A note on the distribution of integrals of geometric Brownian motion

The purpose of this note is to identify an interesting and surprising duality between the equations governing the probability distribution and expected value functional of the stochastic process defined by A_t := ∫^t₀ exp{Z_s} ds, t ≥ 0, where {Z_s: s ≥ 0} is a one-dimensional Brownian motion with drift coefficient μ and diffusion coefficient σ². In particular, both expected values of the form v(t, x) := Ef(x+A_t), f homogeneous, as well as the probability density a(t, y) dy := P(A_t ∈ dy) are shown to be governed by a pair of linear parabolic partial differential equations. Although the equations are not the backward/forward adjoint pairs one would naturally have in the general theory of Markov processes, unifying and remarkably simple derivations of these equations are provided.