# Nonlinear Expectation, Nonlinear PDE and Stochastic Calculus under Knightian Uncertainty

# Nonlinear Expectation, Nonlinear PDE and Stochastic Calculus under Knightian Uncertainty

JOINT PACM COLLOQUIUM & ANALYSIS SEMINAR: A. N. Kolmogorov's "Foundations of the Theory of Probability" published in 1933, has established the modern axiomatic foundations of probability theory. Since then this theory has been profoundly developed and widely applied to situations where uncertainty cannot be neglected. But in 1921 Frank Knight has been already clearly classified two types of uncertainties: the first one is for which the probability is known; the second one, now called Knightian uncertainty, is for cases where the probability itself is also uncertain. The situation with Knightian uncertainty has become one of main concerns in the domain of data processing, economics, statistics, and specially in measuring and controlling financial risks. A long time challenging problem is how to establish a theoretical framework comparable to the Kolmogorov's one, to treat these more complicated situations with Knightian uncertainties. Tthe objective of the theory of nonlinear expectation rapidly developed in recent years is to solve this problem. This is an important program. Some fundamental results have been established such as law of large numbers, central limit theorem, martingales, G-Brownian motions, G-martingales and the corresponding stochastic calculus of Itˆo's type, nonlinear Markov processes, as well as the calculation of measures of risk in finance. But still so many deep problems are still to be explored. This new framework of nonlinear expectation is naturally and deeply linked to nonlinear partial differential equations (PDE) of parabolic and elliptic types. These PDEs appear in the law of large numbers, central limit theorem, and nonlinear diffusion processes in the new theory, and inversely, almost all solutions of linear, quasilinear and/or fully nonlinear PDEs can be expressed in term of the nonlinear expectation of a function of the corresponding (nonlinear) diffusion processes. Moreover, a new type of 'path-dependent partial differential equations' have been introduced which provide a PDE tool to study a stochastic process under a nonlinear expectation. Numerical calculations of these path dependent PDE will provide the corresponding backward stochastic calculations.