Stochastic Analysis

Branched Signature Model ArXiv ID: 2511.00018 “View on arXiv” Authors: Munawar Ali, Qi Feng Abstract In this paper, we introduce the branched signature model, motivated by the branched rough path framework of [“Gubinelli, Journal of Differential Equations, 248(4), 2010”], which generalizes the classical geometric rough path. We establish a universal approximation theorem for the branched signature model and demonstrate that iterative compositions of lower-level signature maps can approximate higher-level signatures. Furthermore, building on the existence of the extension map proposed in [“Hairer-Kelly. Annales de l’Institue Henri Poincaré, Probabilités et Statistiques 51, no. 1 (2015)”], we show how to explicitly construct the extension of the original paths into higher-dimensional spaces via a map $Ψ$, so that the branched signature can be realized as the classical geometric signature of the extended path. This framework not only provides an efficient computational scheme for branched signatures but also opens new avenues for data-driven modeling and applications. ...

Duality and Policy Evaluation in Distributionally Robust Bayesian Diffusion Control ArXiv ID: 2506.19294 “View on arXiv” Authors: Jose Blanchet, Jiayi Cheng, Hao Liu, Yang Liu Abstract We consider a Bayesian diffusion control problem of expected terminal utility maximization. The controller imposes a prior distribution on the unknown drift of an underlying diffusion. The Bayesian optimal control, tracking the posterior distribution of the unknown drift, can be characterized explicitly. However, in practice, the prior will generally be incorrectly specified, and the degree of model misspecification can have a significant impact on policy performance. To mitigate this and reduce overpessimism, we introduce a distributionally robust Bayesian control (DRBC) formulation in which the controller plays a game against an adversary who selects a prior in divergence neighborhood of a baseline prior. The adversarial approach has been studied in economics and efficient algorithms have been proposed in static optimization settings. We develop a strong duality result for our DRBC formulation. Combining these results together with tools from stochastic analysis, we are able to derive a loss that can be efficiently trained (as we demonstrate in our numerical experiments) using a suitable neural network architecture. As a result, we obtain an effective algorithm for computing the DRBC optimal strategy. The methodology for computing the DRBC optimal strategy is greatly simplified, as we show, in the important case in which the adversary chooses a prior from a Kullback-Leibler distributional uncertainty set. ...