Mathematical problems in the theory of incomplete markets.

摘要

We study some mathematical problems that arise in studying the financial mathematics of incomplete markets. In the first chapter, we consider the pricing of derivative securities when markets are incomplete. In this case, the prices of all derivatives cannot be uniquely determined based on only the absence of arbitrage. We propose a method for determining the value of a derivative to a given investor based on the dual variables of the investor's utility maximization problem. We also derive results on the stability of these prices with respect to changes in the model inputs.; In the second chapter, we study some properties of the mean-reverting Ornstein-Uhlenbeck process that appears in many places in applied mathematics, and in particular in financial modelling. We study the estimation of the parameters of the model based on a constrained maximum likelihood problem (a hybrid of method of moments estimation and the method of maximum likelihood). We also derive the distribution of the range of the process over an interval.; The third and fourth chapters study the application of hidden Markov models to financial modelling. In particular, we study two dimensional stochastic differential equations driven by Brownian motion with one hidden variable. In the third chapter, we study the case where only the drift coefficient of the observed variable depends on the hidden variable, while in the fourth chapter we allow the diffusion coefficient of the observed variable to depend on the hidden variable. In each chapter, we study the problem under the following headings: marginal distributions of the observed process, estimation of the path of the hidden process based on observations, and estimation of the parameters of the hidden process.