Construction of a shadow price for discrete infinite horizon discounted functionals

Life-cycle assessments (LCAs) were conducted to evaluate sculptured cement mortar tiles, proposed by Hershcovich et al. (2021), and conventional cement mortar flat tiles for thermal insulation of a typical residential building in Mediterranean climate. The production (P) and operational energy (OE) stages were compared between the sculptured tiles and the conventional flat tiles. The P stage used Portland cement with 95% clinker (CEM I) and Portland limestone cement with 65% clinker (CEM II). The OE stage used 31% coal, 56% natural gas, and 13% photovoltaic (PV) (adopted in 2020) and 8% coal, 57% natural gas, and 35% PV (planned for 2025). The ReCiPe2016 single-score method was used to assess environmental damage over short (20 years), long (100 years), and infinite (1000 years) time horizons of living pollutants. The results show that the use of sculptured tiles caused environmental damage in the short time horizon and environmental benefits in the long and infinite time horizons in the 2020 scenario, while it led to environmental benefits only in the infinite time horizon in the 2025 scenario.

This paper studies convex stochastic dynamic team problems with finite and infinite time horizons under decentralized information structures. First, we introduce the notions of called exchangeable teams and symmetric information structures. We show that, in convex exchangeable team problems, an optimal policy exhibits a symmetry structure. We give a characterization for such symmetrically optimal teams for a general class of convex dynamic team problems. In addition, for convex mean-field teams with a symmetric information structure, through concentration of measure arguments, we establish the convergence of optimal policies for mean-field teams with N decision makers to the corresponding optimal policies of mean-field teams with countably infinite number of decision makers. As a by-product, we also present an existence result for convex mean-field teams. While for partially nested LQG team problems with finite time horizon it is known that the optimal policies are linear, for infinite horizon problems the linearity of optimal policies has not been established in full generality and typically not only linearity but also time-invariance and stability properties are imposed apriori in the literature. In this paper, we also study, average cost finite and infinite horizon dynamic team problems with a symmetric partially nested information structure and obtain globally optimal solutions where we establish linearity of optimal policies. Moreover, we discuss average cost infinite horizon problems for LQG dynamic teams with sparsity and delay constraints.

We consider a portfolio optimization problem in a Black-Scholes model with n stocks, in which an investor faces both fixed and proportional transaction costs. The performance of an investment strategy is measured by the average return of the corresponding portfolio over an infinite time horizon. At first, we derive a representation of the portfolio value process which only depends on the relative fractions of the total portfolio value that the investor holds in the different stocks. This representation allows us to consider these so-called risky fractions as the decision variables of the investor. We show a certain kind of stationarity (Harris recurrence) for a quite flexible class of strategies (constant boundary strategies). Then, using renewal theoretic methods, we are able to describe the asymptotic return by the behaviour of the risky fractions in a “typical” period between two trades. Our results generalize those of [irl1], who considered a financial market model with one bond and one stock, to a market with a finite number n>1 of stocks.

In this article, an inventory model for deteriorating items with two components demand and time-varying holding cost has been developed. Demand is assumed to be stock-dependent up to the level of available stock. After which the demand is considered as constant i.e. during stock-out period. Shortages are allowed and are fully backlogged. Profits are maximized in both the infinite and finite time horizon cases. Some special cases are also derived from the main models. Two numerical examples are provided for both finite and infinite time horizon. Sensitivity analysis performed has shown successful effects of various model parameters on net profit.

We consider two-person zero-sum games of stopping: two players sequentially observe a stochastic process with infinite time horizon. Player I selects a stopping time and player II picks the distribution of the process. The pay-off is given by the expected value of the stopped process. Results of Irle (1990) on existence of value and equivalence of randomization for such games with finite time horizon, where the set of strategies for player II is dominated in the measure-theoretical sense, are extended to the infinite time case. Furthermore we treat such games when the set of strategies for player II is not dominated. A counterexample shows that even in the finite time case such games may not have a value. Then a sufficient condition for the existence of value is given which applies to prophet-type games.

Transaction Costs and Shadow Prices in Discrete Time

For portfolio choice problems with proportional transaction costs, we discuss whether or not there exists a shadow price, i.e., a least favorable frictionless market extension leading to the same optimal strategy and utility. By means of an explicit counterexample, we show that shadow prices may fail to exist even in seemingly perfectly benign situations, i.e., for a log-investor trading in an arbitrage-free market with bounded prices and arbitrarily small transaction costs. We also clarify the connection between shadow prices and duality theory. Whereas dual minimizers need not lead to shadow prices in above “global” sense, we show that they always correspond to a “local” version.

Optimal control of parabolic systems with infinite time horizons

ABSTRACTOur purpose of this paper is to study stochastic control problems for systems driven by mean-field stochastic differential equations with elephant memory, in the sense that the system (like the elephants) never forgets its history. We study both the finite horizon case and the infinite time horizon case. In the finite horizon case, results about existence and uniqueness of solutions of such a system are given. Moreover, we prove sufficient as well as necessary stochastic maximum principles for the optimal control of such systems. We apply our results to solve a mean-field linear quadratic control problem.For infinite horizon, we derive sufficient and necessary maximum principles.As an illustration, we solve an optimal consumption problem from a cash flow modelled by an elephant memory mean-field system.

Numerical methods for construction of value functions in optimal control problems with infinite horizon

An iterative direct-backward procedure for construction of optimal trajectories in control problems with infinite horizon

Shadow vs. market prices in explaining land allocation: Subsistence maize cultivation in rural Mexico

We prove the existence of a Radner equilibrium in a model with proportional transaction costs on an infinite time horizon and analyze the effect of transaction costs on the endogenously determined interest rate. Two agents receive exogenous, unspanned income and choose between consumption and investing into an annuity. After establishing the existence of a discrete-time equilibrium, we show that the discrete-time equilibrium converges to a continuous-time equilibrium model. The continuous-time equilibrium provides an explicit formula for the equilibrium interest rate in terms of the transaction cost parameter. We analyze the impact of transaction costs on the equilibrium interest rate and welfare levels.

We address a portfolio optimization problem in a semi-Markov modulated market. We study both the terminal expected utility optimization on finite time horizon and the risk-sensitive portfolio optimization on finite and infinite time horizon. We obtain optimal portfolios in relevant cases. A numerical procedure is also developed to compute the optimal expected terminal utility for finite horizon problem.

A mean-variance model is proposed for portfolio rebalancing optimization problems with transaction costs and minimum transaction lots. The portfolio optimization problems are modeled as a non-smooth nonlinear integer programming problem. A genetic algorithm based on real value genetic operators is designed to solve the proposed model. It is illustrated via a numerical example that the genetic algorithm can solve the portfolio rebalancing optimization problems efficiently.KeywordsGenetic AlgorithmTransaction CostPortfolio OptimizationGenetic OperatorPortfolio Selection ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.