The solutions of the following bilinearstochastic differential equation are stud-ied (X) where Atk, Bt are (deterministic)continuous matrix-valued functions of t and w1(t),..., wm(t) are m independent standard Brownian motions. Conditions are given such thatthe solution is positive if the initial condition is positive.The equation the most probable path must satisfy is also derived and applied to a mathematicalfinance problem.