计量经济分析 = Econometric analysis

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Convex Analysisの二,三の進展について

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

In this paper, we examine the research and results of dynamic pricing policies and their relation to revenue management. The survey is based on a generic revenue management problem in which a perishable and nonrenewable set of resources satisfy stochastic price sensitive demand processes over a finite period of time. In this class of problems, the owner (or the seller) of these resources uses them to produce and offer a menu of final products to the end customers. Within this context, we formulate the stochastic control problem of capacity that the seller faces: How to dynamically set the menu and the quantity of products and their corresponding prices to maximize the total revenue over the selling horizon.

https://pages.stern.nyu.edu/~rcaldent/papers/Abstract-Survey-Paper-MSOM.pdf

Commissioned Paper: An Overview of Pricing Models for Revenue Management

A retailer is endowed with a finite inventory of a non-perishable product. Demand for this product is driven by a price-sensitive Poisson process that depends on an unknown parameter, theta; a proxy for the market size. If theta is high then the retailer can take advantage of a large market charging premium prices, but if theta is small then price markdowns can be applied to encourage sales. The retailer has a prior belief on the value of theta which he updates as time and available information (prices and sales) evolve. We also assume that the retailer faces an opportunity cost when selling this non-perishable product. This opportunity cost is given by the long-term average discounted profits that the retailer can make if he switches and starts selling a different assortment of products. The retailer's objective is to maximize the discounted long-term average profits of his operation using dynamic pricing policies. We consider two cases. In the first case, the retailer is constrained to sell the entire initial stock of the non-perishable product before a different assortment is considered. In the second case, the retailer is able to stop selling the non-perishable product at any time to switch to a different menu of products. In both cases, the retailer's pricing policy trades-off immediate revenues and future profits based on active demand learning. We formulate the retailer's problem as a (Poisson) intensity control problem and derive structural properties of an optimal solution which we use to propose a simple approximated solution. This solution combines a pricing policy and a stopping rule (if stopping is an option) depending on the inventory position and the retailer's belief about the value of theta. We use numerical computations, together with asymptotic analysis, to evaluate the performance of our proposed solution.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID882851_code337731.pdf?abstractid=882851&mirid=3

Dynamic Pricing for Non-Perishable Products with Demand Learning

A retailer is endowed with a finite inventory of a nonperishable product Demand for this product is driven by a price-sensitive Poisson process that depends on an unknown parameter that is a proxy for the market size The retailer has a prior belief on the value of this parameter that he updates as time and available information (prices and sales) evolve The retailer's objective is to maximize the discounted long-term average profits of his operation using dynamic pricing policies We consider two cases In the first case, the retailer is constrained to sell the entire initial stock of the nonperishable product before a different assortment is considered In the second case, the retailer is able to stop selling the nonperishable product at any time and switch to a different menu of products For both cases, we formulate the retailer's problem as a (Poisson) intensity control problem and derive structural properties of an optimal solution, and suggest a simple and efficient approximated solution We use numerical computations, together with asymptotic analysis, to evaluate the performance of our proposed policy

https://pages.stern.nyu.edu/~rcaldent/papers/DemandLearning-Victor.pdf

Dynamic Pricing for Nonperishable Products with Demand Learning

We study the performance of a stylized supply chain where two firms, a retailer and a producer, compete in a Stackelberg game. The retailer purchases a single product from the producer and afterward sells it in the retail market at a stochastic clearance price. The retailer, however, is budget constrained and is therefore limited in the number of units that he may purchase from the producer. We also assume that the retailer's profit depends in part on the realized path or terminal value of some observable stochastic process. We interpret this process as a financial process such as a foreign exchange rate or interest rate. More generally, the process can be interpreted as any relevant economic index. We consider a variation (the flexible contract) of the traditional wholesale price contract that is offered by the producer to the retailer. Under this flexible contract, at t = 0 the producer offers a menu of wholesale prices to the retailer, one for each realization of the financial process up to a future time τ. The retailer then commits to purchasing at time τ a variable number of units, with the specific quantity depending on the realization of the process up to time τ. Because of the retailer's budget constraint, the supply chain might be more profitable if the retailer was able to shift some of the budget from states where the constraint is not binding to states where it is binding. We therefore consider a variation of the flexible contract, where we assume that the retailer is able to trade dynamically between zero and τ in the financial market. We refer to this variation as the flexible contract with hedging. We compare the decentralized competitive solution for the two contracts with the solutions obtained by a central planner. We also compare the supply chain's performance across the two contracts. We find, for example, that the producer always prefers the flexible contract with hedging to the flexible contract without hedging. Depending on model parameters, however, the retailer might or might not prefer the flexible contract with hedging.

/pdf/supply-contracts-with-financial-hedging-3x8sr4jwyt.pdf

Supply Contracts with Financial Hedging

In this paper we propose a methodology to set prices of perishable items in the context of a retail chain with coordinated prices among its stores and compare its performance with actual practice in a real case study. We formulate a stochastic dynamic programming problem and develop heuristic solutions that approximate optimal solutions satisfactorily. To compare this methodology with current practices in the industry, we conducted two sets of experiments using the expertise of a product manager of a large retail company in Chile. In the first case, we contrast the performance of the proposed methodology with the revenues obtained during the 1995 autumn-winter season. In the second case, we compare it with the performance of the experienced product manager in a "simulation-game" setting. In both cases, our methodology provides significantly better results than those obtained by current practices.

http://people.stern.nyu.edu/rcaldent/papers/ClearanceOR98.pdf

René Caldentey

Papers

Commissioned Paper: An Overview of Pricing Models for Revenue Management

Dynamic Pricing for Non-Perishable Products with Demand Learning

Dynamic Pricing for Nonperishable Products with Demand Learning

Supply Contracts with Financial Hedging

Coordinating Clearance Markdown Sales of Seasonal Products in Retail Chains