On solutions of one-dimensional stochastic differential equations without drift

Abstract

We consider the stochastic differential equation dX t =b(X t )dW t , t≧0, where b is a real-valued (universally) measurable function and W is a Wiener process. In the previous paper [8] the authors have shown that a nontrivial weak solution of this equation exists for all initial distributions if and only if b−2 is locally integrable. However, the uniqueness in law fails in general. In the present paper we give a complete description of all solutions by construction from a so-called fundamental solution. The fundamental solution has no sojourn time in the zeros of b and the general solution can be obtained from it by time delay in the zeros of b. Furthermore, some properties of solutions are investigated. Thus we characterize the set of all strong Markov solutions and a certain class of Markov solutions. We construct examples of Markov solutions which are not strong Markov. Finally, we study the representation property of solutions. In the Appendix, a few results on the time change of arbitrary strong Markov continuous local martingales and perfect additive functionals of them are collected. The basic method of the paper consists in a systematic use of random time change.