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Markov chain approximation is a general method for option pricing and hedging in Markovian models with continuous-state spaces. A key issue for its efficiency is how to design the grid for the Markov chain. In "Analysis of Markov Chain Approximation for Option Pricing and Hedging: Grid Design and Convergence Behavior," Zhang and Li investigated how grid design affects the convergence behavior of barrier and European options in general diffusion models. They obtain sharp estimates for the convergence rate of option price, delta, and gamma and develop two principles that are sufficient and necessary for designing nonuniform grids that can achieve second-order convergence. They further propose a novel class of nonuniform grids that ensure that convergence is not only second order but also, smooth. This allows extrapolation to be applied to achieve even higher convergence rate. Their grids enable the continuous time Markov chain approximation method to price and hedge a large number of options with different strikes fast and accurately. Continuous time Markov chain (CTMC) approximation is an intuitive and powerful method for pricing options in general Markovian models. This paper analyzes how grid design affects the convergence behavior of barrier and European options in general diffusion models. Using the spectral method, we obtain sharp estimates for the convergence rate of option price for nonuniform grids. We propose to calculate an option's delta and gamma by taking central difference of option prices on the grid. For this simple method, we prove that, surprisingly, delta and gamma converge at the same rate as option price does. Our analysis allows us to develop principles that are sufficient and necessary for designing nonuniform grids that can achieve second-order convergence for option price, delta, and gamma. Based on these principles, we propose a novel class of nonuniform grids that ensure that convergence is not only second order but also, smooth. This further allows extrapolation to be applied to achieve even higher convergence rate. Our grids enable the CTMC approximation method to price and hedge a large number of options with different strikes fast and accurately. Applicability of our results to jump models is discussed through numerical examples. The e-companion is available at https://doi.org/10.1287/opre.2018.1791. [ABSTRACT FROM AUTHOR]

In public policy decision making and in capital planning fractional criterion functions occur. For a given set of desirable target values (goals) τi, this paper develops an algorithm for solving a nonconvex programming problem of the type: Minx∈S Maxi{&phii(fi(x)/gi(x) - τi), i = 1,..., m} where fi are convex functions, gi, are concave functions over the convex subset S of Rn and φi are nondecreasing gauge functions. Here φi(&middot) is the penalty incurred whenever the fractional objective fi/gi deviates from the target value τi, the problem is then to choose an x that minimizes the maximum penalty incurred. [ABSTRACT FROM AUTHOR]

Following a successful idea generation exercise, a company might easily be left with hundreds of ideas generated by experts, employees, or consumers. The next step is to screen these ideas and identify those with the highest potential. In this paper we propose a practical approach to involving consumers in idea screening. Although the number of ideas may potentially be very, large, it would be unreasonable to ask each consumer to evaluate more than a few ideas. This raises the challenge of efficiently selecting the ideas to be evaluated by each consumer. We describe several idea-screening algorithms that perform this selection adaptively based on the evaluations made by previous consumers. We use simulations to compare and analyze the performance of the algorithms as well as to understand their behavior. The best-performing algorithm focuses on the ideas that are the most likely to have been misclassified (as "top" or "bottom" ideas) based on the previous evaluations, and avoids discarding ideas too fast by adding random perturbations to the misclassification probabilities. We demonstrate the convergent validity of this algorithm using a field experiment, which also confirms the convergence pattern predicted by simulations. [ABSTRACT FROM AUTHOR]

This paper develops a technique for analyzing dynamic inventory problems with discrete demand using the theory of weak convergence. We treat the dynamic fixed life perishable inventory problem with discrete demand. The optimal structure has been established assuming continuous demand. We construct an infinite sequence of continuous demand distributions that converge weakly to a discrete distribution, and show that the key results carry over in the limit. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to furnish a computational scheme for a class of programming problems with nonlinear constraints. The algorithm is sequential in nature, producing a sequence of feasible solutions whose limit points are optimal solutions of the original problem. Further, with a view to accelerating convergence we suggest two variants of the solution procedure. [ABSTRACT FROM AUTHOR]

