Optimal consumption and investment with bounded downside risk for power utility functions

We investigate optimal consumption and investment problems for a Black-Scholes market under uniform restrictions on Value-at-Risk and Expected Shortfall. We formulate various utility maximization problems, which can be solved explicitly. We compare the optimal solutions in form of optimal value, optimal control and optimal wealth to analogous problems under additional uniform risk bounds. Our proofs are partly based on solutions to Hamilton-Jacobi-Bellman equations, and we prove a corresponding verification theorem. This work was supported by the European Science Foundation through the AMaMeF programme.Comment: 36 page

Projection-based reduced order models for a cut finite element method in parametrized domains

This work presents a reduced order modeling technique built on a high fidelity embedded mesh finite element method. Such methods, and in particular the CutFEM method, are attractive in the generation of projection-based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains and in general to handle topological changes. The combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems, avoiding remeshing as well as the reference domain formulation, often used in the reduced order modeling for boundary fitted finite element formulations. The resulting novel methodology is presented on linear elliptic and Stokes problems, together with several test cases to assess its capability. The role of a proper extension and transport of embedded solutions to a common background is analyzed in detail. \ua9 2019 Elsevier Lt

Influence of Noise on Force Measurements

We demonstrate how the ineluctable presence of thermal noise alters the measurement of forces acting on microscopic and nanoscopic objects. We quantify this effect exemplarily for a Brownian particle near a wall subjected to gravitational and electrostatic forces. Our results demonstrate that the force measurement process is prone to artifacts if the noise is not correctly taken into account.Comment: 4 Pages, 4 Figures, Accepte

Cut finite element error estimates for a class of nonliner elliptic PDEs

In the contexts of fluid–structure interaction and reduced order modeling for parametrically–dependent domains, immersed and embedded methods compare favorably to standard FEMs, providing simple and efficient schemes for the numerical approximation of PDEs in both cases of static and evolving geometries. In this note, the a priori analysis of unfitted numerical schemes with cut elements is extended beyond the realm of linear problems. More precisely, we consider the discretization of semilinear elliptic boundary value problems of the form −∆u + f1(u) = f2 with polynomial nonlinearity via the cut finite element method. Boundary conditions are enforced, using a Nitsche–type approach. To ensure stability and error estimates that are independent of the position of the boundary with respect to the mesh, the formulations are augmented with additional boundary zone ghost penalty terms. These terms act on the jumps of the normal gradients at faces associated with cut elements. A–priori error estimates are derived, while numerical examples illustrate the implementation of the method and validate the theoretical findings

Mutual Fund Theorem for continuous time markets with random coefficients

We study the optimal investment problem for a continuous time incomplete market model such that the risk-free rate, the appreciation rates and the volatility of the stocks are all random; they are assumed to be independent from the driving Brownian motion, and they are supposed to be currently observable. It is shown that some weakened version of Mutual Fund Theorem holds for this market for general class of utilities; more precisely, it is shown that the supremum of expected utilities can be achieved on a sequence of strategies with a certain distribution of risky assets that does not depend on risk preferences described by different utilities.Comment: 17 page