5 research outputs found
Geometric Modular Action, Wedge Duality and Lorentz Covariance are Equivalent for Generalized Free Fields
The Tomita-Takesaki modular groups and conjugations for the observable
algebras of space-like wedges and the vacuum state are computed for
translationally covariant, but possibly not Lorentz covariant, generalized free
quantum fields in arbitrary space-time dimension d. It is shown that for the condition of geometric modular action (CGMA) of Buchholz, Dreyer, Florig
and Summers \cite{BDFS}, Lorentz covariance and wedge duality are all
equivalent in these models. The same holds for d=3 if there is a mass gap. For
massless fields in d=3, and for d=2 and arbitrary mass, CGMA does not imply
Lorentz covariance of the field itself, but only of the maximal local net
generated by the field
Asymptotic ruin probabilities and optimal investment
We study the infinite time ruin probability for an insurance company in the classical Cramér-Lundberg model with finite exponential moments. The additional non-classical feature is that the company is also allowed to invest in some stock market, modeled by geometric Brownian motion. We obtain an exact analogue of the classical estimate for the ruin probability without investment, i.e., an exponential inequality. The exponent is larger than the one obtained without investment, the classical Lundberg adjustment coefficient, and thus one gets a sharper bound on the ruin probability. A surprising result is that the trading strategy yielding the optimal asymptotic decay of the ruin probability simply consists in holding a fixed quantity (which can be explicitly calculated) in the risky asset, independent of the current reserve. This result is in apparent contradiction to the common believe that 'rich' companies should invest more in risky assets than 'poor' ones. The reason for this seemingly paradoxical result is that the minimization of the ruin probability is an extremely conservative optimization criterion, especially for 'rich' companies. (author's abstract)Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science