In the generation of an additional constraint or cut for integer programming introduced by Ralph Gomory, there is a certain parameter whose value is customarily chosen so that one of the original variables has a zero coefficient in the cut-equation. In this paper we consider alternative values of the parameter and refer to any linear combination of the original source equation and this cut as a "generalised cut." Attention is then focused, as customary, on those cuts of this type for which the original variable has a zero coefficient. In particular, we show how to determine values for Gomory's parameter so that only a subset of the feasible solutions to the cuts used by Gomory's method of integer forms are feasible for the generalized cuts. [ABSTRACT FROM AUTHOR]

We obtain a closed-form solution for pricing European options under a general jump-diffusion model that can incorporate arbitrary discrete jump-size distributions, including nonparametric distributions such as an empirical distribution. The flexibility in the jump-size distribution allows the model to better capture leptokurtic features found in real-world data. The model uses a discrete-time framework and leads to a pricing formula that is provably convergent to the continuous-time price as the discretization is increased. The solution is easy to implement with fast convergence properties. Numerical results illustrate the efficiency and accuracy of the proposed model and highlight its robustness and flexibility. This paper was accepted by Noah Gans, stochastic models and simulation. [ABSTRACT FROM AUTHOR]

This paper gives numerical illustrations of the behavior of stochastic approximation, combined with different derivative estimation techniques, to optimize a steady-state system. It is a companion paper to L'Ecuyer and Glynn (1993), which gives convergence proofs for most of the variants experimented here. The numerical experiments are made with a simple M/M/1 queue, which while simple, serves to illustrate the basic convergence properties and possible pitfalls of the various techniques. [ABSTRACT FROM AUTHOR]

The optimal allocation process is of the following type. A sequence of functions (production functions) f[sub 1], f[sub 2],..., is given, each mapping R[sup n, sub +] into [0, ∞). Along with it a vector q > 0 in R[sup n, sub +]. (vector of resources) is provided. For each integer k we consider the problem maximize (Multiple lines cannot be represented in ASCII text) subject to (Multiple lines cannot be represented in ASCII text) Here R[sup n, sub +] is the nonnegative orthant of the n-dimensional Euclidean space. Inequalities between vectors are taken coordinatewise. In this paper we consider the case where the functions f[sub i] are subject to randomness. Specifically, they are generated as independent random drawings from a common distribution. In particular f[sub i] = f[sub i](x, ω) with ω being in a probability space, and the solutions x[sub i] might depend on ω, as well as on k and q. The detailed technical conditions are displayed in §2. We are interested in limit properties, as k → ∞, of the optimal values of the allocation process. [ABSTRACT FROM AUTHOR]

In this paper we propose an efficient Newton-SOR algorithm for solving the separable spatial price equilibrium problem. Although an active set strategy is implemented for solving for blocks of variables associated with origin nodes, global convergence of the iterates is not dependent on nondegeneracy or convexity assumptions. Numerical experiments indicate the excellent performance of the method on both degenerate and nondegenerate problems. [ABSTRACT FROM AUTHOR]

In this paper we propose a network equilibration algorithm for the solution of a class of constrained matrix problems with transportation-type constraints. The algorithm decomposes the problem into two series of supply and demand network equilibrium problems with special structure which can then be solved exactly and in parallel. The theoretical results obtained include the proof of convergence, the rate of convergence, and computational complexity analysis, and are obtained by interpreting the algorithm as a dual method. Computational results on datasets illustrate the theory and the efficiency of this approach. [ABSTRACT FROM AUTHOR]

This paper examines convergence criteria of an algorithm for the computation of Cournot-Nash economic equilibria. The method is based on formulating the equilibrium problem as that of finding a solution to a nonlinear complementarity problem, solved by sequential linearization and Lemke's algorithm. Conditions for local and global convergence are developed and the technique is applied to homogeneous, segmented and differentiated product markets. [ABSTRACT FROM AUTHOR]

Let F be a distribution function on (0, infinity), and let S = 1 - F be its corresponding survival function. F is New Better than Used (NBU) if S(x)S(y) is greater than or equal to S(x + y) for all x and y. Let S.(x) be the empirical survival function based on a random sample of size n from an NBU distribution function F. This paper studies the estimator S[sub n](x) defined as sup{S[sub n](x + y)/S[sub n](y)}, where the supremum is taken over all y for which S[sub n](y) > 0. We show that S[sub n] is an NBU survival curve, and that it is strongly uniformly consistent for S when the underlying distribution has compact support (for example, when sampling is subject to type I censoring). Moreover, in such problems, we show that the rate of convergence of S[sub n] is optimal. [ABSTRACT FROM AUTHOR]

