An Example of Non-Uniqueness of the Solution of the Stochastic Equation of K. Ito

In this article it is proved that the stochastic integral equation \[ \kappa _j ( \omega ) = \kappa _s ( \omega ) + \int_s^t {\left| {\kappa _k ( \omega )} \right|^\alpha } d\xi _u ( \omega ) \] has many solutions for $0 < \alpha < \tfrac{1}{2}$.