Transition probabilities in a problem of stochastic process switching

Flood and Garber (1983), Smith (1991), and Froot and Obstfeld (1991a,b) examined the return
of the United Kingdom to the gold standard in 1925 as an example of state-contingent process
switching. They calculated the exchange rate via the density function of the …rst-passage time
through the announced parity (Flood and Garber, 1983; Smith, 1991) or via solving a di¤erential
equation under suitable boundary conditions (Froot and Obstfeld, 1991a,b). We alternatively
employ the underlying transition probabilities and con…rm the solution obtained in the literature.
In addition, our approach allows us to critically evaluate intuitive arguments in the literature that
actually relied on transition probabilities without the latter actually having been derived. The
transition probabilities also have obvious appeal for econometric analyses, derivative pricing, and
decision making under the potential of “extinction”.
Keywords: Absorption, asset price equation, Brownian motion, stochastic process switching,
transition probability.