On stochastic differential equations without drift

Abstract

Consider a one-dimensional stochastic differential equation without drift. In Section 3 we give a necessary and sufficient condition for the existence of a solution with initial value 0. In Section 4 we prove a criterion for the existence of a so called fundamental solution with initial value 0. It will be shown that this fundamental solution is unique in law. This and the fact that under certain conditions any solution is a fundamental solution of an equation with another diffusion coefficient will be used to prove a criterion for the uniqueness in law of a solution

Keywords:

• Brownian motion
• continuous martingales
• stochastic differential equation
• existence
• uniqueness
• time change