Mathematics

The course introduces the basic theories and methods of stochastic analysis and its application in Finance. It discusses how to apply the basic theories and methods in stochastic analysis to financial pricing and derives the famous Black-Scholes formula. Other topics include Brownian motion, stochastic integral, and Ito formula. This course is taught in both English and Chinese.

The basic content of the course includes complex planes and extended complex planes, Cauchy-Riemann equations and holomorphic functions, fractional linear transformations and basic elementary functions, single-valued branches of complex integrals and multivalued functions, Cauchy integral theory, power series and Laurent levels Number, isolated singularity and residue theorem, biholomorphic mapping and Riemann mapping theorem, preliminary analysis and extension, etc.

The course describes how non-linear systems can be treated through analysis, simulation, and controller design. Lectures cover non-linear phenomena; mathematical modeling of nonlinear systems; stationary points; linearization around stationary points and trajectories; phase plane analysis; stability analysis using the Lyapunov method; circle criterion; small-gain and passivity; computer tools for simulation and analysis; effects of saturation; backlash and dead-zones in control loops; describing functions for analysis of limit cycles; high-gain methods and relay feedback; optimal control; and nonlinear synthesis and design. Laboratory exercises include analysis using the describing function and control design with dead-zone compensation for an air throttle used in car motors; energy-based design of a swing-up algorithm for an inverted pendulum; and trajectory generation using optimal control for the pendulum-on-a-cart process.

This course provides a study of the history and development of mathematics, from its origins to the present. Topics include: Math in ancient Mesopotamia and Egypt; math in ancient China and India; math in classical Greece; Hellenistic and Roman math; the Middle Ages in the Near East and Europe; the Renaissance; 17th century and mathematical revolution; 18th century-- calculus and mechanics; 19th and 20th centuries-- differential geometry and geometrization of space, algebra and algebraic geometry, foundations of math, foundations of analysis, foundations of probability.

This engineering mathematics course covers ordinary and series solutions of differential equations, vectors and vector space, matrix, determinants, linear systems, eigenvalues, diagonalization, system of linear differential equations, and Laplace transformation. 

This course introduces the basics of financial mathematics. Topics include basic mathematical theory of interest, term structure of interest rates, fixed income securities, risk aversion, basic utility theory, single-period portfolio optimization, basic option theory. Mathematical rigor is emphasized.