It is well known that the gradient descent algorithm converges linearly when applied to a strongly convex function with Lipschitz gradient. In this case, the algorithm's rate of convergence is determined by the condition number of the function. In a similar vein, it has been shown that a variant of the Frank–Wolfe algorithm with away steps converges linearly when applied to a strongly convex function with Lipschitz gradient over a polytope. In a nice extension of the unconstrained case, the algorithm's rate of convergence is determined by the product of the condition number of the function and a certain condition number of the polytope. We shed new light on the latter type of polytope conditioning. In particular, we show that previous and seemingly different approaches to define a suitable condition measure for the polytope are essentially equivalent to each other. Perhaps more interesting, they can all be unified via a parameter of the polytope that formalizes a key premise linked to the algorithm's linear convergence. We also give new insight into the linear convergence property. For a convex quadratic objective, we show that the rate of convergence is determined by a condition number of a suitably scaled polytope. [ABSTRACT FROM AUTHOR]

We propose a framework for targeting and selection (T&S), a new problem class in simulation optimization where the objective is to select a simulation alternative whose mean performance matches a prespecified target as closely as possible. T&S resembles the more well-known problem of ranking and selection but presents unexpected challenges: for example, a one-step look-ahead method may produce statistically inconsistent estimates of the values, even under very standard normality assumptions. We create a new and fundamentally different approach, based on a Brownian local time model, that exhibits characteristics of two widely studied methodologies, namely expected value of information and optimal computing budget allocation. We characterize the asymptotic sampling rates of this method and relate them to the convergence rates of metrics of interest. The local time method outperforms benchmarks in experiments, including problems where the modeling assumptions of T&S are violated. [ABSTRACT FROM AUTHOR]

We consider a two-sided assignment market with agent types and a stochastic structure, similar to models used in empirical studies. We characterize the size of the core in such markets. Each agent has a randomly drawn productivity with respect to each type of agent on the other side. The value generated from a match between a pair of agents is the sum of the two productivity terms, each of which depends only on the type (but not the identity) of one of the agents, and a third deterministic termdriven by the pair of types.We prove, under reasonable assumptions, that when the number of agent types is kept fixed, the relative size of the core vanishes rapidly as the number of agents grows. Numerical experiments confirm that the core is typically small. Our results provide justification for the typical assumption of a unique core outcome in such markets, which is close to a limit point. Further, our results suggest that, given the market composition, wages are almost uniquely determined in equilibrium. [ABSTRACT FROM AUTHOR]

Our era is one of increasingly pervasive digital technologies, which penetrate deeply into the very core of the products, services, and operations of many organizations and radically change the nature of product and service innovations. The fundamental properties of digital technology are reprogrammability and data homogenization. Together, they provide an environment of open and flexible affordances that are used in creating innovations characterized by convergence and generativity. An analysis of convergence and generativity observed in innovations with pervasive digital technologies reveals three traits: (1) the importance of digital technology platforms, (2) the emergence of distributed innovations, and (3) the prevalence of combinatorial innovation. Each of the six articles in this special issue relates to one or more of these three traits. In this essay, we explore the organizational research implications of these three digital innovation traits and identify research opportunities for organization science scholars. Examples from the articles in this special issue on organizing for innovation in the digitized world are used to demonstrate the kind of organizational scholarship that can faithfully reflect and inform innovation in a world of pervasive digital technologies. [ABSTRACT FROM AUTHOR]

The look-ahead estimator is used to compute densities associated with Markov processes via simulation. We study a framework that extends the look-ahead estimator to a broader range of applications. We provide a general asymptotic theory for the estimator, where both L1 consistency and L2 asymptotic normality are established. The L2 asymptotic normality implies √n convergence rates for L2 deviation. [ABSTRACT FROM AUTHOR]

