Dynamic risk measures

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.Comment: 30 page

Characterization of max-continuous local martingales vanishing at infinity

We provide a characterization of the family of non-negative local martingales that have continuous running supremum and vanish at infinity. This is done by describing the class of random times that identify the times of maximum of such processes. In this way we extend to the case of general filtrations a result proved by Nikeghbali and Yor [NY06] for continuous filtrations. Our generalization is complementary to the one presented by Kardaras [Kar14], and is obtained by means of similar tools

Modeling and pricing cyber insurance: Idiosyncratic, systematic, and systemic risks

The paper provides a comprehensive overview of modeling and pricing cyber insurance and includes clear and easily understandable explanations of the underlying mathematical concepts. We distinguish three main types of cyber risks: idiosyncratic, systematic, and systemic cyber risks. While for idiosyncratic and systematic cyber risks, classical actuarial and financial mathematics appear to be well-suited, systemic cyber risks require more sophisticated approaches that capture both network and strategic interactions. In the context of pricing cyber insurance policies, issues of interdependence arise for both systematic and systemic cyber risks; classical actuarial valuation needs to be extended to include more complex methods, such as concepts of risk-neutral valuation and (set-valued) monetary risk measures

Introduction to special section: Outstanding problems in quantifying the radiative impact of mineral dust

International audienceThis paper provides an introduction to the special section of the Journal of Geophysical Research on mineral dust. We briefly review the current experimental and theoretical approaches used to quantify the dust radiative impacts, highlight the outstanding issues, and discuss possible strategies to overcome the emerging problems. We also introduce the contributing papers of this special section. Despite the recent notable advances in dust studies, we demonstrate that the radiative effects of dust remain poorly quantified due to both limited data and incomplete understanding of relative physical and chemical processes. The foremost needs are (1) to quantify the spatial and temporal variations of dust burden in the atmosphere and develop a predictive capability for the size‐ and composition‐resolved dust particle distribution; (2) to develop a quantitative description of the processes that control the spatial and temporal variabilities of dust physical and chemical properties and radiative effects; (3) to develop new instrumentation (especially to measure the dust particle size distribution in a wide range from about 0.01 μm to 100 μm, scattering phase function and light absorption by dust particles); and (4) to develop new techniques for interpreting and merging the diverse information from satellite remote sensing, in situ and ground‐based measurements, laboratory studies, and model simulations. Because dust distribution and effects are heterogeneous, both spatially and temporally, a promising strategy to advance our knowledge is to perform comprehensive studies at the targeted regions affected by mineral dust of both natural and anthropogenic origin

