A wide variety of problems in engineering and applied science can be formulated as continuous-time stochastic processes, i.e., differential equations with noisy forcing. Noise also enters prominently in information theory, which is concerned with communication, statistical estimation, and the extraction of patterns in the presence of noise. The topic of Lie groups also enters into engineering and the physical sciences in a variety of ways. For example, problems involving the motion of rigid bodies are naturally described using methods from differential geometry and the theory of Lie groups. The subjects of stochastic processes, information theory, and Lie groups are usually treated separately from each other. This unique textbook presents these topics in a unified setting, thereby building bridges between fields that are rarely studied by the same individuals. Volume I establishes the geometric and statistical foundations required to understand the fundamentals of continuous-time stochastic processes, differential geometry, and the statistical branch of information theory. Volume 2 delves into deeper relationships between these topics, including stochastic geometry, geometrical aspects of the theory of communications and coding, multivariate statistical analysis, and error propagation on Lie groups. Key features and topics of Volume I: * Includes a review of stochastic processes and basic differential geometry in an accessible way for applied mathematicians, scientists, and engineers. * Extensive exercises make the work suitable as a textbook for use in graduate courses that emphasize applied stochastic processes or differential geometry. * The concept of Lie groups as continuous sets of symmetry operations is introduced. * The Fokker–Planck Equation for diffusion processes in Euclidean space and on differentiable manifolds is derived in a way that can be understood by nonspecialists. * Concrete presentation, making it easy for readers to obtain numerical solutions for their own problems. Stochastic Models, Information Theory, and Lie Groups will be of interest to graduate students and researchers working in applied mathematics, the sciences, and engineering.