We study an oligopoly consisting of M leaders and N followers that supply a homogeneous product (or service) noncooperatively. Leaders choose their supply levels first, knowing the demand function only in distribution. Followers make their decisions after observing the leader supply levels and the realized demand function. We term the resulting equilibrium a stochastic multiple-leader Stackelberg-Nash-Cournot (SMS) equilibrium. We show the existence and uniqueness of SMS equilibrium under mild assumptions. We also propose a computational approach to find the equilibrium based on the sample average approximation method and analyze its rate of convergence. Finally, we apply this framework to model competition in the telecommunication industry. [ABSTRACT FROM AUTHOR]

This article presents an investigation of performance and aspiration as drivers of convergence-divergence in strategic positioning in terms of resource-allocation decisions. The study, based on the behavioral theory of the firm, suggests two unexplored kinked-curve relationships between performance gaps and strategic convergence and divergence. Empirical analysis of dyad-level data in the United States food processing industry reveals that the direction of a firm's strategic change is shaped by its aspiration level, its performance level, and the performance gap between it and the firm it selects as a model. This study offers indications of when and how a firm moves closer to or further away from other competing firms in its strategic position.

Technological developments combine previously distinct technologies that result in converging markets. In converging markets, firms from different industries compete against each other, often for the first time. We propose that firms introducing new products in converging markets will learn vicariously from other firms in the market. Further, we propose that this learning will vary across the dual-technology frontier (DTF), where the high-technology frontier (HTF) and low-technology frontier (LTF) map onto innovative activities driven by technological opportunity and user needs. We propose that at the HTF, local search will dominate and firms will be influenced by HTF product introductions of similarly sized, successful firms. At the LTF, learning will occur across the DTF, vary by origin industry of the firm, and be affected by complementarities in routines and capabilities and market competition among firms. We test the proposed model of vicarious learning using panel data on new product introductions of 67 firms in the U.S. digital camera market in the 1990s. Findings generally support our proposed model of vicarious learning in this market. They show heterogeneity in vicarious learning across the technology frontier and firm characteristics--including the origin industry of target firms. Our results show that vicarious learning in new product introductions in converging markets--which includes both mimetic and nonmimetic learning--is similar in some ways, but different from more traditional markets. We conclude with a discussion of the implications of our findings for theories of organizational learning, new product development, and converging markets. [ABSTRACT FROM AUTHOR]

During the last two decades the concept of epi-convergence was introduced and then was used in various investigations in optimization and related areas. The aim of this review is to show in an elementary way how closely the arguments in the epi-convergence approach are related to those of the classical theory of convergence of functions. [ABSTRACT FROM AUTHOR]

New iterative separable programming techniques based on two-segment, piecewise-linear approximations are described for the minimization of convex separable functions over convex sets. These techniques have two advantages over traditional separable programming methods. The first is that they do not require the cumbersome "fine grid" approximations employed to achieve high accuracy in the usual separable programming approach. In addition, the new methods yield feasible solutions with objective values guaranteed to be within any specified tolerance of optimality. In computational tests with real-world problems of up to 500 "nonlinear" variables the approach has exhibited rapid convergence and yielded very close bounds on the optimal value. [ABSTRACT FROM AUTHOR]

The rate at which Markov decision processes converge as the horizon length increases can be important for computations and judging the appropriateness of models. The convergence rate is commonly associated with the discount factor a. For example, the total value function for a broad set of problems is known to converge 0(α[sup n]), i.e., geometrically with the discount factor. But the rate at which the finite horizon optimal policies converge depends on the convergence of the relative value function. (Relative value at a given state is the difference between total value at that state and total value at some fixed reference state.) Relative value convergence in turn depends both on the discount factor and on ergodic properties of the underlying nonhomogeneous Markov chains. We show in particular that for the stationary finite state space Compact action space Markov decision problem, the relative value function converges 0((αλ)n) for all λ > r(P), the argument of the subdominant eigenvalue of the optimal infinite horizon policy (assumed unique). Easily obtained bounds for r(P) are also given which are related to those of A. Brauer. Under additional restrictions, policy convergence is shown to be of the same order as relative value convergence, generalizing work of Shapiro, Schweitzer, and Odoni. The same result gives convergence properties for the undiscounted problem and for the case α > 1. If αr(P) > 1 the problem does not converge. As a by-product of the analysis, necessary conditions are given for the relative value function to converge 0((αλ)n), 0 < αλ < 1, for the nonstationary problem. [ABSTRACT FROM AUTHOR]