STRONG APPROXIMATION OF SOME PARTICULAR ONE-DIMENSIONAL DIFFUSIONS

. We develop a new technique for the path approximation of onedimensional stochastic processes. Our results apply to the Brownian motion and to some families of stochastic differential equations whose distributions could be represented as a function of a time-changed Brownian motion (usually known as L and G-classes). We are interested in the epsilon-strong approximation. We propose an explicit and easy-to-implement procedure that jointly constructs, the sequences of exit times and corresponding exit positions of some well-chosen domains. In our main results, we prove the convergence of our scheme and how to control the number of steps, which depends on the covering of a fixed time interval by intervals of random sizes. The underlying idea of our analysis is to combine results on Brownian exit times from timedepending domains (one-dimensional heat balls) and classical renewal theory. Numerical examples and issues are also developed in order to complete the theoretical results.