Stochastic Bass diffusion : model properties, exact simulation and statistical inference

Abstract

The need for modeling diffusion of innovation among individuals of a social system has motivated relevant research both from theoretical and applied perspective since the 1960s. Modeling diffusion of new products, services, or social behaviors can be useful in marketing strategies in the private sector, as well as in decision making processes by public authorities in social or health fields. Among pioneering works, a preeminent role is played by the aggregate model proposed by Bass (1969). According to this deterministic model, the diffusion dynamics is represented by an ordinary differential equation (ODE) of logistic-type, where, at a given time, the instantaneous rate of individuals adopting the innovation (adopters) is a quadratic function of the number of individuals who at that time have already adopted it. The logistic dynamics is suitable for the modeling of diffusion phenomena since it captures the two elements most influencing the adopter behavior: 1) the individual propensity to adopt (self-adoption) due to external factors (e.g., advertisement), 2) influence exerted by “already adopter” via imitation or “word by mouth”. In some applications it is also useful to include a further term to take into account the possibility of receding from adopting the innovation (disadoption). In this thesis, a stochastic extension of the classical Bass model is studied. The extension is a diffusion process defined as a solution of a stochastic differential equation (SDE) obtained by the classical equation by adding a suitable diffusion term. The diffusion coefficient used to introduce randomness in the model is defined so that the resulting stochastic process satisfies some “natural” properties which allow to interpret it as diffusion of an innovation. In particular, the trajectories of the process are a.s. non-negative and bounded by the constant K representing the number of potential adopters (regularity with respect to the interval [0,K]). In this thesis the result on regularity is extended to include the case where the number of potential adopters is a deterministic (non-increasing) function of time. This extension can be useful in situations where effects of the population dynamics are to be included into the model. Part of the thesis is devoted to the study of the process invariant distribution and to the analysis of the ergodicity properties for different values of the model parameters. More over, explicit expressions for the stationary distributions have been derived. The theoretical part of the thesis also includes the analysis of stochastic stability of singular stationary solutions based on use of appropriate Lyapunov functions. The last part of the thesis is devoted to numerical applications. Problems concerning simulation and inference for the stochastic Bass model are considered. To this regard, major difficulties arise from the fact that the analytic expression of process transition density is not available and cannot be directly used for simulation of process sample paths and in inferential approaches based on likelihood. In particular, in case of Bass model without self-innovation, it is illustrated how to draw realizations from the “true” law of the process using an exact simulation method recently introduced by Beskos et al. (2006), based on retrospective sampling. A version of this method allows to obtain realizations also from the law of the process conditioned on taking given values at the extremal points of a time interval (diffusion bridge). This is of particular interest because simulation of conditioned diffusions is often needed in some inferential approaches for incomplete data (EM algorithm, data augmentation) and it is not easy to obtain with traditional methods like the Eulerian scheme. Exact simulation method is based on the Girsanov Theorem and involves change of measure in infinite dimensional spaces. In the same framework, approximations of the transition density via Monte Carlo averages of certain Brownian functionals are derived. An approximate likelihood function and approximate maximum likelihood (aml) estimators of the model parameters can be obtained on the basis of the approximate transition density. In this thesis, aml estimators are evaluated through a Monte Carlo study where they are compared with classical estimators based on Gaussian approximation. It results that, especially for low sampling frequency, the aml estimators perform better than the competing Gaussian estimators.