A Note on One-Dimensional Stochastic Equations

Abstract

We consider the stochastic equation $$X_t = x_0 + \int_0^t {b(u,X_u ){\text{d}}B_u ,{\text{ }}t \geqslant 0,} $$ where B is a one-dimensional Brownian motion, x 0∈ℝ is the initial value, and b [0,∞)×ℝ→ℝ is a time-dependent diffusion coefficient. While the existence of solutions is well-studied for only measurable diffusion coefficients b, beyond the homogeneous case there is no general result on the uniqueness in law of the solution. The purpose of the present note is to give conditions on b ensuring the existence as well as the uniqueness in law of the solution.