Dynamic convex risk measures

In dieser Arbeit werden verschiedene Eigenschaften von dynamischen konvexen Risikomaßen für beschränkte Zufallsvariablen untersucht. Dabei gehen wir vor allem der Frage nach, wie die Risikobewertungen in verschiedenen Zeitpunkten von einander abhängen, und wie sich solche Zeitkonsistenzeigenschaften in der Dynamik der Penalty-Funktionen und Risikoprozesse widerspiegeln. Im Kapitel 2 widmen wir uns zunächst der starken Zeitkonsistenz und charakterisieren diese mithilfe von Akzeptanzmengen, Penalty-Funktionen und einer gemeinsamen Supermartingaleigenschaft des Risikoprozesses und seiner Penalty-Funktion. Die Charakterisierung durch Penalty-Funktionen liefert eine explizite Form der Doob- und der Riesz-Zerlegung des Prozesses der Penalty-Funktionen. Anschließend führen wir einen schwächeren Begriff der Zeitkonsistenz ein, den wir Besonnenheit nennen. In Analogie zu dem zeitkonsistenten Fall charakterisieren wir Besonnenheit durch Akzeptanzmengen, Penalty-Funktionen und eine bestimmte Supermartingaleigenschaft. Diese Supermartingaleigenschaft gilt allgemeiner für alle beschränkten adaptierten Prozesse, die sich ohne zusätzliches Risiko aufrechterhalten lassen. Wir nennen solche Prozesse nachhaltig und beschreiben Nachhaltigkeit durch eine gemeinsame Supermartingaleigenschaft des Prozesses und der schrittweisen Penalty-Funktionen. Dieses Resultat kann als eine verallgemeinerte optionale Zerlegung unter konvexen Restriktionen gesehen werden. Mithilfe der Supermartingaleigenschaft identifizieren wir das stark zeitkonsistente dynamische Risikomaß, das aus jedem beliebigen Risikomaß rekursiv konstruiert werden kann, als den kleinsten Prozeß, der nachhaltig ist und den Endverlust minimiert. Diese Beschreibung liefert ein neues Argument für den Einsatz von zeitkonsistenten Risikomaßen. Im Kapitel 3 diskutieren wir das asymptotische Verhalten von zeitkonsistenten und von besonnenen Risikomaßen hinsichtlich der asymptotischen Sicherheit und der asymptotischen Präzision. Im Kapitel 4 werden die allgemeinen Ergebnisse aus den Kapiteln 2 und 3 anhand des entropischen Risikomaßes und des Superhedging-Preisprozesses veranschaulicht.In this thesis we study various properties of a dynamic convex risk measure for bounded random variables. The main subject is to investigate possible interdependence of conditional risk assessments at different times and the manifestation of these time consistency properties in the dynamics of corresponding penalty functions and risk processes. In Chapter 2 we focus first on the strong notion of time consistency and characterize it in terms of penalty functions, acceptance sets and a joint supermartingale property of the risk measure and its penalty function. The characterization in terms of penalty functions provides the explicit form of the Doob and of the Riesz decomposition of the penalty function process for a time consistent risk measure. Then we introduce and study a weaker notion of time consistency, that we call prudence. Similar to the time consistent case, we characterize prudent dynamic risk measures in terms of acceptance sets, of penalty functions and by a certain supermartingale property. This supermartingale property holds more generally for any bounded adapted process that can be upheld without any additional risk. We call such processes sustainable, and we give an equivalent characterization of sustainability in terms of a combined supermartingale property of a process and one-step penalty functions. This result can be viewed as a generalized optimal decomposition under convex constraints. The supermartingale property allows us to characterize the strongly time consistent risk measure arising from any dynamic risk measure via recursive construction as the smallest process that is sustainable and covers the final loss. Thus our discussion provides a new reason for using strongly time consistent risk measures. In Chapter 3 we discuss the limit behavior of time consistent and of prudent risk measures in terms of asymptotic safety and of asymptotic precision. In the final Chapter 4 we illustrate the general results of Chapter 2 and Chapter 3 by examples. In particular we study the entropic dynamic risk measure and the superhedging price process under convex constraints

Consistent Risk Measures and a non-linear Extension of Backwards Martingale Convergence.

We study the behavior of conditional risk measures along decreasing σ-fields. Under a condition of consistency, we prove a non-linear extension of backwards martingale convergence. In particular we show the existence of a limiting conditional risk measure with respect to the tail field, we describe its dual representation in terms of a limiting penalty function, and we show that consistency extends to the tail field. Moreover, we clarify the structure of global risk measures which are consistent with the given sequence of conditional risk measures

Hedging of claims with physical delivery under convex transaction costs

We study superhedging of contingent claims with physical delivery in a discrete-time market model with convex transaction costs. Our model extends Kabanov's currency market model by allowing for nonlinear illiquidity effects. We show that an appropriate generalization of Schachermayer's robust no arbitrage condition implies that the set of claims hedgeable with zero cost is closed in probability. Combined with classical techniques of convex analysis, the closedness yields a dual characterization of premium processes that are sufficient to superhedge a given claim process. We also extend the fundamental theorem of asset pricing for general conical models.

Polycyclic Aromatic Hydrocarbons in the Atmosphere of the Southern Baikal Region (Russia): Sources and Relationship with Meteorological Conditions

This article presents the results of the long-term studies at two stations located in the city of Irkutsk and the Listvyanka settlement of the southern Baikal region (East Siberia) concerning the concentration of polycyclic aromatic hydrocarbons (PAHs) in atmospheric aerosol. The studies revealed the seasonal and interannual dynamics in the distribution of PAHs in aerosols from urban (source) and rural (receptor) areas. We carried out a comprehensive analysis of weather conditions such as wind direction, relative humidity, air temperature, and atmospheric pressure. The analysis determined high correlations between air temperature, atmospheric pressure, temperature inversions, and PAHs at the monitoring stations. The average annual concentrations of PAHs in the abnormally warm 2020 were three times lower than the average values obtained in the cold 2016. The toxic equivalent concentrations (BaPeq) increased from summer to winter with an increase in the contribution from benzo(a)pyrene, one of the most toxic and hazardous compounds of this class of organic substances. Four-, five- and six-ring PAHs mainly predominated in aerosol; the proportion of two- and three-ring PAHs increased from the warm season to the cold season. Diagnostic ratios of PAHs identified the main sources of air pollution by this class of compounds: combustion of coal, liquid fuel and firewood, vehicle emissions, and wildfires. The percentage of the transport of anthropogenic aerosol containing PAHs from industrial sources of the Southern Baikal region towards Lake Baikal was 65 to 71%