Modelling, analysis and simulation of stochastic innovation diffusion
- Schurz, Henri
- Innovation diffusion, Bass model, Stochastic differential equations, Algebraic constraints, Regularity, Lyapunov-type methods, Numerical methods, Balanced implicit methods, Numerical regularization, Boundedness, Mean square convergence
The well-known BASS model for description of diffusion of innovations has been extensively investigated within deterministic framework. One of the basic processes in modelling of these diffusions concerns with the propagation through word of mouth which is inherently nonlinear. As a more realistic modelling, the diffusion of an innovation in the presence of uncertainty is generally formulated in terms of nonlinear stochastic differential equations (SDEs). At first we discuss well-posedness, regularity (boundedness) and uniqueness of solutions of these SDEs. However, an explicit expression for analytical solution itself is not available. Accordingly one has to resort to numerical solution of SDEs for studying various aspects like the time-development of growth patterns, exit frequencies, mean passage times and impact of advertising policies. In this respect we present some basic aspects of numerical analysis of these random extensions of the BASS model, e.g. numerical regularity and mean square convergence. Therein the problem of numerical movement within reasonable boundaries (numerical solution on bounded manifolds) plays a significant role, in particular on intervals with reflecting or absorbing barriers, whereas the discretization of the state space (continuous time version of the set of possible adopters of the innovation) is circumvented. Such a study brings out salient features of the stochastic models (as e.g. boundedness of equilibria, faster initial adoption or earlier peak sales in comparison with deterministic model). To this end we shall provide discrete time estimations of the moment evolution and pathwise solutions based on Balanced implicit methods (see Mil'shtein et al. (1992)).