LMU Christmas Workshop in Stochastics and Finance

This paper proposes a simple descriptive model of discrete-time double auction markets for divisible assets. As in the classical models of exchange economies, we consider afinite set of agents described by their initial endowments and preferences. Instead of the classical Walrasian-type market models, however, we assume that all trades take place in a centralized double auction where the agents communicate through sealed limit orders for buying and selling. We find that, under nonstrategic bidding, double auction clears with zero trades precisely when the agents’current holdings are on the Pareto frontier. More interestingly, the double auctions implement Adam Smith’s "invisible hand" in the sense that, when starting from disequilibrium, repeated double auctions lead to a sequence of allocations that converges to individually rational Pareto allocations.

The talk features nested, and conditional risk measures. We derive their properties, and investigate, if they are suitable for dynamic optimization. In analogy to the generator in stochastic differential equations, we define a risk-averse generator. The generator is a non-linear differential operator, the non-linear term accounting for risk-aversion. It turns out that the generator is independent of the risk measure, which is employed to construct the nested analogue. The results generalize classical models in financial and actuarial sciences by involving a new, risk-averse aspect.

Machine learning typically presupposes classical probability theory which implies that aggregation is built upon expectation. There are now multiple reasons to motivate looking at richer alternatives to classical probability theory as a mathematical foundation for machine learning. We systematically examine a powerful and rich class of such alternatives, known variously as spectral risk measures, Choquet integrals or Lorentz norms. We present a range of characterization results, and demonstrate what makes this spectral family so special. In doing so we demonstrate a natural stratification of all coherent risk measures in terms of the upper probabilities that they induce by exploiting results from the theory of rearrangement invariant Banach spaces. We empirically demonstrate how this new approach to uncertainty helps tackling practical machine learning problems.

We develop a novel procedure for estimating the optimizer of general convex stochastic optimization problems from an iid sample. This procedure is the first one that exhibits the optimal statistical performance in heavy tailed situations and also applies in highdimensional settings. We discuss the portfolio optimization problem and if time permits, the estimation of risk measures. Based on joint works with Stephan Eckstein, Felix Liebrich, and Shahar Mendelson.