This book explores the critical area of approximating probabilistic characteristics in stochastic differential equations, focusing on elements such as probability density functions, hitting probabilities, and large deviation principles. These characteristics are fundamental for understanding the behavior of stochastic systems, which are prevalent in fields ranging from finance to physics and engineering. Despite extensive theoretical research on these probabilistic characteristics, the effects of numerical discretizations on them have not been thoroughly investigated. This gap is significant because accurate numerical approximations are essential for practical applications where analytical solutions are often unattainable.   This book addresses this need by examining discrete approximations that preserve the probabilistic characteristics of stochastic ordinary and partial differential equations. We delve into the probabilistic and asymptotic behaviors of key characteristics, including probability density functions, hitting probabilities, large deviations of invariant measures, and weak intermittency. By providing a deeper understanding of these elements, our work offers valuable insights for accurately modelling and analyzing complex stochastic systems, thereby advancing both theoretical knowledge and practical applications.   The topics covered in this book are essential to several research areas, including numerical analysis, stochastic calculus, large deviation theory, and probabilistic potential theory. This book offers valuable insights that will engage researchers interested in these fields, ultimately contributing to the advancement of both theoretical understanding and practical applications.