A theory of two-point boundary value problems analogousto the theory of initial value problems for stochastic ordinarydifferential equations whose solutions form Markov processes isdeveloped. The theory of initial value problems consists ofthree main parts: the proof that the solution process ismarkovian and diffusive; the construction of the Kolmogorovor Fokker-Planck equation of the process; and the proof thatthe transistion probability density of the process is a uniquesolution of the Fokker-Planck equation.
It is assumed here that the stochastic differential equationunder consideration has, as an initial value problem, a diffusivemarkovian solution process. When a given boundary value problemfor this stochastic equation almost surely has unique solutions,we show that the solution process of the boundary value problemis also a diffusive Markov process. Since a boundary valueproblem, unlike an initial value problem, has no preferreddirection for the parameter set, we find that there are twoFokker-Planck equations, one for each direction. It is shownthat the density of the solution process of the boundary valueproblem is the unique simultaneous solution of this pair ofFokker-Planck equations.
This theory is then applied to the problem of a vibratingstring with stochastic density.
