Advertising a new product in a segmented market

We bring some market segmentation concepts into the statement of the “new product introduction” problem with Nerlove-Arrow’s linear goodwill dynamics. In fact, only a few papers on dynamic quantitative advertising models deal with market segmentation, although this is a fundamental topic of marketing theory and practice. In this way we obtain some new deterministic optimal control problems solutions and show how such marketing concepts as “targeting” and “segmenting” may find a mathematical representation. We consider two kinds of situations. In the first one, we assume that the advertising process can reach selectively each target group. In the second one, we assume that one advertising channel is available and that it has an effectiveness segment-spectrum, which is distributed over a non-trivial set of segments. We obtain the explicit optimal solutions of the relevant problems.     

Optimal Goodwill Path to Introduce a New Product

We consider the problem of determining an optimal goodwill path for the introduction of a new product in a market, while looking for the maximum foreseen profit. The foreseen revenue depends on the product introduction time and on the goodwill level at the same time. We focus on the advertising costs associated with the goodwill evolution and assume that the cost function possesses some rather general features which are shared by the cost functions of the Nerlove-Arrow type models. The dynamic optimization problem is discussed in the calculus of variations framework. A few examples associated with special cost functions are discussed in detail.     

一类随机模型的最优捕获问题研究

在Volterra两种群竞争模型的基础上,构造了随机的具有捕获的两种群竞争模型,研究讨论了捕获对种群生长过程的影响和如何实现最优捕获等问题.从确定性模型入手,深入讨论随机竞争模型的收获最优问题.通过对捕获强度E和贴现率等的估计与讨论,计算出了最优捕获强度最优捕获量最优经济收益.     

Continuous-Time Mean-Variance Portfolio Selection with Random Horizon

This paper examines the continuous-time mean-variance optimal portfolio selection problem with random market parameters and random time horizon. Treating this problem as a linearly constrained stochastic linear-quadratic optimal control problem, I explicitly derive the efficient portfolios and efficient frontier in closed forms based on the solutions of two backward stochastic differential equations. Some related issues such as a minimum variance portfolio and a mutual fund theorem are also addressed. All the results are markedly different from those in the problem with deterministic exit time. A key part of my analysis involves proving the global solvability of a stochastic Riccati equation, which is interesting in its own right.     

The asymptotic behaviors of a stochastic social epidemics model with multi-perturbation

Alcohol abuse is a major social problem, which is often called social epidemic, for the some similarities to the classical infectious diseases. In this paper, we formulated a new stochastic alcoholism model based on the deterministic model proposed in \cite{Wangxy}, with the mortalities of all populations as well as the contact infected coefficient are all perturbed. Based on this model, we investigate the long-term stochastic dynamics behaviors of two equilibria of the corresponding deterministic model and point out the effect of random disturbance on the stability of the system. Finally, we carry out numerical simulations to support our theoretical results.     

Non-linear advertising capital model with time delayed feedback between advertising and stock of goodwill

According to the classical Nerlove-Arrow model, advertising expenditure can be considered as a capital investment to create present and future demand for the firm's products and, hence, to create present and future revenues for the firm. Advertising is assumed to influence via stock of goodwill which cumulatively counts for the effects of the firm's current and past advertising outlays. The paper presents a time delayed feedback model describing the relations between advertising and goodwill. Three different types of effects of advertising upon the dynamics of goodwill are modelled. The advertising policy of the management is incorporated into the model via a non-linear advertising function. The advertising function controls the advertising outlay e.g. by budget constraint and by the actual and target values of goodwill. The behavior of the model is analysed both analytically and numerically. Special attention is given for deriving the stability conditions for the limiting solution. The cases of repelling or chaotic limiting solutions are analysed by bifurcation and state space diagrams. Several numerical examples are given.     

Delayed effects of cooperative advertising in goodwill dynamics

This paper proposes a tool to control cooperative advertising which increases the goodwill of companies operating in a competitive market. We introduce the lag between advertising exposure and customer reaction in the goodwill dynamics evolved à la Nerlove–Arrow. As a result, we obtain a cooperative differential game with immediate and delayed effects of control variables for which we investigate the optimal solution. We examine the role the pre-coalition programmes and the length of delayed response in generating goodwill.     

Stochastic differential portfolio games for an insurer in a jump-diffusion risk process

We discuss an optimal portfolio selection problem of an insurer who faces model uncertainty in a jump-diffusion risk model using a game theoretic approach. In particular, the optimal portfolio selection problem is formulated as a two-person, zero-sum, stochastic differential game between the insurer and the market. There are two leader-follower games embedded in the game problem: (i) The insurer is the leader of the game and aims to select an optimal portfolio strategy by maximizing the expected utility of the terminal surplus in the “worst-case” scenario; (ii) The market acts as the leader of the game and aims to choose an optimal probability scenario to minimize the maximal expected utility of the terminal surplus. Using techniques of stochastic linear-quadratic control, we obtain closed-form solutions to the game problems in both the jump-diffusion risk process and its diffusion approximation for the case of an exponential utility.     

