The Probability Research Group

 

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

·  Eric Key (Cornell University 1983)

·  Richard Stockbridge (University of Wisconsin –Madison 1987)

·  Chao Zhu (Wayne State University 2007)

 
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 applications 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 impulse and 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 focuses on stochastic analysis and stochastic control. In particular, he is interested in continuous time stochastic processes such as regime switching diffusions with jumps and Lévy 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 arising in 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.

 

Seven students have completed their PhDs in Probability and related areas:

- J. Wood. May, 2012. Modeling Processes with Heavy Tails. (Eric Key)

- 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. Vachadze. 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)


Current and former members of the Probability group have been very active in directing MS students as well, with over 35 theses completed since 2001.Those completing since 2005 are: 


 

v  T. Bielert (2013)

v  M. Wahl (2013)

v  S. Klein (2012)

v  S. Schreiber (2012)

v  M. Wagner (2012)

v  Z. Wang (2012)

v  D. Bhatt (2011)

v  J. Bluhm (2011)

v  M. Serbiné (2011)

v  J. Wirfs (2011)

v  J. Eisenmann (2010)

v  M. Groeger (2010)

v  H.P. Steiner (2010)

v  F. Thieme (2010)

v  X. Xie (2010)

v  L. Zhuo (2010)

v  P. Heinrich (2009)

v  C. Rolser (2009)

v  Y. Sheng (2009)

v  F. Hauessler (2008)

v  K. Jensen (2008)

v  A. Klein (2008)

v  S. Rieger (2008)

v  E. Schumacher (2008)

v  W. Knoke (2007)

v  A. Linder (2007)

v  M. Lutz (2007)

v  G. Schreier (2007)

v  P. Kaczmarek (2006)

v  P. Orth (2006)

v  A. Glaser (2005)

v  R. Reeb (2005)

v  M. Schneider (2005)

v  L. Zettler (2005)

  

Courses in Probability

 

The Probability Faculty offer the courses Math 571 Introduction to Probability Models, Math 768 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.”

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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 basic Markov 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.

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Math 873 is typically a continuation of Math 771 in which the fundamental results in Probability are completed (1/4 to 1/3 of the 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 768/771 or 771/873 may be used for the PhD Preliminary Examination in the Probability and Statistics area. The Probability courses are offered in the following semesters:

 

Fall Semester

Spring Semester

Mthstat 361

X

X

Math 571

 

X

Math 768

X

 

Math 771

 

X

Math 873

X

 

 

In particular, Math 771Theory of Probability is offered in the Spring Semester so that sudents 
may take Math 711  Real Analysis in the preceding Fall semester so as to have the necessary prerequisites.