  This paper develops a new technique for the path approximation of
one-dimensional stochastic processes, more precisely the Brownian motion and
families of stochastic differential equations sharply linked to the Brownian
motion (usually known as L and G-classes). We are interested here in the
$\epsilon$-strong approximation. We propose an explicit and easy to implement
procedure that constructs jointly, the sequences of exit times and
corresponding exit positions of some well chosen domains. The main results
control the number of steps to cover a fixed time interval and the convergence
theorems for our scheme. We combine results on Brownian exit times from
time-depending domains (one-dimensional heat balls) and classical renewal
theory. Numerical examples and issues are also described in order to complete
the theoretical results.
