The Probability Research Group

Faculty are currently interested in problems arising from Probability, Stochastic Analysis and their applications. The members of the group are:

• Runhuan Feng (University of Waterloo 2008)
  
• Eric Key (Cornell University 1983)
  
• Richard Stockbridge (University of Wisconsin- Madison 1987)
  
• Chao Zhu (Wayne State University 2007)
  

Runhuan Feng's research interests involve stochastic modeling and analysis of insurance and financial risks arising from actuarial applications. His main research area, ruin theory, at the interface of actuarial science and applied probability, is concerned with quantifying and assessing insolvency risks associated with the insurance business. His recent work has garnered particular attention to operator-based analytical approaches that provide a unified treatment for a wide variety of default or ruin-related quantities. He also has keen interests in modeling and valuation of embedded financial derivatives in combined investment-insurance products in the rapidly developing industry.

Eric Key's work lies at the intersection of classical probability, linear algebra and analysis. Recent investigations include copulas and optimal design for sampling from Gaussian processes.

Dick Stockbridge's interests lie in the area of optimal stopping, control and application of continuous-time stochastic processes. He investigates the solution of such problems by use of an imbedding as a linear program over a space of measures representing the expected occupation measure(s) arising from the processes. Recent work has concentrated on singular stochastic control problems and their applications and on various methods for numerical approximation of the solutions, including the use of moments to characterize the measures and the approximation of the measures using finite elements to determine approximating densities. A variety of applications have been considered including the pricing of options and other topics in financial mathematics, optimal harvesting policies, and investment and disinvestment problems.

Chao Zhu's research has been focused on stochastic analysis and stochastic control. In particular, he is interested in continuous time stochastic processes such as regime switching diffusions with jumps and Levy processes. He studies the long time behavior of such processes and their applications in ecosystem modeling and mathematical finance. He is also interested in stochastic control problems rising from areas like optimal harvesting, finance, and risk management.

The Probability Group runs an active seminar that meets twice per week. Topics range from discussion of well-known results that are not typically covered in classes to presentation of current research. 

Six students have completed their PHDs in Probability and related areas:

• G.A. Rus. August, 2009. Finite Element Methods for Control of Singular Stochastic Processes. (Richard Stockbridge and Bruce Wade)
• K. Zaglauer. August 2009. Fair Pricing of Participating Life Insurance Contracts in a Regime-Switching Market Environment.  (Richard Stockbridge)
• H. He. August 2008. Utility Maximization of a Portfolio that Includes an Illiquid Asset. (Richard Stockbridge)
• G. Vachadeze. May 2003. Finite Mixture Models and Their Applications in Finance. (Eric Key)
• S. Karmarkar. August 1992. Compositions of Random Moebius Transformations and Their Applications. (Eric Key)
• A. Abeyratne. May 1991. Limiting Distributions for Multitype Branching Processes with Immigrations in a Random Environment. (Eric Key)

This group has been very active in directing MS students as well, with over 20 theses completed since 2001. Those completing since 2005 are:

• J. Eisenmann (2010)
• M. Groeger (2010)
• H.P. Steiner (2010)
• F. Thieme (2010)
• X. Xie (2010)
• L. Zhuo (2010)
• P. Heinrich (2009)
• C. Rolser (2009)
• Y. Sheng (2009)
• F. Hauessler (2008)
• K. Jensen (2008)
• A. Klein (2008)
• S. Rieger (2008)
• E. Schumacher (2008)
• W. Knoke (2007)
• A. Linder (2007)
• M. Lutz (2007)
• G. Schreier (2007)
• P. Kaczmarek (2006)
• P. Orth (2006)
• A. Glaser (2005)
• R. Reeb (2005)
• M. Schneider (2005)
• L. Zettler (2005)

Courses in Probability

The Probability Faculty offer the courses Math 571 Introduction to Probability Models, Math768 Applied Stochastic Models, Math 771 Theory of Probability, Math 873 Advanced Topics in Probability and participate in the teaching of Mthstat 361,362 Introduction to Mathematical Statistics I, II. Mthstat 361 provides an introduction to Probability at the undergraduate level and serves as a prerequisite for Math 571; it examines the basic theory concerning discrete and continuous probability distributions and one and two random variables representing the outcomes of a single or two random "experiments." Math 571 develops basic Markov models for phenomena that evolve in time and are subject to random influences, and investigates the probabilistic behavior of these models. Math 768 examines these and other models from a more mathematically sophisticated point of view. Math 771 develops the modern theory of probability using measure theory and provides the theoretical level appropriate for research in Probability. Math 873 is typically a continuation of Math 771 in which the fundamental results in Probability are completed (1/4 to 1/3 semester) and then topics of interest to the students and/or the instructor are discussed. When a sufficient number of students are available, additional Math 873 Topics courses may be offered with changes in topics. Math 771/873 may be used for the PhD Preliminary Examination in the Probability and Statistics area. These courses are offered on a two-year rotation as follows:  

 

Fall Year 1        Spring Year 1        Fall Year 2         Spring Year 2

Mthstat 361          Math571              Mthstat 361            Math 571

Math 771              Math 873             Math 768