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Lévy-Type Processes Under Uncertainty and Related Nonlocal Equations

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Lévy-Type Processes Under Uncertainty and Related Nonlocal Equations Technische Universitat¨ Dresden Institute of Mathematical Stochastics Dissertation Levy-Type´ Processes under Uncertainty and Related Nonlocal Equations Author: Julian Hollender Birthdate: 27th October 1987 Birthplace: Dresden, Germany Degree Aimed At: Doctor Rerum Naturalium (Dr. rer. nat.) Institution: Technische Universitat¨ Dresden Faculty: School of Science Supervisor: Prof. Dr. Rene´ L. Schilling Referees: Prof. Dr. Rene´ L. Schilling Prof. Dr. Jiang-Lun Wu Submission: 13th July 2016 Defence: 12th October 2016 We should not write so that it is possible for our readers to understand us, but so that it is impossible for them to misunderstand us. Marcus Fabius Quintilian, Institutio Oratoria Contents Abstract 7 Introduction 9 1. Approximation Theory 19 1.1. Friedrichs Mollification . 19 1.2. Supremal Convolutions . 25 1.3. Monotone Functions . 44 1.4. Miscellaneous . 52 2. Integro-Differential Equations 57 2.1. Viscosity Solutions . 58 2.2. Uniqueness of Solutions . 70 2.3. Hamilton{Jacobi{Bellman Equations . 113 3. Sublinear Expectations 135 3.1. Expectation Spaces . 135 3.2. Important Distributions . 141 4. Stochastic Processes 147 4.1. Markov Processes . 149 4.2. Construction of Processes . 155 4.3. Stochastic Integration . 196 4.4. Stochastic Differential Equations . 206 A. Appendix 219 A.1. Convex Analysis . 219 A.2. Functional Analysis . 221 Outlook 225 Bibliography 227 List of Symbols 237 List of Statements 244 Index of Subjects 245 Abstract In recent years, the theoretical study of nonlinear expectations became the focus of attention for applications in a variety of different fields { often with the objective to model systems under incomplete information. Especially in mathematical finance, advancements in the theory of sublinear expectations (also known as coherent risk measures) laid the theoretical foundation for many improved or novel approaches to evaluations under the presence of Knightian uncertainty. In his innovative work [102], Shige Peng studied sublinear expectations as an intrinsic generalization of classical probability spaces: One of his main contributions in this article is the introduction of stochastic processes with continuous paths and independent, stationary increments for sublinear expectations, which have a one-to-one correspondence to nonlinear local equations 1 2 @tu(t; x) − sup tr cαD u(t; x) = 0 α2A 2 d×d for families (cα)α2A ⊂ S+ of positive, symmetric matrices with arbitrary index sets A. These so-called G-Brownian motions can be interpreted as a generalization of classical Brownian motions (and their relation to the heat equation) under volatility uncertainty, and are therefore of particular interest for applications in mathematical finance. Shige Peng's approach was later extended by Mingshang Hu and Shige Peng in [67] and by Ariel Neufeld and Marcel Nutz in [93] to introduce stochastic jump-type processes with independent, stationary increments for sublinear expectations, which correspond to nonlinear nonlocal equations 1 2 @tu(t; x) − sup bαDu(t; x) + tr cαD u(t; x) α2A 2 Z + u(t; x + z) − u(t; x) − Du(t; x)z1jz|≤1 Fα(dz) = 0 Rd d d×d d for families (bα; cα;Fα)α2A ⊂ R × S+ × L(R ) of L´evytriplets, and can be viewed as a generalization of classical L´evyprocesses under uncertainty in their L´evytriplets. In the latter half of this thesis, we pick up these ideas to introduce a broad class of stochastic processes with jumps for sublinear expectations, that can also exhibit spatial inhomogeneities. More precisely, we introduce the concept of Markov processes for sublinear expectations and relate them to nonlinear nonlocal equations @tu(t; x) − G u(t; · ) (x) = 0 for some sublinear generalization G(·) of classical (pointwise) Markov generators. 7 Abstract In addition, we give a construction for a large subclass of these Markov processes for sublinear expectations, that corresponds to equations of the special form @tu(t; x) − sup Gα u(t; · ) (x) = 0 α2A for families (Gα)α2A of L´evy-type generators. These processes can be interpreted as generalizations of classical L´evy-type processes under uncertainty in their semimartingale characteristics. Furthermore, we establish a general stochastic integration and differential equation theory for processes with jumps under sublinear expectations. In the first half of this thesis, we give a purely analytical proof, showing that a large family of fully nonlinear initial value problems (including the ones from above) d @tu(t; x) − G u (t; x) = 0 in (0;T ) × R d u(t; x) = '(x) on f0g × R have at most one viscosity solution, where G : C2((0;T ) × Rd) ! C((0;T ) × Rd) are second-order, nonlocal degenerate elliptic operators { i.e. they satisfy