An introduction to general theories of stochastic processes and modern martingale theory. The volume focuses on consistency, stability and contractivity under geometric invariance in numerical analysis, and discusses problems related to implementation, simulation, variable step size algorithms, and random number generation.

Markov processes and their applications; semimartingale theory and stochastic calculus; white noise theory; stochastic differential equations and its applications; large deviations and applications; a brief introduction to numerical analysis of (ordinary) stochastic differential equations without tears; stochastic differential games and applications; stability and stabilizing control of stochastic systems; stochastic approximation - theory and applications; stochastic manufacturing systems; optimization by stochastic methods; stochastic control methods in asset pricing.