On explicit solutions to stochastic differential equations

This note is concerned with the study of explicit solutions to stochastic differential equations. previously Doss and Sussman showed that the unique strong solution to the scaler It (o) over cap equation X can be represented as a function rho of a Brownian motion and an auxiliary stochastic process Yr determined, for every path of {W-t(omega); t greater than or equal to 0}, by an ordinary differential equation (ODE). rho itself is determined by a second differential equation. Now, it will be shown that X can be solved explicitly as X-t = rho(x, t, integral(0)(t)(s)dW(s)). with f(.) being a continuous real valued function, provided (t,u) --> rho(x, t, u) solves a differential equation related to the one defining rho as well as a simple reaction-diffusion equation strongly In particular, for a given dispersion coefficient sigma(.), there will be a class of drift coefficients b(.) for which there is such an explicit solution. Examples illustrating how to find suitable drift b(.) are provided. The corresponding explicit solution X-t for any given dispersion sigma(.) is also supplied.