The authors consider the stochastic version of the Bass model described by the following Ito stochastic differential equation \[ dx(t)= [a(1- x(t))+ bx(t)(1- x(t))]\,dt+\sigma a(1- x(t))\,dw(t), \] where \(x\in\mathbb{R}\) is the state variable, \(w(t)\) is the standard Wiener process, \(a\), \(b\), \(\sigma\) are positive constants.
The authors studied two problems: the optimal stopping problem and a combined optimal stopping and control problem for the optimization of the advertisement effectiveness. The first problem by reducing it to a free boundary problem was solved numerically by an iterative procedure while the second problem by applying a finite difference scheme. In both problems the obtained results are illustrated by numerical examples.