One-dimensional stochastic differential equations involving a singular increasing process

Abstract

We study questions of existence and uniqueness for one-dimensional stochastic differential equations driven by a Brownian motion and an increasing process. It is shown that under fairly general conditions on the diffusion coefficient, if the drift coefficient is cadlag in x and has only positive jumps, then maximal and minimal strict solutions exist. If the drift coefficient has negative jumps, then the stochastic differential equation need not have a solution on any space. We give an example showing that the maximal and minimal solutions may be distinct as soon as the classical Lipschitz condition on the drift coefficient is weakened.