Dual representations of risk measures on Orlicz spaces
The standard theory of risk measures, developed for bounded positions, asserts that any coherent risk measure with the Fatou property can be represented as the worst expectation over a class of probabilities. In this talk, we will discuss possible extensions of this result when the space of financial positions is taken to be an Orlicz space. We show that the representation fails in general and remains valid if the risk measure possess additional properties (e.g., law-invariance, strong Fatou property).
Probability models for discretization uncertainty with adaptive grid designs for systems of differential equations
When models are defined implicitly by systems of differential equations without a closed form solution, small local errors in finite-dimensional solution approximations can propagate into large deviations from the true underlying state trajectory. Inference for such models relies on a likelihood approximation constructed around a numerical solution, which underestimates posterior uncertainty. This talk will introduce and discuss progress in a new adaptive formalism for modeling and propagating discretization uncertainty through the Bayesian inferential framework, allowing exact inference and uncertainty quantification for discretized differential equation models.