sparse coarse subregions with altered density, and (ii) predicting clinically significant prostate cancers stemming from the disproportion between high- and low- diffusivity gland components.

Effective field theories and their phenomenological applications

Effective field theories (EFTs) are ubiquitous in theoretical physics and in particular in field theory descriptions of quantum systems probed at energies much lower than one or few characterizing scales. More recently, EFTs have gained a prominent role in the study of fundamental interactions and in particular in the parametriasation of new physics beyond the Standard Model, which would occur at scales Λ, much larger than the electroweak scale. In this thesis, EFTs are employed to study three different physics cases. First, we consider light-by-light scattering as a possible probe of new physics. At low energies it can be described by dimension-8 operators, leading to the well-known Euler-Heisenberg Lagrangian. We consider the explicit dependence of matching coefficients on type of particle running in the loop, confirming the sensitiveness to the spin, mass, and interactions of possibly new particles. Second, we consider EFTs to describe Dark Matter (DM) interactions with SM particles. We consider a phenomenologically motivated case, i.e., a new fermion state that couples to the Hypercharge through a form factor and has no interactions with photons and the Z boson. Results from direct, indirect and collider searches for DM are used to constrain the parameter space of the model. Third, we consider EFTs that describe axion-like particles (ALPs), whose phenomenology is inspired by the Peccei-Quinn solution to strong CP problem. ALPs generically couple to ordinary matter through dimension-5 operators. In our case study, we investigate the rather unique phenomenological implications of ALPs with enhanced couplings to the top quark.

A Unified Model for XVA, including Interest Rates and Rating

We start in Chapter 2 to investigate linear matrix-valued SDEs and the Itô-stochastic Magnus expansion. The Itô-stochastic Magnus expansion provides an efficient numerical scheme to solve matrix-valued SDEs. We show convergence of the expansion up to a stopping time τ and provide an asymptotic estimate of the cumulative distribution function of τ. Moreover, we show how to apply it to solve SPDEs with one and two spatial dimensions by combining it with the method of lines with high accuracy. We will see that the Magnus expansion allows us to use GPU techniques leading to major performance improvements compared to a standard Euler-Maruyama scheme. In Chapter 3, we study a short-rate model in a Cox-Ingersoll-Ross (CIR) framework for negative interest rates. We define the short rate as the difference of two independent CIR processes and add a deterministic shift to guarantee a perfect fit to the market term structure. We show how to use the Gram-Charlier expansion to efficiently calibrate the model to the market swaption surface and price Bermudan swaptions with good accuracy. We are taking two different perspectives for rating transition modelling. In Section 4.4, we study inhomogeneous continuous-time Markov chains (ICTMC) as a candidate for a rating model with deterministic rating transitions. We extend this model by taking a Lie group perspective in Section 4.5, to allow for stochastic rating transitions. In both cases, we will compare the most popular choices for a change of measure technique and show how to efficiently calibrate both models to the available historical rating data and market default probabilities. At the very end, we apply the techniques shown in this thesis to minimize the collateral-inclusive Credit/ Debit Valuation Adjustments under the constraint of small collateral postings by using a collateral account dependent on rating trigger.

MR in vivo tractography for the reconstruction of cranial nerves course

Aim The aim of my Ph.D. was to implement a diffusion tensor tractography (DTT) pipeline to reconstruct cranial nerve I (olfactory) to study COVID-19 patients, and anterior optic pathway (AOP, including optic nerve, chiasm, and optic tract) to study patients with sellar/parasellar tumors, and with Leber’s Hereditary Optic Neuropathy (LHON). Methods We recruited 23 patients with olfactory dysfunction after COVID-19 infection (mean age 37±14 years, 12 females); 27 patients with sellar/parasellar tumors displacing the optic chiasm eligible for endonasal endoscopic surgery (mean age 53. ±16.4 years, 13 female) and 6 LHON patients (mutation 11778/MT-ND4, mean age 24.9±15.7 years). Sex- and age-matched healthy control were also recruited. In LHON patients, optical coherence tomography (OCT) was performed. Acquisitions were performed on a clinical high field 3-T MRI scanner, using a multi-shell HARDI (High Angular Resolution Diffusion Imaging) sequence (b-values 0-300-1000-2000 s/mm2, 64 maximum gradient directions, 2mm3 isotropic voxel). DTT was performed with a multi-tissue spherical deconvolution approach and mean diffusivity (MD) DTT metrics were compared with healthy controls using an unpaired t-test. Correlations of DTT metrics with clinical data were sought by regression analysis. Results In all 23 hypo/anosmic patients with previous COVID-19 infection the CN I was successfully reconstructed with no DTT metrics alterations, thus suggesting the pathogenetic role of central olfactory cortical system dysfunction. In all 27 patients with sellar/parasellar tumors the AOP was reconstructed, and in 11/13 (84.7%) undergoing endonasal endoscopic surgery the anatomical fidelity of the reconstruction was confirmed; a significant decrease in MD within the chiasma (p<0.0001) was also found. In LHON patients a reduction of MD in the AOP was significantly associated with OCT parameters (p=0.036). Conclusions Multi-shell HARDI diffusion-weighted MRI followed by multi-tissue spherical deconvolution for the DTT reconstruction of the CN I and AOP has been implemented, and its utility demonstrated in clinical practice.