We propose a simple model for the behaviour of longterm investors on a stock market. It consists of three particles that represent the stock's current price and the buyers', respectively sellers', opinion about the right trading price. As time evolves, both groups of traders update their opinions with respect to the current price. The speed of updating is controled by a parameter; the price process is described by a geometric Brownian motion. We consider the market's stability in terms of the distance between the buyers' and sellers' opinion, and prove that the distance process is recurrent/transient in dependence on the parameter.
"Escaping the Brownian stalkers." Electron. J. Probab. 14 139 - 160, 2009. https://doi.org/10.1214/EJP.v14-594