Mathematics Colloquium
Jin Ma
USC
Title:New Developments in Set-Valued Stochastic Analysis and Applications
Date: Friday, April 5, 2024
Place and Time: Love 101, 3:05-3:55 pm
Abstract. Abstract/Desc: Set-valued functions have been used extensively in various fields including optimization, control theory, economics, and finance. The concepts of set-valued random variable and set-valued stochastic processes, appear also naturally in measuring risks of multivariate positions in finance. However, due to some fundamental differences between the (metric) space of sets and that of vectors, there are many important and delicate issues in vector-valued stochastic analysis still cannot find the appropriate counterparts in the set- valued cases; and many natural questions regarding the set-valued processes (especially set-valued martingales), albeit elementary, remain unanswered in the current literature. In this talk I present some new results that answer some of the questions that are beyond the existing theory. These particularly include the path- regularity and martingale properties of a new set-valued stochastic integral that has some fundamental differences from the well-known Aumann-Ito integral, and is more suitable for representing set-valued martingales, hence potentially useful in the study of set-valued backward SDEs. Similar to the Aumann-Ito integral, the new integral is only a set-valued submartingale in general, we shall provide sufficient conditions under with a set-valued stochastic integral yields a martingale. We shall also have a close look at the set-valued martingale in the form of the conditional expectation of a set-valued random variable, especially in the case of a random polytope. Finally, I will talk about some new developments in set-valued stochastic analysis on a special space of unbounded sets, and the potential applications in set-valued dynamic risk measures and super-hedging problems in finance. This talk is based on the joint works with Cagin Ararat, Wenqian Wu, and Atiqah Almuzaini