Die Dissertation widmet sich dem inversen Problem der Bestimmung der fünf Parameter eines Jump-Diffusions-Prozesses aus einer Preistrajektorie. Numerische Rechnungen zu statistischen Standardverfahren haben gezeigt, dass Stabilitätsprobleme insbesondere dann auftreten, wenn die Parameter aus einer relativ kleinen Zahl beobachteter Assetpreise bestimmt werden. Daher untersuchen wir das Problem der Parameterschätzung in dieser Arbeit unter Einbeziehung von Methoden aus der Theorie inverser Probleme, da deren zentrales Anliegen die Analyse und Regularisierung inkorrekter und instabiler inverser Aufgaben ist. In dieser Arbeit werden Phänomene der Instabilität der Parameterbestimmung herausgearbeitet und analysiert. Hierfür leiten wir eine entsprechende nichtlineare Operatorgleichung her, die den Zusammenhang zwischen einer von den Parametern abhängigen Trajektorie des Jump-Diffusions-Prozesses und der Dichtefunktion der Returns beschreibt. Diese Operatorgleichung untersuchen wir bezüglich ihrer Korrektheit. Wir zeigen anhand einer Fallstudie mit simulierten Daten, dass bei der numerischen Lösung Inkorrektheitsphänomene auftreten, sobald die Daten mit kleinen Datenfehlern behaftet sind. Um diese Stabilitätsprobleme zu überwinden, diskutieren wir einen Multiparameter-Regularisierungszugang, bei dem zusätzlich zur Least-Squares Anpassung der empirischen Dichtefunktion die Semiinvarianten berücksichtigt werden. / This thesis deals with the inverse problem of estimating simultaneously the five parameters of a jump diffusion process based on return observations of a price trajectory. It is well known that there occur instability effects using conventional statistical methods, particularly if only a small number of data are available. Therefore we apply the theory of inverse problems for parameter estimation. We analyse the forward operator mapping the parameters to the density function of the returns with respect to well-posedness and ill-posedness of the problem. We show that there occur some ill-posedness phenomena in the parameter estimation problem in case of noisy data and illustrate the instability effect by a numerical case study. To obtain stable approximate solutions of the estimation problem, we use a multi-parameter regularization approach, where a least-squares fitting of empirical densities is superposed by a quadratic penalty term of fitted semi-invariants with weights.

Die Dissertation widmet sich dem inversen Problem der Bestimmung der fünf Parameter eines Jump-Diffusions-Prozesses aus einer Preistrajektorie. Numerische Rechnungen zu statistischen Standardverfahren haben gezeigt, dass Stabilitätsprobleme insbesondere dann auftreten, wenn die Parameter aus einer relativ kleinen Zahl beobachteter Assetpreise bestimmt werden. Daher untersuchen wir das Problem der Parameterschätzung in dieser Arbeit unter Einbeziehung von Methoden aus der Theorie inverser Probleme, da deren zentrales Anliegen die Analyse und Regularisierung inkorrekter und instabiler inverser Aufgaben ist. In dieser Arbeit werden Phänomene der Instabilität der Parameterbestimmung herausgearbeitet und analysiert. Hierfür leiten wir eine entsprechende nichtlineare Operatorgleichung her, die den Zusammenhang zwischen einer von den Parametern abhängigen Trajektorie des Jump-Diffusions-Prozesses und der Dichtefunktion der Returns beschreibt. Diese Operatorgleichung untersuchen wir bezüglich ihrer Korrektheit. Wir zeigen anhand einer Fallstudie mit simulierten Daten, dass bei der numerischen Lösung Inkorrektheitsphänomene auftreten, sobald die Daten mit kleinen Datenfehlern behaftet sind. Um diese Stabilitätsprobleme zu überwinden, diskutieren wir einen Multiparameter-Regularisierungszugang, bei dem zusätzlich zur Least-Squares Anpassung der empirischen Dichtefunktion die Semiinvarianten berücksichtigt werden. / This thesis deals with the inverse problem of estimating simultaneously the five parameters of a jump diffusion process based on return observations of a price trajectory. It is well known that there occur instability effects using conventional statistical methods, particularly if only a small number of data are available. Therefore we apply the theory of inverse problems for parameter estimation. We analyse the forward operator mapping the parameters to the density function of the returns with respect to well-posedness and ill-posedness of the problem. We show that there occur some ill-posedness phenomena in the parameter estimation problem in case of noisy data and illustrate the instability effect by a numerical case study. To obtain stable approximate solutions of the estimation problem, we use a multi-parameter regularization approach, where a least-squares fitting of empirical densities is superposed by a quadratic penalty term of fitted semi-invariants with weights.

We revisit in L2-spaces the autoconvolution equation x ∗ x = y with solutions which are real-valued or complex-valued functions x(t) defined on a finite real interval, say t ∈ [0,1]. Such operator equations of quadratic type occur in physics of spectra, in optics and in stochastics, often as part of a more complex task. Because of their weak nonlinearity deautoconvolution problems are not seen as difficult and hence little attention is paid to them wrongly. In this paper, we will indicate on the example of autoconvolution a deficit in low order convergence rates for regularized solutions of nonlinear ill-posed operator equations F(x)=y with solutions x† in a Hilbert space setting. So for the real-valued version of the deautoconvolution problem, which is locally ill-posed everywhere, the classical convergence rate theory developed for the Tikhonov regularization of nonlinear ill-posed problems reaches its limits if standard source conditions using the range of F (x† )∗ fail. On the other hand, convergence rate results based on Hölder source conditions with small Hölder exponent and logarithmic source conditions or on the method of approximate source conditions are not applicable since qualified nonlinearity conditions are required which cannot be shown for the autoconvolution case according to current knowledge. We also discuss the complex-valued version of autoconvolution with full data on [0,2] and see that ill-posedness must be expected if unbounded amplitude functions are admissible. As a new detail, we present situations of local well-posedness if the domain of the autoconvolution operator is restricted to complex L2-functions with a fixed and uniformly bounded modulus function.

We revisit in L2-spaces the autoconvolution equation x ∗ x = y with solutions which are real-valued or complex-valued functions x(t) defined on a finite real interval, say t ∈ [0,1]. Such operator equations of quadratic type occur in physics of spectra, in optics and in stochastics, often as part of a more complex task. Because of their weak nonlinearity deautoconvolution problems are not seen as difficult and hence little attention is paid to them wrongly. In this paper, we will indicate on the example of autoconvolution a deficit in low order convergence rates for regularized solutions of nonlinear ill-posed operator equations F(x)=y with solutions x† in a Hilbert space setting. So for the real-valued version of the deautoconvolution problem, which is locally ill-posed everywhere, the classical convergence rate theory developed for the Tikhonov regularization of nonlinear ill-posed problems reaches its limits if standard source conditions using the range of F (x† )∗ fail. On the other hand, convergence rate results based on Hölder source conditions with small Hölder exponent and logarithmic source conditions or on the method of approximate source conditions are not applicable since qualified nonlinearity conditions are required which cannot be shown for the autoconvolution case according to current knowledge. We also discuss the complex-valued version of autoconvolution with full data on [0,2] and see that ill-posedness must be expected if unbounded amplitude functions are admissible. As a new detail, we present situations of local well-posedness if the domain of the autoconvolution operator is restricted to complex L2-functions with a fixed and uniformly bounded modulus function.