# A Stochastic Formulation of the Bass Model of New-Product Diffusion

For a large variety of new products, the Bass Model (BM) describes
the empirical cumulative-adoptions curve extremely well. The BM postulates
that the trajectory of cumulative adoptions of a new product follows a
deterministic function whose instantaneous growth rate depends on two
parameters, one of which captures an individual's intrinsic tendency to
purchase, independent of the number of previous adopters, and the other
captures a positive force of influence on an individual by previous
adopters. In this paper, we formulate a stochastic version of the BM, which
we call the Stochastic Bass Model (SBM), where the trajectory of cumulative
number of adoptions is governed by a pure birth process. We show that with
an appropriately-chosen set of birth rates, the fractions of individuals who
have adopted the product by time *t* in a family of SBMs indexed by the
size of the target population converge in probability to the deterministic
fraction in a corresponding BM, when the population size approaches
infinity. The formulation therefore supports and expands the BM by allowing
stochastic trajectories.