### Refine

#### Year of publication

- 2016 (13) (remove)

#### Document Type

- Doctoral Thesis (13) (remove)

#### Keywords

- Autoregressive Hilbertian model (1)
- Bootstrap (1)
- Functional autoregression (1)
- Functional time series (1)
- Gröbner bases (1)
- algebraic function fields (1)
- algebraic number fields (1)
- dense univariate rational interpolation (1)
- sparse interpolation of multivariate rational functions (1)
- sparse multivariate polynomial interpolation (1)

#### Faculty / Organisational entity

- Fachbereich Mathematik (13) (remove)

This thesis deals with risk measures based on utility functions and time consistency of dynamic risk measures. It is therefore aimed at readers interested in both, the theory of static and dynamic financial risk measures in the sense of Artzner, Delbaen, Eber and Heath [7], [8] and the theory of preferences in the tradition of von Neumann and Morgenstern [134].
A main contribution of this thesis is the introduction of optimal expected utility (OEU) risk measures as a new class of utility-based risk measures. We introduce OEU, investigate its main properties, and its applicability to risk measurement and put it in perspective to alternative risk measures and notions of certainty equivalents. To the best of our knowledge, OEU is the only existing utility-based risk measure that is (non-trivial and) coherent if the utility function u has constant relative risk aversion. We present several different risk measures that can be derived with special choices of u and illustrate that OEU reacts in a more sensitive way to slight changes of the probability of a financial loss than value at risk (V@R) and average value at risk.
Further, we propose implied risk aversion as a coherent rating methodology for retail structured products (RSPs). Implied risk aversion is based on optimal expected utility risk measures and, in contrast to standard V@R-based ratings, takes into account both the upside potential and the downside risks of such products. In addition, implied risk aversion is easily interpreted in terms of an individual investor's risk aversion: A product is attractive (unattractive) for an investor if its implied risk aversion is higher (lower) than his individual risk aversion. We illustrate this approach in a case study with more than 15,000 warrants on DAX ® and find that implied risk aversion is able to identify favorable products; in particular, implied risk aversion is not necessarily increasing with respect to the strikes of call warrants.
Another main focus of this thesis is on consistency of dynamic risk measures. To this end, we study risk measures on the space of distributions, discuss concavity on the level of distributions and slightly generalize Weber's [137] findings on the relation of time consistent dynamic risk measures to static risk measures to the case of dynamic risk measures with time-dependent parameters. Finally, this thesis investigates how recursively composed dynamic risk measures in discrete time, which are time consistent by construction, can be related to corresponding dynamic risk measures in continuous time. We present different approaches to establish this link and outline the theoretical basis and the practical benefits of this relation. The thesis concludes with a numerical implementation of this theory.

In this thesis, mathematical research questions related to recursive utility and stochastic differential utility (SDU) are explored.
First, a class of backward equations under nonlinear expectations is investigated: Existence and uniqueness of solutions are established, and the issues of stability and discrete-time approximation are addressed. It is then shown that backward equations of this class naturally appear as a continuous-time limit in the context of recursive utility with nonlinear expectations.
Then, the Epstein-Zin parametrization of SDU is studied. The focus is on specifications with both relative risk aversion and elasitcity of intertemporal substitution greater that one. A concave utility functional is constructed and a utility gradient inequality is established.
Finally, consumption-portfolio problems with recursive preferences and unspanned risk are investigated. The investor's optimal strategies are characterized by a specific semilinear partial differential equation. The solution of this equation is constructed by a fixed point argument, and a corresponding efficient and accurate method to calculate optimal strategies numerically is given.

Inflation modeling is a very important tool for conducting an efficient monetary policy. This doctoral thesis reviewed inflation models, in particular the Phillips curve models of inflation dynamics. We focused on a well known and widely used model, the so-called three equation new Keynesian model which is a system of equations consisting of a new Keynesian Phillips curve (NKPC), an investment and saving (IS) curve and an interest rate rule.
We gave a detailed derivation of these equations. The interest rate rule used in this model is normally determined by using a Lagrangian method to solve an optimal control problem constrained by a standard discrete time NKPC which describes the inflation dynamics and an IS curve that represents the output gaps dynamics. In contrast to the real world, this method assumes that the policy makers intervene continuously. This means that the costs resulting from the change in the interest rates are ignored. We showed also that there are approximation errors made, when one log-linearizes non linear equations, by doing the derivation of the standard discrete time NKPC.
We agreed with other researchers as mentioned in this thesis, that errors which result from ignoring such log-linear approximation errors and the costs of altering interest rates by determining interest rate rule, can lead to a suboptimal interest rate rule and hence to non-optimal paths of output gaps and inflation rate.
To overcome such a problem, we proposed a stochastic optimal impulse control method. We formulated the problem as a stochastic optimal impulse control problem by considering the costs of change in interest rates and the approximation error terms. In order to formulate this problem, we first transform the standard discrete time NKPC and the IS curve into their high-frequency versions and hence into their continuous time versions where error terms are described by a zero mean Gaussian white noise with a finite and constant variance. After formulating this problem, we use the quasi-variational inequality approach to solve analytically a special case of the central bank problem, where an inflation rate is supposed to be on target and a central bank has to optimally control output gap dynamics. This method gives an optimal control band in which output gap process has to be maintained and an optimal control strategy, which includes the optimal size of intervention and optimal intervention time, that can be used to keep the process into the optimal control band.
Finally, using a numerical example, we examined the impact of some model parameters on optimal control strategy. The results show that an increase in the output gap volatility as well as in the fixed and proportional costs of the change in interest rate lead to an increase in the width of the optimal control band. In this case, the optimal intervention requires the central bank to wait longer before undertaking another control action.

