Systemic Values-at-Risk: Computation and Convergence
Montag, 26.5.25, 14:00-15:30, Seminarraum 232
We investigate the convergence properties of sample-average approximations (SAA) for set-valued systemic risk measures. We assume that the systemic risk measure is defined using a general aggregation function with some continuity properties and value-at-risk applied as a monetary risk measure. Our focus is on the theoretical convergence of its SAA under Wijsman and Hausdorff topologies for closed sets. After building the general theory, we provide an in-depth study of an important special case where the aggregation function is defined based on the Eisenberg-Noe network model. In this case, we provide mixed-integer programming formulations for calculating the SAA sets via their weighted-sum and norm-minimizing scalarizations. To demonstrate the applicability of our findings, we conduct a comprehensive sensitivity analysis by generating a financial network based on the preferential attachment model and modeling the economic disruptions via a Pareto distribution.
Rough and path-dependent affine models and their path-valued interpretation
Mittwoch, 28.5.25, 16:00-17:30, Seminarraum 232
We first extend results on affine Volterra processes to the inhomogeneous case. This includes moment bounds of solutions of Volterra equations driven by a Brownian motion with an inhomogeneous kernel and inhomogeneous drift and diffusion coefficients and in the case of affine drift and variance we show how the conditional Fourier-Laplace functional can be represented by a solution of an inhomogeneous Riccati-Volterra integral equation. For a time homogeneous kernel of convolution type we establish existence of a solution to the stochastic inhomogeneous Volterra equation. If in addition the coefficients are affine, we prove that the conditional Fourier-Laplace functional is exponential-affine in the past path. Finally, we apply these results to an inhomogeneous extension of the rough Heston model used in mathematical finance. Secondly we investigate the equivalence between affine coefficients of a path dependent continuous stochastic differential equations and the affine structure of the log Fourier-transform. Applications in mathematical finance include e.g. delayed Heston model. In both cases the corresponding path processes are infinite-dimensional affine Markov processes. This is joint work (partially in progress) with Julia Ackermann (Wuppertal), Boris Günther (Gießen) und Thomas Kruse (Wuppertal).
Probability Theory and Monads: From Category Theory to Lean4 Implementation
Donnerstag, 18.9.25, 10:00-11:00, Seminarraum 232
Monads provide an elegant categorical framework for formalizing probabilistic objects. Using the Giry monad as our theoretical foundation, I demonstrate how probability measures can be structured categorically and then implement this in Lean4 through the binomial distribution. Three equivalent definitions—recursive, convolution-based, and closed formula—illustrate both the power of categorical abstraction and the practical challenges of formal verification. Key technical hurdles include working with extended nonnegative real numbers versus standard reals, managing type conversions, and proving equivalence between different implementations. The talk targets mathematically interested audiences with basic probability knowledge—no prior experience with category theory or Lean required. Through concrete code examples, I'll show how elegant mathematical theory meets the gritty reality of formal proof verification.