We deal with locally ill-posed nonlinear operator equations F(x) = y in L^2(0,1), where the Fréchet derivatives A = F'(x_0) of the nonlinear forward operator F are compact linear integral operators A = M ◦ J with a multiplication operator M with integrable multiplier function m and with the simple integration operator J. In particular, we give examples of nonlinear inverse problems in natural sciences and stochastic finance that can be written in such a form with linearizations that contain multiplication operators. Moreover, we consider the corresponding ill-posed linear operator equations Ax = y and their degree of ill-posedness. In particular, we discuss the fact that the noncompact multiplication operator M has only a restricted influence on this degree of ill-posedness even if m has essential zeros of various order.

Finite-difference methods for computing the derivative of a function with respect to an independent variable require knowledge of the perturbation step size for that variable. Although rules of thumb exist for determining the magnitude of the step size, their effectiveness diminishes for complicated functions or when numerically solving difficult optimization problems. This dissertation investigates the problem of determining the step size that minimizes the total error associated with finite-difference derivative approximations. The total error is defined as the sum of errors from numerical sources (roundoff error) and mathematical approximations (truncation error). Several finite-difference approximations are considered, and expressions are derived for the errors associated with each approximation. Analysis of these errors leads to an algorithm that determines the optimal perturbation step size that minimizes the total error. A benefit of this algorithm is that the computed optimal step size, when used with neighboring values of the independent variable, results in approximately the same magnitude of error in the derivative. This allows the same step size to be used for several successive iterations of the independent variable in an optimization loop. A range of independent variable values for which the optimal step size can safely remain constant is also computed. In addition to roundoff and truncation errors within the finite-difference method, numerical errors within the actual function implementation are also considered. It is shown that the optimal step size can be used to compute an upper bound for these condition errors, without any prior knowledge of the function implementation. Knowledge of a function's condition error is of great assistance during the debugging stages of simulation design. Although the fundamental analysis assumes a scalar function of a scalar independent variable, it is later extended to the general case of a vector function of a vector independent variable. Several numerical examples are shown, ranging from simple polynomial and trigonometric functions to complex trajectory optimization problems. In each example, the step size is computed using the algorithm developed herein, a rule-of-thumb method, and an alternative statistical algorithm, and the resulting finite-difference derivatives are compared to the true derivative where available. / text

This work contributes to the methodology of valuation of financial derivative contracts in an incomplete market. It focuses on a special type of incompleteness caused by the presence of a non-traded stochastic risk factor, affecting the value of the contract. The non-traded risk factor may only appear in the payoff of the contract or, in addition, may enter the dynamics of the traded asset. We consider both cases. We suggest a discrete time discrete space binomial model for the traded stock and the non-traded risk factor. We work in the utility maximization framework with dynamically changing agent's preferences. We present a discrete time multi-period analog of the forward and backward utility processes recently developed in continuous time. We use methods of stochastic control and provide the indifference valuation algorithm with both the forward and backward dynamic utilities. We compare the two approaches and provide conditions under which they assign the same value to the contract. We show that unlike the backward dynamic utility, the forward dynamic utility yields prices that do not depend on the end of the investment horizon. We pay attention to the choice of the equivalent martingale measure used for valuation (i.e., the minimal martingale measure and the minimal entropy measure for the forward and the backward utility processes correspondingly). We explicitly characterize both measures and give conditions under which they coincide. We extend our algorithm to the case of American and partial exercise contracts. We illustrate our work with numerical examples, showing that in an incomplete market, a call option on a non-traded risk factor may optimally be exercised early, and that it may be optimal to exercise only a fraction of the total number of contracts held, if partial exercise is allowed. In continuous time we extend the existing results to the case of American contracts with both the backward and the forward utilities. We emphasize the similarities between our discrete time valuation algorithm and the continuous time valuation. The two approaches use the same pricing measures, yield prices through nonlinear functionals of similar form, exhibit a similar relationship between the backward and forward prices, and a similar structure for the aggregate minimal entropy. We believe that our work makes a contribution by exposing the two above mentioned ways of dependence on the non-traded risk factor, and by providing a new dynamic indifference pricing algorithm that allows consistent valuation across different investment horizons.

