The density function of the solution of a two-point boundary value problem containing small stochastic processes

Abstract

This paper concerns a two-point boundary value problem for an m m thorder system of ordinary differential equations containing a vector stochastic process \[ ξ ( t , ω ) = ξ 0 ( t ) + ε ξ 1 ( t , ω ) + ε 2 ξ 2 ( t , ω ) + ⋅ ⋅ ⋅ . \xi \left ( {t, \omega } \right ) = {\xi _0}\left ( t \right ) + \varepsilon {\xi _1}\left ( {t, \omega } \right ) + {\varepsilon ^2}{\xi _2}\left ( {t, \omega } \right ) + \cdot \cdot \cdot . \] When ε \varepsilon is small, the existence and the asymptotic properties of the solution can be obtained by means of the shooting method, and its density function can be determined by solving a sequence of first-order deterministic partial differential equations.