Bass Diffusion: A Stochastic Differential Equation Approach

The need for modeling diffusion of innovation among individuals of a social systemhas motivated relevant research both from theoretical and applied perspective since the 1960s.Among pioneering works, a preeminent role is played by the aggregate model proposed byBass [2]. According to this deterministic model, the diffusion dynamics is represented by anordinary differential equation (ODE) of logistic-type. In this paper, a stochastic extension ofthe classical Bass model is studied. The extension is a diffusion process defined as solution of astochastic differential equation (SDE) obtained by adding to the classical equation a suitablediffusion term. The diffusion term guarantees thatthe resulting stochastic process satisfiessome relevant properties. In particular, the trajectories of the process are almost surely (a.s.)non-negative and bounded by the constant K representing the number of potential adopters(regularity with respect to the interval?0,K?). A known regularity result is extended here toinclude the case where the number of potential adopters is a deterministic (non-decreasing)function of time. This extension can be useful in situations where effects of the populationdynamics need to be included into the model. We also study theoretically when the SDE Bassmodel admits a stationary distribution. In the case when there is no stationary distribution,stochastic stability of the steady state solutions?Yt?K,t>=0?and?Yt?0,t>=0?ofclassical Bass model is studied. A Monte Carlo simulation procedure to approximate thetransition density of the SDE process is proposed and applied here to estimate SDE Bassmodel parameters via approximate maximumlikelihood estimation (AMLE). The resultsobtained are compared to those from a known method based on Gaussian approximation. Anintensive simulation study shows that the proposed method outperforms the classical on