Credit derivative market has experienced an exponential growth during the last 10 years with credit default swap (CDS) as an undoubted leader within this group. CDS contract is a bilateral agreement where the seller of the financial instrument provides the buyer the right to get reimbursed in case of the default in exchange for a continuous payment expressed as a CDS spread multiplied by the notional amount of the underlying debt. Originally invented to transfer the credit risk from the risk-averse investor to that one who is more prone to take on an additional risk, recently the instrument has been actively employed by the speculators betting on the financial health of the underlying obligation. It is believed that CDS contributed to the recent turmoil on financial markets and served as a weapon of mass destruction exaggerating the systematic risk. However, the latest attempts to curb the destructive force of the credit derivative for the market by means of enhancing the regulation over the instrument, bringing it on the stock-exchange and solving the transparency issue might approve CDS in the face of investor who seeks to diminish the risk of his financial portfolio. In our thesis we provide empirical evidence of CDS ability to fulfil the diversification function in the portfolio of such credit sensitive claims as bonds and stocks. Our data for the empirical analysis consist of 12 European companies whose debt underlies the most frequently traded single-name CDS with the maturity of 5 years. Through multivariate vector autoregressive models we have tested the intertemporal relation between stock returns, CDS and bond spreads changes as well as the magnitude of this relation depending on the stock market state.   The results we have achieved for our sample are the following: 1) stock returns are mainly negatively related to the CDS and bond spread changes; 2) stock returns are the least affected by both credit spread changes, whereas changes in bond spreads are the best explained by the stock and CDS market movements; 3) the strength of the relation between three variables differs over the time: the relationship between stock returns and CDS spreads is the most dominant during the pre and post-crisis periods, while during the financial crisis time the relation between stock returns and bond spread changes as well as that of between both credit spreads comes to the foreground.   The above described relations between the three markets serve as a proof of the possibility to work out diversification strategies employing CDS. During the time of turbulence on the markets the investor may exert bigger diversification gains with the help of CDS. Thus, in spite of all the recent blame of the instrument from the investor perspective it is still remains one of the sources of profit.

Several different air pollutions are produced during combustion in a diesel engine, for example nitric oxides, NOx, which can be harmful for humans. This has led to stricter emission legislations for heavy duty trucks. The law requires both lower emissions and an On-Board Diagnosis system for all manufactured heavy duty trucks. The OBD system supervises the engine in order to keep the emissions below legislation demands. The OBD system shall detect malfunctions which may lead to increased emissions. To design the OBD system an automatic model based diagnosis approach has been developed at Scania CV AB where residual generators are generated from an engine model. The main objective of this thesis is to improve the existing methods at Scania CV AB to extract residual generators from a model in order to generate more residual generators. The focus lies on the methods to find possible residual generators given an overdetermined subsystem. This includes methods to estimate derivatives of noisy signals. A method to use both integral and derivative causality has been developed, called mixed causality. With this method it has been shown that more residual generators can be found when designing a model based diagnosis system, which improves the fault isolation. To use mixed causality, derivatives are estimated with smoothing spline approximation.

Firms can use financial derivatives to hedge risks and thereby decrease the probability of bankruptcy and increase total expected tax shields. Firms also can adjust their operational policies in response to fluctuations in prices, a strategy that is often referred to as "operational hedging". In this paper, I investigate the relationship between the optimal financial and operational hedging strategies for a firm, which are endogenously determined together with its capital structure. This allows me to examine how operational hedging affects debt capacity and total expected tax shields and to make quantitative predictions about the relationship between debt issues and hedging policies. I also model the effects of asymmetric information about firms' investment opportunities on their financing and hedging decisions. First, I examine the case in which both debt and hedging contracts are observable. Then, I study the case in which firms' hedging activities are not completely transparent. The models are tested using a data set compiled from the annual reports of North American gold mining companies. Supporting evidence is found for the key predictions of the model under asymmetric information.

