On the time a diffusion process spends along a line

Abstract

For an arbitrary diffusion process X with time-homogeneous drift and variance parameters μ( x) and σ 2( x), let V ε be 1 ε times the total time X( t) spends in the strip [a + bt − 1 2 ε, a + bt + 1 2 ε] . The limit V as ε → 0 is the full halfline version of the local time of X( t) − a − bt at zero, and can be thought of as the time X spends along the straight line x = a + bt. We prove that V is either infinite with probability 1 or distributed as a mixture of an exponential and a unit point mass at zero, and we give formulae for the parameters of this distribution in terms of μ(·), σ(·), a, b, and the starting point X(0). The special case of a Brownian motion is studied in more detail, leading in particular to a full process V( b) with continuous sample paths and exponentially distributed marginals. This construction leads to new families of bivariate and multivariate exponential distributions. Truncated versions of such ‘total relative time’ variables are also studied. A relation is pointed out to a second order asymptotics problem in statistical estimation theory, recently investigated in Hjort and Fenstad (1992a, 1992b).