The thesis consists of two parts. In the first part we consider the stable Auslander--Reiten quiver of a block \(B\) of a Hecke algebra of the symmetric group at a root of unity in characteristic zero. The main theorem states that if the ground field is algebraically closed and \(B\) is of wild representation type, then the tree class of every connected component of the stable Auslander--Reiten quiver \(\Gamma_{s}(B)\) of \(B\) is \(A_{\infty}\). The main ingredient of the proof is a skew group algebra construction over a quantum complete intersection. Also, for these algebras the stable Auslander--Reiten quiver is computed in the case where the defining parameters are roots of unity. As a result, the tree class of every connected component of the stable Auslander--Reiten quiver is \(A_{\infty}\).\[\]
In the second part of the thesis we are concerned with branching rules for Hecke algebras of the symmetric group at a root of unity. We give a detailed survey of the theory initiated by I. Grojnowski and A. Kleshchev, describing the Lie-theoretic structure that the Grothendieck group of finite-dimensional modules over a cyclotomic Hecke algebra carries. A decisive role in this approach is played by various functors that give branching rules for cyclotomic Hecke algebras that are independent of the underlying field. We give a thorough definition of divided power functors that will enable us to reformulate the Scopes equivalence of a Scopes pair of blocks of Hecke algebras of the symmetric group. As a consequence we prove that two indecomposable modules that correspond under this equivalence have a common vertex. In particular, we verify the Dipper--Du Conjecture in the case where the blocks under consideration have finite representation type.

By using Gröbner bases of ideals of polynomial algebras over a field, many implemented algorithms manage to give exciting examples and counter examples in Commutative Algebra and Algebraic Geometry. Part A of this thesis will focus on extending the concept of Gröbner bases and Standard bases for polynomial algebras over the ring of integers and its factors \(\mathbb{Z}_m[x]\). Moreover we implemented two algorithms for this case in Singular which use different approaches in detecting useless computations, the classical Buchberger algorithm and a F5 signature based algorithm. Part B includes two algorithms that compute the graded Hilbert depth of a graded module over a polynomial algebra \(R\) over a field, as well as the depth and the multigraded Stanley depth of a factor of monomial ideals of \(R\). The two algorithms provide faster computations and examples that lead B. Ichim and A. Zarojanu to a counter example of a question of J. Herzog. A. Duval, B. Goeckner, C. Klivans and J. Martin have recently discovered a counter example for the Stanley Conjecture. We prove in this thesis that the Stanley Conjecture holds in some special cases. Part D explores the General Neron Desingularization in the frame of Noetherian local domains of dimension 1. We have constructed and implemented in Singular and algorithm that computes a strong Artin Approximation for Cohen-Macaulay local rings of dimension 1.

Gröbner bases are one of the most powerful tools in computer algebra and commutative algebra, with applications in algebraic geometry and singularity theory. From the theoretical point of view, these bases can be computed over any field using Buchberger's algorithm. In practice, however, the computational efficiency depends on the arithmetic of the coefficient field.
In this thesis, we consider Gröbner bases computations over two types of coefficient fields. First, consider a simple extension \(K=\mathbb{Q}(\alpha)\) of \(\mathbb{Q}\), where \(\alpha\) is an algebraic number, and let \(f\in \mathbb{Q}[t]\) be the minimal polynomial of \(\alpha\). Second, let \(K'\) be the algebraic function field over \(\mathbb{Q}\) with transcendental parameters \(t_1,\ldots,t_m\), that is, \(K' = \mathbb{Q}(t_1,\ldots,t_m)\). In particular, we present efficient algorithms for computing Gröbner bases over \(K\) and \(K'\). Moreover, we present an efficient method for computing syzygy modules over \(K\).
To compute Gröbner bases over \(K\), starting from the ideas of Noro [35], we proceed by joining \(f\) to the ideal to be considered, adding \(t\) as an extra variable. But instead of avoiding superfluous S-pair reductions by inverting algebraic numbers, we achieve the same goal by applying modular methods as in [2,4,27], that is, by inferring information in characteristic zero from information in characteristic \(p > 0\). For suitable primes \(p\), the minimal polynomial \(f\) is reducible over \(\mathbb{F}_p\). This allows us to apply modular methods once again, on a second level, with respect to the
modular factors of \(f\). The algorithm thus resembles a divide and conquer strategy and
is in particular easily parallelizable. Moreover, using a similar approach, we present an algorithm for computing syzygy modules over \(K\).
On the other hand, to compute Gröbner bases over \(K'\), our new algorithm first specializes the parameters \(t_1,\ldots,t_m\) to reduce the problem from \(K'[x_1,\ldots,x_n]\) to \(\mathbb{Q}[x_1,\ldots,x_n]\). The algorithm then computes a set of Gröbner bases of specialized ideals. From this set of Gröbner bases with coefficients in \(\mathbb{Q}\), it obtains a Gröbner basis of the input ideal using sparse multivariate rational interpolation.
At current state, these algorithms are probabilistic in the sense that, as for other modular Gröbner basis computations, an effective final verification test is only known for homogeneous ideals or for local monomial orderings. The presented timings show that for most examples, our algorithms, which have been implemented in SINGULAR [17], are considerably faster than other known methods.