Feedback Stackelberg equilibrium strategies when the private label competes with the national brand

We consider a noncooperative differential game where a retailer sells her own private label in addition to the manufacturer’s brand. We assume that each brand’s goodwill evolves according to a modified Nerlove-Arrow dynamics, in such a way that the advertising effort of one brand hurts the competitor’s goodwill stock. We characterize Feedback-Stackelberg pricing and advertising strategies and employ simulations to analyze their sensitivity to the main model parameters. We thank an anonymous Reviewer for very helpful comments. Research supported by NSERC, Canada, and FQRSC, Quebec. Research completed when the second author was a visiting professor at GERAD, HEC Montréal. The second author’s research was partially supported by MEC and JCYL under projects SEJ2005-03858/ECON and VA045A06, co-financed by FEDER funds.     

TIMBER SUPPLY UNDER DEMAND UNCERTAINTY: WELFARE GAINS FROM PERFECT COMPETITION WITH RATIONAL EXPECTATIONS

ABSTRACT. This paper presents a noneconometric approach to estimating the short‐run timber supply function based on optimal harvest decisions. Determination of optimal harvest levels and estimation of supply function coefficients are integrated into one step by incorporating a parametric short‐run timber supply function into the harvest decision model. In this manner we convert the original harvest decision model into a new optimization problem with the supply function coefficients functioning as “decision variables.” Optimal solution to the new decision model gives the coefficients of the short‐run supply function and, indirectly, the optimal harvest levels. This approach enables us to develop stochastic models of the timber market that are particularly useful for forest sector analysis involving comparison of alternative institutional regimes or policy proposals and when the timber market is affected by stochastic variables. For demonstration purposes, we apply this method to compare the performances of two timber market regimes (perfect competition and monopoly) under demand uncertainty, using the Swedish data. The results show that the expected timber price is 22 percent lower and the expected annual timber supply is 43 percent higher in the competitive market than in the monopoly market. This confirms the theoretical result that monopoly reduces supply and increases price. The expected social welfare gain from perfect competition over monopoly is about 24 percent.     

Effect of stochastic perturbation on a two species competitive model

In this paper first we study the stability and bifurcation of a two species competitive model with a delay effect. Next we extend the deterministic model system to a stochastic delay differential system by incorporating multiplicative white noise terms in growth equations of both species. We consider the stochastic stability of a co-existing equilibrium point in terms of mean square stability by constructing a suitable Lyapunov functional. We perform a numerical simulation to validate our analytical findings.     

Interpreting supply chain dynamics: A quasi-chaos perspective

Chaotic phenomena, chaos amplification and other interesting nonlinear behaviors have been observed in supply chain systems. Chaos can be defined theoretically if the dynamics under study are produced only by deterministic factors. However, deterministic settings rarely present themselves in reality. In fact, real data are typically unknown. How can the chaos theory and its related methodology be applied in the real world? When the demand is stochastic, the interpretation and distribution of the Lyapunov exponents derived from the effective inventory at different supply chain levels are not similar to those under deterministic demand settings. Are the observed dynamics of the effective inventory random, chaotic, or simply quasi-chaos? In this study, we investigate a situation whereby the chaos analysis is applied to a time series as if its underlying structure, deterministic or stochastic, is unknown. The result shows clear distinction in chaos characterization between the two categories of demand process, deterministic vs. stochastic. It also highlights the complexity of the interplay between stochastic demand processes and nonlinear dynamics. Therefore, caution should be exercised in interpreting system dynamics when applying chaos analysis to a system of unknown underlying structure. By understanding this delicate interplay, decision makers have the better chance to tackle the problem correctly or more effectively at the demand end or the supply end.     

Advertising for a new product introduction: A stochastic approach

We formulate a stochastic extension of the Nerlove and Arrow’s advertising model in order to analyze the problem of a new product introduction. The main idea is to introduce some uncertainty aspects in connection both with the advertising action and the goodwill decay, in order to represent the random consequences of the advertising messages and of the word-of-mouth publicity, respectively. The model is stated in terms of the stochastic optimal control theory and a general study is attempted using the stochastic Maximum Principle. Closed form solutions are obtained under linear quadratic assumptions for the cost and the reward functions. Such optimal policies suggest that the decision-maker considers both the above mentioned phenomena as opportunities to increase her/his final reward. After stating some general features of the optimal solutions, we analyze in detail three extreme cases, namely the deterministic model and the stochastic models with either the word-of-mouth effect only, or the lure/repulsion effect only. The optimal policies provide us with some insight on the general effects of the advertising action. Supported by MIUR and University of Padua.     