In many practical problems in applied sciences, the features of most interest cannot be observed directly, but have to be inferred from other, observable quantities. In particular, many important data acquisition devices provide an access to the measurement of the partial derivatives of a feature of interest rather than sensing its values in a direct way. In this case, the feature has to be recovered through integration which is known to be an ill-posed problem in the presence of noises. Moreover, the problem becomes even less trivial to solve when only a portion of a complete set of partial derivatives is available. In this case, the instability of numerical integration is further aggravated by the loss of information which is necessary to perform the reconstruction in a unique way. As formidable as it may seem, however, the above problem does have a solution in the case when the partial derivatives can be sparsely represented in the range of a linear transform. In this case, the derivatives can be recovered from their incomplete measurements using the theory of compressive sampling (aka compressed sensing), followed by reconstruction of the associated feature/object by means of a suitable integration method. It is known, however, that the overall performance of compressive sampling largely depends on the degree of sparsity of the signal representation, on the one hand, and on the degree of incompleteness of data, on the other hand. Moreover, the general rule is the sparser the signal representation is, the fewer measurements are needed to obtain a useful approximation of the true signal. Thus, one of the most important questions to be addressed in such a case would be of how much incomplete the data is allowed to be for the signal reconstruction to remain useful, and what additional constraints/information could be incorporated into the estimation process to improve the quality of reconstruction in the case of extremely under-sampled data. With these questions in mind, the present proposal introduces a way to augment the standard constraints of compressive sampling by additional information related to some natural properties of the signal to be recovered. In particular, in the case when the latter is defined to be the partial derivatives of a multidimensional signal (e.g. image), such additional information can be derived from some standard properties of the gradient operator. Consequently, the resulting scheme of derivative compressive sampling (DCS) is capable of reliably recovering the signals of interest from much fewer data samples as compared to the case of the standard CS. The signal recovery by means of DCS can be used to improve the performance of many important applications which include stereo imaging, interferometry, coherent optical tomography, and many others. In this proposal, we focus mainly on the application of DCS to the problem of phase unwrapping, whose solution is central to all the aforementioned applications. Specifically, it is shown both conceptually and experimentally that the DCS-based phase unwrapping outperforms a number of alternative approaches in terms of estimation accuracy. Finally, the proposal lists a number of research questions which need to be answered in order to attach strong theoretical guarantees to the practical success of DCS.

Šiame darbe išnagrinėta išsigimstanti keturių pirmos eilės dalinių išvestinių diferencialinių lygčių sistema. Dalinių išvestinių diferencialinių lygčių sistemai spręsti pritaikytas apibendrintų laipsninių eilučių metodas. Rasti analiziniai šios sistemos sprendiniai ir ištirtos jų savybės išsigimimo daugdaros taškų aplinkoje. Apibendrintų laipsninių eilučių metodas gali būti pritaikytas sprendžiant panašios struktūros dalinių išvestinių diferencialines lygtis, kurių eilė išsigimsta. Darbe gauti rezultatai gali būti pritaikomi modeliuojant ir tiriant realius procesus. / In this work the system of four degenerating differential equations with partial derivatives of first order was studied. For the solution of system of differential equations with partial derivatives the method of generalized power series was applied. Analytical solutions of this system were found and properties of solutions on neighbourhood of points of degeneration manifold were investigated. The method of generalized power series can be applied to the solution of systems of differential equations with partial derivatives of similar structure, which order is degenerating. The results, which were obtained in this work, can be applied to modelling and studying the real processes.

Buvo tirta išsigimstančių dalinių išvestinių diferencialinių lygčių sistema, sudaryta iš dviejų lygčių. Gauti formalūs tos sistemos sprendiniai, kurių struktūra yra tokia: kintamojo, pagal kurį išsigimsta sistemos eilė, laipsnis, padaugintas iš eksponentės, kurios rodiklyje yra to kintamojo polinomas neigiamais kintamojo laipsniais, ir ši sandauga padauginta iš laipsninės eilutės teigiamais kintamojo laipsniais. / A degenerative partial derivatives differential equations system consisting of two equations were analyzed. Formal solutions of a system were formed. The structure of solutions is as follows: variable, by which the system degenerates, degree is multiplied by the exponent, which subscript is the variable’s polynome with variable’s negative degrees, and this multiplication is multiplied by power series with positive variable degrees.

Wadley’s problem frequently emerges in dosage-mortality data and is one in which the number of surviving organisms is observed but the number initially treated is unknown. Data in this setting are also often overdispersed, that is the variability within the data exceeds that described by the distribution modelling it. The aim of this thesis is to explore distributions that can accommodate overdispersion in a Wadley’s problem setting. Two methods are essentially considered. The first considers adapting the beta-binomial and multiplicative binomial models that are frequently used for overdispersed binomial-type data to a Wadley’s problem setting. The second strategy entails modelling Wadley’s problem with a distribution that is suitable for modelling overdispersed count data. Some of the distributions introduced can be used for modelling overdispersed count data as well as overdispersed doseresponse data from a Wadley context. These models are compared using goodness of fit tests, deviance and Akaike’s Information Criterion and their properties are explored. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2009.