Robust risk aggregation with neural networks

We consider settings in which the distribution of a multivariate random variable is partly ambiguous. We assume the ambiguity lies on the level of the dependence structure, and that the marginal distributions are known. Furthermore, a current best guess for the distribution, called reference measure, is available. We work with the set of distributions that are both close to the given reference measure in a transportation distance (e.g. the Wasserstein distance), and additionally have the correct marginal structure. The goal is to find upper and lower bounds for integrals of interest with respect to distributions in this set. The described problem appears naturally in the context of risk aggregation. When aggregating different risks, the marginal distributions of these risks are known and the task is to quantify their joint effect on a given system. This is typically done by applying a meaningful risk measure to the sum of the individual risks. For this purpose, the stochastic interdependencies between the risks need to be specified. In practice the models of this dependence structure are however subject to relatively high model ambiguity. The contribution of this paper is twofold: Firstly, we derive a dual representation of the considered problem and prove that strong duality holds. Secondly, we propose a generally applicable and computationally feasible method, which relies on neural networks, in order to numerically solve the derived dual problem. The latter method is tested on a number of toy examples, before it is finally applied to perform robust risk aggregation in a real world instance.Comment: Revised version. Accepted for publication in "Mathematical Finance

Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn's Algorithm

We study the stability of entropically regularized optimal transport with respect to the marginals. Lipschitz continuity of the value and H\"older continuity of the optimal coupling in $p$-Wasserstein distance are obtained under general conditions including quadratic costs and unbounded marginals. The results for the value extend to regularization by an arbitrary divergence. As an application, we show convergence of Sinkhorn's algorithm in Wasserstein sense, including for quadratic cost. Two techniques are presented: The first compares an optimal coupling with its so-called shadow, a coupling induced on other marginals by an explicit construction. The second transforms one set of marginals by a change of coordinates and thus reduces the comparison of differing marginals to the comparison of differing cost functions under the same marginals.Comment: Forthcoming in 'SIAM Journal on Mathematical Analysis

Estimating the Rate-Distortion Function by Wasserstein Gradient Descent

In the theory of lossy compression, the rate-distortion (R-D) function $R(D)$ describes how much a data source can be compressed (in bit-rate) at any given level of fidelity (distortion). Obtaining $R(D)$ for a given data source establishes the fundamental performance limit for all compression algorithms. We propose a new method to estimate $R(D)$ from the perspective of optimal transport. Unlike the classic Blahut--Arimoto algorithm which fixes the support of the reproduction distribution in advance, our Wasserstein gradient descent algorithm learns the support of the optimal reproduction distribution by moving particles. We prove its local convergence and analyze the sample complexity of our R-D estimator based on a connection to entropic optimal transport. Experimentally, we obtain comparable or tighter bounds than state-of-the-art neural network methods on low-rate sources while requiring considerably less tuning and computation effort. We also highlight a connection to maximum-likelihood deconvolution and introduce a new class of sources that can be used as test cases with known solutions to the R-D problem.Comment: Accepted as conference paper at NeurIPS 202

Robust pricing and hedging of options on multiple assets and its numerics

We consider robust pricing and hedging for options written on multiple assets given market option prices for the individual assets. The resulting problem is called the multi-marginal martingale optimal transport problem. We propose two numerical methods to solve such problems: using discretisation and linear programming applied to the primal side and using penalisation and deep neural networks optimisation applied to the dual side. We prove convergence for our methods and compare their numerical performance. We show how adding further information about call option prices at additional maturities can be incorporated and narrows down the no-arbitrage pricing bounds. Finally, we obtain structural results for the case of the payoff given by a weighted sum of covariances between the assets.Comment: Forthcoming in SIAM Journal on Financial Mathematic

Real-time motion and retrospective coil sensitivity correction for CEST using volumetric navigators (vNavs) at 7T

Purpose To explore the impact of temporal motion-induced coil sensitivity changes on CEST-MRI at 7T and its correction using interleaved volumetric EPI navigators, which are applied for real-time motion correction. Methods Five healthy volunteers were scanned via CEST. A 4-fold correction pipeline allowed the mitigation of (1) motion, (2) motion-induced coil sensitivity variations, Delta B1-, (3) motion-induced static magnetic field inhomogeneities, Delta B-0, and (4) spatially varying transmit RF field fluctuations, Delta B1+. Four CEST measurements were performed per session. For the first 2, motion correction was turned OFF and then ON in absence of voluntary motion, whereas in the other 2 controlled head rotations were performed. During post-processing Delta B1- was removed additionally for the motion-corrected cases, resulting in a total of 6 scenarios to be compared. In all cases, retrospective increment B-0 and -Delta B1+ corrections were performed to compute artifact-free magnetization transfer ratio maps with asymmetric analysis (MTRasym). Results Dynamic Delta B1- correction successfully mitigated signal deviations caused by head motion. In 2 frontal lobe regions of volunteer 4, induced relative signal errors of 10.9% and 3.9% were reduced to 1.1% and 1.0% after correction. In the right frontal lobe, the motion-corrected MTRasym contrast deviated 0.92%, 1.21%, and 2.97% relative to the static case for Delta omega = 1, 2, 3 +/- 0.25 ppm. The additional application of Delta B1- correction reduced these deviations to 0.10%, 0.14%, and 0.42%. The fully corrected MTRasym values were highly consistent between measurements with and without intended head rotations. Conclusion Temporal Delta B1- cause significant CEST quantification bias. The presented correction pipeline including the proposed retrospective Delta B1- correction significantly reduced motion-related artifacts on CEST-MRI.Peer reviewe