The LASSO is a widely used L^2 statistical methodology for simultaneous estimation and variable selection. In the last years, many authors analyzed this technique from a theoretical and applied point of view. In the first part of the seminar, we introduce and study the adaptive LASSO problem for discretely observed ergodic diffusion processes We prove oracle properties also deriving the asymptotic distribution of the LASSO estimator. In the second part of the seminar we present general L^p approach for stochastic differential equations with small diffusion noise. Finally, we present simulated and real data analysis to provide some evidence on the applicability of this method.