Suchergebnisse

"This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the Martingale problem and therefore the existence of the associated Markov process. Charles Epstein and Rafe Mazzeo use an "integral kernel method" to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. Epstein and Mazzeo establish the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hèolder spaces. They show that the semigroups defined by these operators have holomorphic extensions to the right half-plane. Epstein and Mazzeo also demonstrate precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations."--

The final publication is available at Springer via http://dx.doi.org/10.1007/s00224-017-9803-8 ; The state complexity of a regular language is the number of states in a minimal deterministic finite automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worst-case syntactic complexity taken as a function of the state complexity n of languages in that class. We prove that nn−1, nn−1 + n − 1, and nn−2 + (n − 2)2n−2 + 1 are tight upper bounds on the syntactic complexities of right ideals and prefix-closed languages, left ideals and suffix-closed languages, and two-sided ideals and factor-closed languages, respectively. Moreover, we show that the transition semigroups meeting the upper bounds for all three types of ideals are unique, and the numbers of generators (4, 5, and 6, respectively) cannot be reduced. ; Natural Sciences and Engineering Research Council of Canada (NSERC) grant No. OGP000087 ; National Science Centre, Poland project number 2013/09/N/ST6/01194 ; NSERC Postgraduate Scholarship and a Graduate Award from the Department of Computer Science ; co-financed by the European Union under the European Social Fund's project "International computer science and applied mathematics for business study programme at the University of Wrocław

1. Introduction. 1.1. Questions about model uncertainty. 1.2. Ten papers about model uncertainty -- 2. Discounted linear exponential quadratic Gaussian control. 2.1. Cost formulation. 2.2. Cost recursions and aggregator functions. 2.3. Infinite horizon costs. 2.4. Arbitrary time-invariant linear control laws. 2.5. Solution to the infinite horizon discounted problem. 2.6. Summary -- 3. Robust permanent income and pricing / Thomas D. Tallarini -- 4. A quartet of semigroups for model specification, robustness, prices of risk, and model detection / Evan W. Anderson -- 5. Robust control and model uncertainty. 5.1. Introduction. 5.2. A benchmark resource allocation problem. 5.3. Model misspecification. 5.4. Two robust control problems. 5.5. Recursivity of the multiplier formulation. 5.6. Two preference orderings. 5.7. Recursivity of the preference orderings. 5.8. Concluding remarks -- 6. Robust control and model misspecification / Gauhar A. Turmuhambetova and Noah Williams -- 7. Doubts or variability? / Francisco Barillas -- 8. Robust estimation and control without commitment. 8.1. Introduction. 8.2. A control problem without model uncertainty. 8.3. Using Martingales to represent model misspecifications. 8.4. Two pairs of operators. 8.5. Control problems with model uncertainty. 8.6. The [symbol] = [symbol] case. 8.7. Implied worst case model of signal distortion. 8.8. A recursive multiple priors model. 8.9. Risk sensitivity and compound lotteries. 8.10. Another example. 8.11. Concluding remarks -- 9. Fragile beliefs and the price of uncertainty. 9.1. Introduction. 9.2. Stochastic discounting and risks. 9.3. Three information structures. 9.4. Risk prices. 9.5. A full-information perspective on agents' learning. 9.6. Price effects of concerns about robustness. 9.7. Illustrating the mechanism. 9.8. Concluding remarks -- 10. Beliefs, doubts and learning: Valuing macroeconomic risk / Lars Peter Hansen -- 11. Three types of ambiguity. 11.1. Illustrative model. 11.2. No concern about robustness. 11.3. Representing probability distortions. 11.4. The first type of ambiguity. 11.5. Heterogeneous beliefs without robustness. 11.6. The second type of ambiguity. 11.7. The third type of ambiguity. 11.8. Comparisons. 11.9. Numerical example. 11.10. Concluding remarks.

Financial markets have experienced a precipitous increase in complexity over the past decades, posing a significant challenge from a risk management point of view. This complexity motivates the application and development of sophisticated models based on the theory of stochastic processes and in particular stochastic calculus. In this regard, the contribution of this thesis is twofold, namely by extending the class if tractable stochastic processes in form of polynomial processes and polynomial semimartingales and by showing how efficient calibration of local stochastic volatility models is possible by applying machine learning techniques. In the first part - the main part - we extend the class of polynomial processes that has previously been established to include beyond stochastic discontinuity. This extension is motivated by the fact that certain events in financial markets take place at a deterministic time point but without foreseeable outcome. Such events consist e.g. of decisions regarding interest rates of central banks or political elections/votes. Since the outcome has a significant impact on markets, it is therefore desirable to consider stochastic processes, that can reproduce such jumps at previously specified time points. Such an extension has already been introduced in the affine framework. We will show that similar modifications hold true in the polynomial case. In particular, we will show how after this extension, computation of mixed moments in a multivariate setting reduces to solving a measure ordinary differential equation, posing a significant reduction in complexity to the measure partial differential case in the context of Kolmogorow equations. A central role in the theory of time-homogeneous polynomial processes is played by the theory of one parameter matrix semigroups. Hence, we will develop a two parameter version of the matrix semigroup theory under lower regularity then what exists in the literature. This accounts for time-inhomogeneity of the stochastic processes we consider. While in the one parameter case, full regularity follows already from very mild assumptions, we will see that this is not the case anymore in the two parameter case. In the second part of this thesis we investigate a more applied topic, namely the exact calibration of local stochastic volatility models to financial data. We show how this computationally challenging problem can be efficiently solved by applying machine learning techniques in form of deep neural networks. These methods have dramatically surged in the literature. Since this surge was accompanied by the development of highly efficient machine learning libraries, we can exploit this and make use of sophisticated computational tools such as gpu accelerated numerical computation. We will provide a short exposition to the underlying concepts and give numerical examples in form of toy models. We will further show how this high dimensional problem can be made tractable by the application of an auxiliary machine learning method in the context of variance reduction for Monte-Carlo pricing methods.