Tao Chen Ph.D. Thesis Defense

Dynamic Conic Finance Via Backward Stochastic Difference Equations and Recursive Construction of Confidence Regions

This thesis consists of two major parts, and it contributes to the fields of mathematical finance and statistics.

The contribution to mathematical finance is made via developing new theoretical results in the area of conic finance. Specifically, it has advanced dynamic aspects of conic finance by developing arbitrage free theoretical framework for modeling bid and ask prices of dividend paying securities using the theory of dynamic acceptability indices. This has been done within the framework of general probability spaces and discrete time. In the process, it has progressed the theory of dynamic subscale invariant performance measures. In particular, Chen proved a representation theorem of such measures in terms of a family of dynamic convex risk measures, and provided a representation of dynamic risk measures in terms of BS∆Es.

The contribution to statistics is of fundamental importance as it initiates the theory underlying recursive computation of confidence regions for finite dimensional parameters in the context of stochastic dynamical systems. In the field of engineering, particularly in the field of control engineering, the area of recursive point estimation came to the great prominence in the last forty years. However, there has been no work done with regard to recursive computation of confidence regions. To partially fill this gap, the second part of the thesis is devoted to recursive construction of confidence regions for parameter characterizing one-step transition kernel of a time-homogeneous Markov chain.