(2002). Modelling Extremal Events for Insurance and Finance. Journal of the American Statistical Association: Vol. 97, No. 457, pp. 360-360.

Modelling Extremal Events for Insurance and Finance

Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

We introduce the notion of a convex measure of risk, an extension of the concept of a coherent risk measure defined in Artzner et al. (1999), and we prove a corresponding extension of the representation theorem in terms of probability measures on the underlying space of scenarios. As a case study, we consider convex measures of risk defined in terms of a robust notion of bounded shortfall risk. In the context of a financial market model, it turns out that the representation theorem is closely related to the superhedging duality under convex constraints.

/pdf/convex-measures-of-risk-and-trading-constraints-v5q4n25dqw.pdf

Convex measures of risk and trading constraints

We introduce the notion of a convex measure of risk, an extension of the concept of a coherent risk measure defined in Artzner et aL (1999), and we prove a corresponding extension of the representation theorem in terms of probability measures on the underlying space of scenarios. As a case study, we consider convex measures of risk defined in terms of a robust not ion of bounded shortfall risk. In the context of a financial market model, it turns out that the representation theorem is closely related to the superhedging duality under convex constraints.

The relation between coherent risk measures, valuation bounds, and certain classes of portfolio optimization problems is established. One of the key results is that coherent risk measures are essentially equivalent to generalized arbitrage bounds, named “good deal bounds” by Cerny and Hodges (1999). The results are economically general in the sense that they work for any cash stream spaces, be it in dynamic trading settings, one-step models, or deterministic cash streams. They are also mathematically general as they work in (possibly infinite-dimensional) linear spaces.

Coherent risk measures and good-deal bounds

Natural Exponential Families of Leevy Processes.- Definitions and Examples.- First Properties.- Random Time Transformations.- Exponential Families of Markov Processes.- The Envelope Families.- Likelihood Theory.- Linear Stochastic Differential Equations with Time Delay.- Sequential Methods.- The Semimartingale Approach.- Alternative Definitions.

Exponential families of stochastic processes

Strong discrete time approximation of stochastic differential equations with time delay

The stochastic differential equation is a generalization of Lange-vin's equation, which is obtained if b = 0. Necessary and sufficient conditions on a, b and r are given under which a stationary so...

Langevins stochastic differential equation extended by a time-delayed term

https://www.mathematik.hu-berlin.de/~kuechler/preprints/Bilateral_Paper.pdf

Uwe Küchler

Papers

Coherent risk measures and good-deal bounds

Exponential families of stochastic processes

Strong discrete time approximation of stochastic differential equations with time delay

Langevins stochastic differential equation extended by a time-delayed term

Bilateral gamma distributions and processes in financial mathematics