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We study the risk assessment of uncertain cash flows in terms of dynamic convex risk measures for processes as introduced in Cheridito, Delbaen, and Kupper (2006). These risk measures take into account not only the amounts but also the timing of a cash flow. We discuss their robust representation in terms of suitably penalized probability measures on the optional $\sigma$-field. This yields an explicit analysis both of model and discounting ambiguity. We focus on supermartingale criteria for time consistency. In particular we show how ``bubbles'' may appear in the dynamic penalization, and how they cause a breakdown of asymptotic safety of the risk assessment procedure.
Dynamic risk measures
(2010)
This paper gives an overview of the theory of dynamic convex risk measures for random variables in discrete time setting. We summarize robust representation results of conditional convex risk measures, and we characterize various time consistency properties of dynamic risk measures in terms of acceptance sets, penalty functions, and by supermartingale properties of risk processes and penalty functions.
The classical valuation of an uncertain cash flow in discrete time consists in taking the expectation of the sum of the discounted future payoffs under a fixed probability measure, which is assumed to be known. Here we discuss the valuation problem in the context of Knightian uncertainty. Using results from the theory of convex risk measures, but without assuming the existence of a global reference measure, we derive a robust representation of concave valuations with an infinite time horizon, which specifies the interplay between model uncertainty and uncertainty about the time value of money.
Recently, there is a growing trend to offer guarantee products where the investor is allowed to shift her account/investment value between multiple funds. The switching right is granted a finite number per year, i.e. it is American style with multiple exercise possibilities. In consequence, the pricing and the risk management is based on the switching strategy which maximizes the value of the guarantee put option. We analyze the optimal stopping problem in the case of one switching right within different model classes and compare the exact price with the lower price bound implied by the optimal deterministic switching time. We show that, within the class of log-price processes with independent increments, the stopping problem is solved by a deterministic stopping time if (and only if) the price process is in addition continuous. Thus, in a sense, the Black & Scholes model is the only (meaningful) pricing model where the lower price bound gives the exact price. It turns out that even moderate deviations from the Black & Scholes model assumptions give a lower price bound which is really below the exact price. This is illustrated by means of a stylized stochastic volatility model setup.
In this paper we consider the optimal stopping problem for general dynamic monetary utility functionals. Sufficient conditions for the Bellman principle and the existence of optimal stopping times are provided. Particular attention is payed to representations which allow for a numerical treatment in real situations. To this aim, generalizations of standard evaluation methods like policy iteration, dual and consumption based approaches are developed in the context of general dynamic monetary utility functionals. As a result, it turns out that the possibility of a particular generalization depends on specific properties of the utility functional under consideration.
Boolean modeling frameworks have long since proved their worth for capturing and analyzing essential characteristics of complex systems.
Hybrid approaches aim at exploiting the advantages of Boolean formalisms while refining expressiveness. In this paper, we present a formalism that augments Boolean models with stochastic aspects. More specifically, biological reactions effecting a system in a given state are associated
with probabilities, resulting in dynamical behavior represented as a Markov chain. Using this approach, we model and analyze the cytokinin
response network of Arabidopsis thaliana with a focus on clarifying the character of an important feedback mechanism.
Bovine fertility is the subject of extensive research in animal sciences,
especially because fertility of dairy cows has declined during the last
decades. The regulation of estrus is controlled by the complex interplay
of various organs and hormones. Mathematical modeling of the bovine
estrous cycle could help in understanding the dynamics of this complex
biological system. In this paper we present a mechanistic mathematical
model of the bovine estrous cycle that includes the processes of follicle
and corpus luteum development and the key hormones that interact to
control these processes. The model generates successive estrous cycles of
21 days, with three waves of follicle growth per cycle. The model contains
12 differential equations and 54 parameters. Focus in this paper is on
development of the model, but also some simulation results are presented,
showing that a set of equations and parameters is obtained that describes
the system consistent with empirical knowledge. Even though the majority
of the mechanisms that are included in the model are based on relations
that in literature have only been described qualitatively (i.e. stimulation
and inhibition), the output of the model is surprisingly well in line with
empirical data. This model of the bovine estrous cycle could be used
as a basis for more elaborate models with the ability to study effects of
external manipulations and genetic differences.
We consider simple models of financial markets with less and better
informed investors described by a smaller and a larger filtration on a
general stochastic basis that describes the market dynamics, including
continuous and jump components. We study the relation between different forms of non existance of arbitrage and the characteristics of the stochastic basis under the different filtrations. This is achieved through the analysis of the properties of the numéraire portfolio. Furthermore, we focus on the problem of calculating the additional logarithmic utility of the better informed investor in terms of the Shannon antropy of is additional information. The information drift, i.e. the drift to eliminate in order to preserved the martingale property in the larger filtration terms out to be the crucial quantity needed to tackle these problems. We show that the expected
ed logarithmic utility increment due to better information equals its Shannon
entropy also in case of a pure jump basis with jumps that are quadratically
hedgeable, and so extend a similar result known for bases consisting of
continuous semimartingales. An example illustrates that the equality may
not persist if both continuous and jump components are present in the
underlying.
Motivated by the analysis of passive control systems, we undertake a detailed perturbation analysis of Hamiltonian matrices that have eigenvalues on the imaginary axis. We construct minimal Hamiltonian perturbations that move and coalesce eigenvalues of opposite sign characteristic to form multiple eigenvalues with mixed sign characteristics, which are then moved from the imaginary axis to specific locations in the complex plane by small Hamiltonian perturbations. We also present a numerical method to compute upper bounds for the minimal perturbations that move all eigenvalues of a given Hamiltonian matrix outside a vertical strip along the imaginary axis.