Gφ(t; x) ≤ G (t; x) whenever φ − has a global maximum in (t; x) 2 (0;T ) × Rd with (φ − )(t; x) ≥ 0. The notion of viscosity solutions, which we are using, was originally introduced by Michael Crandall and Pierre-Louis Lions in [36] for local equations. We prove a comparison principle for unbounded viscosity solutions of parabolic, second-order, nonlocal degenerate elliptic equations. In order to cover the equations needed for the stochastic part, we have to generalize the maximum principle from Espen Jakobsen and Kenneth Karlsen in [75], since our operators have significantly weaker continuity properties. [36] Crandall, M., Lions, P. Viscosity solutions of Hamilton-Jacobi equations. Transactions of the American Mathematical Society 277, 1 (1983), 1{42. [67] Hu, M., Peng, S. G-L´evy processes under sublinear expectations. arXiv:0911.3533v1, 2009. [75] Jakobsen, E., Karlsen, K. A \maximum principle for semicontinuous functions" applicable to integro-partial differential equations. Nonlinear Differential Equations and Applications 13, 2 (2006), 137{165. [93] Neufeld, A., Nutz, M. Nonlinear L´evyprocesses and their characteristics. arXiv:1401.7253v2, 2015. [102] Peng, S. G-expectation, G-Brownian motion and related stochastic calculus of It^o type. In Stochastic Analysis and Applications - The Abel Symposium 2005 (2007), vol. 2, Springer, 541{567. 8 Introduction Motivation In his pioneering work [82], the economist Frank Knight postulated a distinction between two inherently different types of threats in the evaluation of unknown parameters: On the one hand, there are parameters under so-called Knightian risk (or risk for short), for which one can reasonably assign probabilities to the possible values of the unknown parameter, despite the lack of knowledge of its true value. Typical examples include the tossing of fair coins or the rolling of perfect dice. On the other hand, there are parameters under so-called Knightian uncertainty (or uncertainty for short), for which it seems not feasible to assign distinct probabilities to each observable value of the unknown parameter, often caused by the absence of relevant information or due to a lack of full understanding of the underlying dependencies. Typical examples include the pricing of contingent claims and their underpinning financial modeling of stock prices. Experiment 1 Experiment 2 Color Option 1A Option 1B Option 2A Option 2B Ratio Red $100 - $100 - 33% Black - $100 - $100 66% Yellow - - $100 $100 Figure 0.1.: Payoff matrix for Ellsberg's experiments In order to clarify the difference between Knightian risk and uncertainty, and to motivate its relevance for the related mathematical modeling, let us take a look at the following illustrative example: Daniel Ellsberg popularized in [49] the evaluation of two experiments in the context of decision theory, which were originally proposed by John Maynard Keynes in [81]. In both experiments a ball is drawn from an urn that contains a mix of nine red, black and yellow balls. The participants are asked to choose between two different options for each experiment in advance, which determine the payoffs according to the color of the ball drawn, as described in Figure 0.1. Before the participants pick their options, however, they are provided with the information that the urn contains three red and six other balls, which are either black or yellow. As a result of this, the unknown payoff parameter is under Knightian risk for options 1A and 2B, since it only depends on the event that a red ball is drawn. In fact, it seems reasonable to assume that the chances for a red ball to be drawn are exactly one third. For options 1B and 2A, on the other hand, the unknown payoff parameter is under Knightian uncertainty, since the ratio between the black and yellow balls is unknown and it is therefore not 9 Introduction feasible to assign probabilities to the different payoffs. Empirical data suggests (cf. [23, Section 3.1] for a literature review) that the majority of participants prefer the options under Knightian risk over those under Knightian uncertainty { an effect that is often attributed to the ambiguity aversion of the participants. Decision theory, as a subfield of economics, is concerned with models that describe certain characteristics in the behavior of rational agents, such as the ambiguity aversion of participants in Ellsberg's two experiments. One idea at the heart of decision theory is the expected utility hypothesis, which asserts that rational agents prefer one option over another if they assign a higher expected utility to it. Suppose that u : R −! R is a function (usually referred to as utility function), which assigns the utility (for rational agents) to possible payoff values. The preference of option 1A over 1B by the majority of participants in Ellsberg's first experiment translates to E (u(π1A)) > E (u(π1B)) under
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