A basic model in mathematical finance theory is the celebrated geometric Brownian motion. Moreover, the geometric telegraph process is a simpler model to describe the alternating dynamics of the price of risky assets. In this note we consider a more general stochastic process that combines the characteristics of such two models. Precisely, we deal with a geometric Brownian motion with alternating trend. It is defined as the exponential of a standard Brownian motion whose drift alternates randomly between a positive and a negative value according to a generalized telegraph process. We express the probability law of this process as a suitable mixture of Gaussian densities, where the weighting measure is the probability law of the occupation time of the underlying telegraph process.
|Titolo:||On the geometric Brownian motion with alternating trend|
|Data di pubblicazione:||2014|
|Appare nelle tipologie:||2.1.2 Articolo su libro con ISBN|