FZ: Dynamic Risk- E11: "Beyond Value at Risk: Dynamic Risk Measures and Applications"

Monetary risk measures allow countries and regulating institutions to impose restrictions on financial positions with random future outcome for financial and insurance companies. By now, coherent and convex risk measures that are defined on spaces of random variables are theoretically well understood and are widely used in practice. Dynamic risk measures additionally take into account time aspects and exclude for instance positions which seem to be acceptable for a period of one year but include risky liabilities thereafter. While in continuous time, there is literature on risk measures for random variables (mathematically this is the link to backward stochastic differential equations (BSDE)), the analogue literature on dynamic risk measures depending on stochastic processes is rather limited. We will study the transition from discrete to continuous time, when the risk capital is computed with an increasing frequency over the same period of time. One problem is that the optimal risk capital modeled as a stochastic processes typically has infinite variation which in practice is not realizable. We will therefore focus on the analysis of risk measures in continuous time which penalize the variation of capital requirement strategies. As an illustration of the theory, we will derive continuous time type optimized certainty equivalents (OCE) having a BSDE representation which makes them analytically tractable. We will further investigate on the robust representations for quasi-convex risk measures. In cooperation with Samuel Drapeau, we are working on a new approach for this duality, which is motivated from monetary risk measure representation theory. Our approach allows to study convolutions of quasi-convex functions with the goal of characterizing Pareto optimal allocations for general (quasi-)convex preferences. The theory will be extended to dynamic quasi-convex functions and several applications and examples will be studied.