In this work we propose a general framework for the structured perturbation
analysis of several classes of structured matrix polynomials in homogeneous
form, including complex symmetric, skew-symmetric, even and odd matrix polynomials. We introduce structured backward errors for approximate eigenvalues and eigenvectors and we construct minimal structured perturbations such that an approximate eigenpair is an exact eigenpair of an appropriately perturbed matrix polynomial. This work extends previous work for the non-homogeneous case (we include infinite eigenvalues) and we show that the structured backward errors improve the known unstructured backward errors.
Mathematical programs in which the constraint set is partially defined by the solutions of an elliptic variational inequality, so-called ``elliptic MPECs'', are formulated in reflexive Banach spaces. With the goal of deriving explicit first order optimality conditions amenable to the development of numerical procedures, variational analytic concepts are both applied and further developed. The paper is split into two main parts. The first part concerns the derivation of conditions in which the state constraints are assumed to be polyhedric sets. This part is then completed by two examples, the latter of which involves pointwise bilateral bounds on the gradient of the state. The second part begins with the derivation of a formula for the second order (Mosco) epiderivative of the indicator function of a general convex set. This result is then used to derive analogous conditions to those which are presented in the first part. Finally, an elliptic MPEC is considered important to the study of elasto-plasticity in which the pointwise Euclidean norm of the gradient of the state is bounded. Explicit strong stationarity conditions are provided for this problem.
This paper is devoted to the numerical approximation of Lyapunov and Sacker-Sell spectral intervals for linear differential-algebraic equations (DAEs). The spectral analysis for DAEs is improved and the concepts of leading directions and solution subspaces associated with spectral intervals are extended to DAEs. Numerical methods
based on smooth singular value decompositions are introduced for computing all or only some spectral intervals and their associated leading directions. The numerical algorithms as well as implementation issues are discussed in detail and numerical examples are presented to illustrate the theoretical results.
This paper proposes a new mathematical model for the open pit mine planning problem,
based on continuous functional analysis. The traditional models for this problem have been
constructed by using discrete 0-1 decision variables, giving rise to large-scale combinatorial
and Mixed Integer Programming (MIP) problems. Instead, we use a continuous approach
which allows for a refined imposition of slope constraints associated with geotechnical stability.
The model introduced here is posed in a suitable functional space, essentially the
real-valued functions that are Lipschitz continuous on a given two dimensional bounded region.
We derive existence results and investigate some qualitative properties of the solutions
Cross–derivatives are mixed partial derivatives that are obtained by differentiating at most
once in every coordinate direction. They are a computational tool in combinatorics and high–
dimensional integration. Here we present two methods of computing exact values of all cross–
derivatives at a given point both following the general philosophy of automatic differentiation.
Implementation details are discussed and numerical results given.
We consider the behavior of a modulated wave solution to
an $\mathbb{S}^1$-equivariant autonomous system of differential equations under an external
forcing of modulated wave type. The modulation frequency of the forcing is assumed to be close to the
modulation frequency of the modulated wave solution, while the wave frequency of the forcing is supposed to be far
from that of the modulated wave solution. We describe the domain in the three-dimensional
control parameter space (of frequencies and amplitude of the forcing)
where stable locking of the modulation frequencies of the forcing and the modulated wave solution
occurs.
Our system is a simplest case scenario for the behavior of self-pulsating lasers under the influence of external
periodically modulated
optical signals.
We show that the coupled balance equations for a large class of dissipative materials
can be cast in the form of GENERIC (General Equations for Non-Equilibrium
Reversible Irreversible Coupling). In dissipative solids, also called generalized standard
materials, the state of a material point is described by dissipative internal variables in addition to the elastic deformation and the temperature. The framework GENERIC allows
for an efficient derivation of thermodynamically consistent coupled field equations,
while revealing additional underlying physical structures, like the role of the free energy
as the driving potential for reversible effects and the role of the free entropy (Massieu potential) as the driving potential for dissipative effects.
Applications to large and small-strain thermoplasticity is given. Moreover, for the
quasistatic case, where the deformation can be statically eliminated, we derive a generalized
gradient structure for the internal variable and the temperature with a reduced
entropy as driving functional.
In this paper we consider the first exit problem of an overdamped
Lévy driven particle in a confining potential. We survey results
obtained in recent years from our work on the Kramers' times for
dynamical systems of this type with Lévy perturbations containing
heavy, and exponentially light jumps, and compare them to the well
known case of dynamical systems with Gaussian perturbations. It
turns out that exits induced by Lévy processes with jumps are
always essentially faster than Gaussian exits.
In this paper we study BSDEs arising from a special class of backward stochastic partial differential equations (BSPDEs) that is intimately related to utility maximization problems with respect to arbitrary utility functions. After providing existence and uniqueness we discuss the numerical realizability. Then we study utility maximization problems on incomplete financial markets whose dynamics are governed by continuous semimartingales. Adapting standard methods that solve the utility maximization problem using BSDEs, we give solutions for the portfolio optimization problem which involve the delivery of a liability at maturity. We illustrate our study by numerical simulations for selected examples. As a byproduct we prove existence of a solution to a very particular quadratic growth BSDE with unbounded terminal condition. This complements results on this topic obtained in [6,7,8].