This project develops a novel class of analytical and computational methods based on spectral analysis to solve first passage and optimal stopping problems for a class of Markov processes. The objective of an optimal stopping problem is to determine optimal decision timing to maximize reward in the face of uncertainty modeled by a stochastic process. The objective of a first passage problem is to determine the probability distribution of the first time a stochastic process passes through a boundary. These mathematical problems arise in a wide variety of applications in financial mathematics, including modeling credit risk (the risk of default of a borrower on its debt, such as a corporate bond), evaluating financial contracts with early exercise rights, such as American-style options and callable and convertible bonds, and real options. The class of Markov processes under study are jump-diffusion processes that can be constructed by stochastic time changes of one-dimensional diffusions. The methods developed in this project are based on representing conditional expectation operators associated with Markov processes by eigenfunction expansions. Efficient computational algorithms will be developed for Markov processes whose eigenfunctions are expressed in terms of orthogonal polynomials. Many of the most important stochastic processes in financial mathematics belong to this category, including the Ornstein-Uhlenbeck, Cox-Ingersoll-Ross, and constant elasticity of variance (CEV) diffusions, as well as pure jump and jump-diffusion processes arising from time changing these diffusions.
The novel analytical methods and computational algorithms developed in this project will be applied to a range of problems in financial mathematics in the areas of modeling credit risk and evaluating financial contracts with early exercise rights in a variety of markets, including bond markets, equity markets, commodities and energy markets, and to real options and irreversible investment decisions. The mathematical methods developed in this project will help financial institutions to accurately evaluate and manage the risk of a variety of financial transactions. They will also help non-financial firms make better managerial decisions by facilitating applications of real options analysis. The project is expected to have a broader mathematical impact on research on optimal stopping and first passage problems. The project will also have an impact on education and human resources development. It is part of the long-term effort at Northwestern in financial mathematics and engineering, including the Ph.D. concentration in this area. It will train highly qualified researchers for academia and industry.