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We introduce a novel framework to account for sensitivity to rewards uncertainty in sequential decision-making problems. While risk-sensitive formulations for Markov decision processes studied so far focus on the distribution of the…
Respecting the laws of thermodynamics is crucial for ensuring that numerical simulations of dynamical systems deliver physically relevant results. In this paper, we construct a structure-preserving and thermodynamically consistent finite…
A novel data-driven method for formal verification is proposed to study complex systems operating in safety-critical domains. The proposed approach is able to formally verify discrete-time stochastic dynamical systems against temporal logic…
This paper presents a new method to approximate the time-dependent convection-diffusion equations using conforming finite element methods, ensuring that the discrete solution respects the physical bounds imposed by the differential…
The porous medium equation (PME) is a typical nonlinear degenerate parabolic equation. We have studied numerical methods for PME by an energetic variational approach in [C. Duan et al, J. Comput. Phys., 385 (2019) 13-32], where the…
We introduce some approximation schemes for linear and fully non-linear diffusion equations of Bellman-Isaacs type. Although they are not monotone one can prove their convergence to the viscosity solution of the problem. Effective…
In the second part of our series we suggest new definitions of credit bond duration and convexity that remain consistent across all levels of credit quality including deeply distressed bonds and introduce additional risk measures that are…
In this paper, we study a system of second order integro-partial differential equations with interconnected obstacles with non-local terms, related to an optimal switching problem with the jump-diffusion model. Getting rid of the…
We show via counterexamples that relative entropy between the solution of a Markovian master equation and the steady state is not a convex function of time. We thus let down a curtain on a possible formulation of a principle of…
Semi-Lagrangian methods have traditionally been developed in the framework of hyperbolic equations, but several extensions of the Semi-Lagrangian approach to diffusion and advection--diffusion problems have been proposed recently. These…
We consider portfolio selection when decisions based on a dynamic risk measure are affected by the use of a moving horizon, and the possible inconsistencies that this creates. By giving a formal treatment of time consistency which is…
Using results from our companion article [arXiv:1112.4824v2] on a Schauder approach to existence of solutions to a degenerate-parabolic partial differential equation, we solve three intertwined problems, motivated by probability theory and…
The aim of this paper is to investigate risk-averse and distributionally robust modeling of Stochastic Optimal Control (SOC) and Markov Decision Process (MDP). We discuss construction of conditional nested risk functionals, a particular…
In this paper, a new approach based on convex analysis is introduced to solve the $H_\infty$ problem for discrete-time nonlinear stochastic systems. A stochastic version of bounded real lemma is proved and the state feedback $H_\infty$…
Optimization under uncertainty and risk is indispensable in many practical situations. Our paper addresses stability of optimization problems using composite risk functionals which are subjected to measure perturbations. Our main focus is…
A classic reachability problem for safety of dynamic systems is to compute the set of initial states from which the state trajectory is guaranteed to stay inside a given constraint set over a given time horizon. In this paper, we leverage…
In this work, we provide conditions for nonlinear monotone semigroups on locally convex vector lattices to give rise to a generalized notion of viscosity solutions to a related nonlinear partial differential equation. The semigroup needs to…
In this work we investigate the long-time behavior, that is the existence and characterization of invariant measures as well as convergence of transition probabilities, for Markov processes obtained as the unique mild solution to stochastic…
We consider a class of sampling-based decomposition methods to solve risk-averse multistage stochastic convex programs. We prove a formula for the computation of the cuts necessary to build the outer linearizations of the recourse…
We formulate a probabilistic Markov property in discrete time under a dynamic risk framework with minimal assumptions. This is useful for recursive solutions to risk-sensitive versions of dynamic optimisation problems such as optimal…