The Filtering Equations of Forward-Backward Stochastic Systems with Random Jumps and Applications to Partial Information Stochastic Optimal Control

In this article, we consider a filtering problem for forward-backward stochastic systems that are driven by Brownian motions and Poisson processes. This kind of filtering problem arises from the study of partially observable stochastic linear-quadratic control problems. Combining forward-backward stochastic differential equation theory with certain classical filtering techniques, the desired filtering equation is established. To illustrate the filtering theory, the theoretical result is applied to solve a partially observable linear-quadratic control problem, where an explicit observable optimal control is determined by the optimal filtering estimation.     

Optimal dynamic advertising policies in the presence of continuously distributed time lags

The Nerlove-Arrow model of optimal dynamic advertising policies is generalized by incorporating a continuously distributed lag between advertising expenditures and increases in the stock of goodwill. This leads to a control problem where the equation of motion is given by an integro-differential equation. The transitory and steady-state properties of the optimal policies are examined, both for a general lag function and for a gamma distributed lag. The dependence of the steady-state solution on the parameters of the gamma distribution is also investigated. An example is given using specific demand and cost functions.     

Analysis of a Kind of Stochastic Dynamics Model with Nonlinear Function

In this paper, we establish stochastic differential equations on the basis of a nonlinear deterministic model and study the global dynamics. For the deterministic model, we show that the basic reproduction number $\Re _0$ determines whether there is an endemic outbreak or not: if $\Re _0< 1$, the disease dies out; while if $\Re _0> 1$, the disease persists. For the stochastic model, we provide analytic results regarding the stochastic boundedness, perturbation, permanence and extinction. Finally, some numerical examples are carried out to confirm the analytical results. One of the most interesting findings is that stochastic fluctuations introduced in our stochastic model can suppress disease outbreak, which can provide us some useful control strategies to regulate disease dynamics.     

Differential game approach to pricing and advertising decisions

This study proposes a model to make concurrent decisions on dynamic pricing and advertising to maximise firms' profitability over an infinite time horizon in a duopoly market. To this end, the Nerlove-Arrow pricing and advertising model is designed in the presence of shifting costs in a dynamic duopolistic competition as a differential game. The Nash equilibrium solution is defined based upon a set of Hamilton–Jacobi–Bellman. Four scenarios are applied for economic interpretations and the efficacy of the model.     

Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients

In Monte Carlo methods quadrupling the sample size halves the error. In simulations of stochastic partial differential equations (SPDEs), the total work is the sample size times the solution cost of an instance of the partial differential equation. A Multi-level Monte Carlo method is introduced which allows, in certain cases, to reduce the overall work to that of the discretization of one instance of the deterministic PDE. The model problem is an elliptic equation with stochastic coefficients. Multi-level Monte Carlo errors and work estimates are given both for the mean of the solutions and for higher moments. The overall complexity of computing mean fields as well as k-point correlations of the random solution is proved to be of log-linear complexity in the number of unknowns of a single Multi-level solve of the deterministic elliptic problem. Numerical examples complete the theoretical analysis.     

Dynamic traveling salesman problem with stochastic release dates

The dynamic traveling salesman problem with stochastic release dates (DTSP-srd) is a problem in which a supplier has to deliver parcels to its customers. These parcels are delivered to its depot while the distribution is taking place. The arrival time of a parcel to the depot is called its release date. In the DTSP-srd, release dates are stochastic and dynamically updated as the distribution takes place. The objective of the problem is the minimization of the total time needed to serve all customers, given by the sum of the traveling time and the waiting time at the depot. The problem is represented as a Markov Decision Process and is solved through a reoptimization approach. Two models are proposed for the problem to be solved at each stage. The first model is stochastic and exploits the entire probabilistic information available for the release dates. The second model is deterministic and uses an estimation of the release dates. An instance generation procedure is proposed to simulate the evolution of the information to perform computational tests. The results show that a more frequent reoptimization provides better results across all tested instances and that the stochastic model performs better than the deterministic model. The main drawback of the stochastic model lies in the computational time required to evaluate a solution, which makes an iteration of the heuristic substantially more time-consuming than in the case where the deterministic model is used.     

随机扰动下的库存投资决策模型

本文讨论的是库存投资的最优决策问题.不同于确定性q理论,对于引入了市场不确定性扰动的库存控制系统,文章建立了库存投资随机优化决策模型.从市场利率波动的角度对库存决策模型进行分析,得出的结论是:小的市场利率的扰动能够提高企业折现利润的预期,进而导致公司库存投资的上升.