This thesis is concerned with interest rate modeling by means of the potential approach. The contribution of this work is twofold. First, by making use of the potential approach and the theory of affine Markov processes, we develop a general class of rational models to the term structure of interest rates which we refer to as "the affine rational potential model". These models feature positive interest rates and analytical pricing formulae for zero-coupon bonds, caps, swaptions, and European currency options. We present some concrete models to illustrate the scope of the affine rational potential model and calibrate a model specification to real-world market data. Second, we develop a general family of "multi-curve potential models" for post-crisis interest rates. Our models feature positive stochastic basis spreads, positive term structures, and analytic pricing formulae for interest rate derivatives. This modeling framework is also flexible enough to accommodate negative interest rates and positive basis spreads.

A vehicles fatigue damage is a highly relevant figure in the complete vehicle design process.
Long term observations and statistical experiments help to determine the influence of differnt parts of the vehicle, the driver and the surrounding environment.
This work is focussing on modeling one of the most important influence factors of the environment: road roughness. The quality of the road is highly dependant on several surrounding factors which can be used to create mathematical models.
Such models can be used for the extrapolation of information and an estimation of the environment for statistical studies.
The target quantity we focus on in this work ist the discrete International Roughness Index or discrete IRI. The class of models we use and evaluate is a discriminative classification model called Conditional Random Field.
We develop a suitable model specification and show new variants of stochastic optimizations to train the model efficiently.
The model is also applied to simulated and real world data to show the strengths of our approach.

We investigate the long-term behaviour of diffusions on the non-negative real numbers under killing at some random time. Killing can occur at zero as well as in the interior of the state space. The diffusion follows a stochastic differential equation driven by a Brownian motion. The diffusions we are working with will almost surely be killed. In large parts of this thesis we only assume the drift coefficient to be continuous. Further, we suppose that zero is regular and that infinity is natural. We condition the diffusion on survival up to time t and let t tend to infinity looking for a limiting behaviour.

Advantage of Filtering for Portfolio Optimization in Financial Markets with Partial Information
(2016)

In a financial market we consider three types of investors trading with a finite
time horizon with access to a bank account as well as multliple stocks: the
fully informed investor, the partially informed investor whose only source of
information are the stock prices and an investor who does not use this infor-
mation. The drift is modeled either as following linear Gaussian dynamics
or as being a continuous time Markov chain with finite state space. The
optimization problem is to maximize expected utility of terminal wealth.
The case of partial information is based on the use of filtering techniques.
Conditions to ensure boundedness of the expected value of the filters are
developed, in the Markov case also for positivity. For the Markov modulated
drift, boundedness of the expected value of the filter relates strongly to port-
folio optimization: effects are studied and quantified. The derivation of an
equivalent, less dimensional market is presented next. It is a type of Mutual
Fund Theorem that is shown here.
Gains and losses eminating from the use of filtering are then discussed in
detail for different market parameters: For infrequent trading we find that
both filters need to comply with the boundedness conditions to be an advan-
tage for the investor. Losses are minimal in case the filters are advantageous.
At an increasing number of stocks, again boundedness conditions need to be
met. Losses in this case depend strongly on the added stocks. The relation
of boundedness and portfolio optimization in the Markov model leads here to
increasing losses for the investor if the boundedness condition is to hold for
all numbers of stocks. In the Markov case, the losses for different numbers
of states are negligible in case more states are assumed then were originally
present. Assuming less states leads to high losses. Again for the Markov
model, a simplification of the complex optimal trading strategy for power
utility in the partial information setting is shown to cause only minor losses.
If the market parameters are such that shortselling and borrowing constraints
are in effect, these constraints may lead to big losses depending on how much
effect the constraints have. They can though also be an advantage for the
investor in case the expected value of the filters does not meet the conditions
for boundedness.
All results are implemented and illustrated with the corresponding numerical
findings.