One-Dimensional Homogeneous Diffusions

Abstract

When constructing a model defined by a stochastic differential equation (SDE) the basic problem is whether the equation has a solution and if so, when an initial condition is given, whether the solution is unique. Once the existence and uniqueness of the solution has been established so that the model is well-defined one may then proceed to study specific properties of the solution such as its long term behaviour, stationarity and the form of the invariant distribution, boundedness or positivity and whatever other properties are needed for the problem at hand. The solution to an SDE is a stochastic process, i.e, a randomly generated function of time so that formally the solution may be viewed as a typically huge collection of ordinary functions of time. It is this that makes SDEs much more difficult to deal with than ordinary differential equations where a unique solution is just one function of time.