Backward Stochastic Differential Equation, Nonlinear Expectation and Their Applications

We give a survey of the developments in the theory of Backward Stochastic Differential Equations during the last 20 years, including the solutions' existence and uniqueness, comparison theorem, nonlinear Feynman-Kac formula, g-expectation and many other important results in BSDE theory and their applications to dynamic pricing and hedging in an incomplete financial market. We also present our new framework of nonlinear expectation and its applications to financial risk measures under uncertainty of probability distributions. The generalized form of law of large numbers and central limit theorem under sublinear expectation shows that the limit distribution is a sublinear G normal distribution. A new type of Brownian motion, G-Brownian motion, is constructed which is a continuous stochastic process with independent and stationary increments under a sublinear expectation (or a nonlinear expectation). The corresponding robust version of ItO's calculus turns out to be a basic tool for problems of risk measures in finance and, more general, for decision theory under uncertainty. We also discuss a type of "fully nonlinear" BSDE under nonlinear expectation.