![]() The paper is concluded with explicit results for profit functions of Cobb–Douglas type and CES type. As a remarkable by-product we prove the continuity of the optimal investment boundary. ![]() With positive probability we show that the integral equation for the investment boundary is uniquely satisfied by the unique solution of another equation which is easier to handle. In case the underlying Lévy process hits any point in Such a relation and the Wiener–Hopf factorization allow us to derive an integral equation for the optimal investment boundary. In line with the results recently obtained in a diffusive setting, we show that the optimal boundary is intimately linked to the unique optional solution of an appropriate Bank–El Karoui representation problem. The problem is set as an infinite time-horizon, two-dimensional degenerate singular stochastic control problem. We derive a new equation for the optimal investment boundary of a general irreversible investment problem under exponential Lévy uncertainty. The proposed numerical technique is employed in well-understood problems to assess its efficiency. Hence, instead of solving the problem analytically, we use a collocation technique: the value function is replaced by a truncated series of polynomials with unknown coefficients that, together with the boundary points, are determined by forcing the series to satisfy the boundary conditions and, at fixed points, the integro-differential equation. Due to the form of the Levy measure of a gamma process, determining the solution of this equation and the boundaries is not an easy task. The initial optimal stopping problem is reduced to a free-boundary problem where, at the unknown boundary points separating the stopping and continuation set, the principles of the smooth and/or continuous fit hold and the unknown value function satisfies on the continuation set a linear integro-differential equation. We study the Bayesian problem of sequential testing of two simple hypotheses about the parameter alpha > 0 of a Levy gamma process. This paper was accepted by Baris Ata, stochastic models and simulation. ![]() The solution carries structural properties analogous to those obtained under continuous-time models, and it provides a useful tool for making new discoveries through discrete-time models. In the infinite-horizon setting, with the aid of a set of signal quality indices, the extreme points on the efficient frontier can be linked through a set of difference equations and solved analytically. This framework accommodates different optimality criteria simultaneously. We adopt a relatively new solution framework from the POMDP literature based on the backward construction of the efficient frontier(s) of continuation-value vectors. As a special partially observable Markov decision process (POMDP), this model unifies several types of learning-and-doing problems such as sequential hypothesis testing, dynamic pricing with demand learning, and multiarmed bandits. We study an infinite-horizon discrete-time model with a constant unknown state that may take two possible values. Problems concerning dynamic learning and decision making are difficult to solve analytically.
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