A note on the inhomogeneous linear stochastic differential equation

The inhomogeneous linear SDE X = C + ∫ 0 + X -   d R , where X and C are càdlàg processes and R is a semimartingale, is solved. We give the solution in a “nice” form, which is also more general than that of Yoeurp and Yor [Espace orthogonal à une semi-martingale, Unpublished, 1977]. This SDE has a very natural interpretation in finance. If C is a stochastic cash flow and R is the return process of a money market account (that is, N t = N 0 ɛ ( R ) t is the value of the money market account at time t), then the solution Xt is the time-t value of the cash flow C accumulated in the money market account (at the stochastic interest “rate” dR) over the time interval [0,t].