## Research Seminar

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.

## Research Interests

### Prof. Dick Stockbridge

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.

#### Selected Publications

*SIAM Journal on Control and Optimization*56. (2018): 4336-4364.

*Annals of Applied Probability*27. (2017): 1831--1885.

*SIAM Journal on Control and Optimization*53. (2015): 2100-2140.

*Metrika, Springer*77.1 (2013): 137-162.

*Stochastics: An International Journal of Probability and Stochastic Processes*84.1 (2011): 55-84.

*SIAM Journal on Control and Optimization*49.2 (2011): 830--858.

*Journal of Economic Dynamics and Control*35.1 (2011): 25-39.

*Advances in Applied Probability*42.1 (2010): 158-182.

### Prof. Wei Wei

Wei Wei’s research interests focus on modeling dependence structures and stochastic comparisons. In particular he uses these tools to model risks in the fields of finance and actuarial science, and study risk management and optimization problems. He is also interested in ruin analysis of insurance companies, which is essentially exiting problems of a special class of stochastic processes. His recent work includes comparison of dispersion between multivariate random vectors and capital allocation principles based on ruin-related criteria.

#### Selected Publications

*ASTIN Bulletin: The Journal of IAA*48.3 (2018): 1025-1047.

*Risks*6. (2018): 49.

*Insurance: Mathematics and Economics*76. (2017): 141-148.

*North American Actuarial Journal*. (2017).

*Annals of Actuarial Science*. (2016): 1-16.

*Insurance: Mathematics and Economics*68. (2016): 84-91.

*Journal of Multivariate Analysis*138. (2015): 156-169.

*Insurance: Mathematics and Economics*55.1 (2014): 200-209.

*Insurance: Mathematics and Economics*50.1 (2012): 43-49.

*Insurance: Mathematics and Economics*50.1 (2012): 57-63.

### Prof. Chao Zhu

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.

#### Selected Publications

*SIAM J. Control and Optimization*55.3 1789â-1818.

*Journal of Differential Equations*. (2018).

*Stochastic Process. Appl.*12.12 (2018).

*SIAM J. Control and Optimization*55.6 (2017): 3458â3488.

*Annals of Applied Probability*27. (2017): 1831--1885.

*Stochastic Process. Appl.*127.10 (2017): 3135-3158.

*Journal of Mathematical Analysis and Applications*451. (2017): 448-472.

*Automatica*70. (2016): 66-73.

*Stochastic Process. Appl.*126. (2016): 1585-1621.

*SIAM Journal on Control and Optimization*53. (2015): 2100-2140.

*Stochastics*87.6 (2015): 1000-1032.

*SIAM J. Control Optim.*50.2 (2012): 629--651.

*Automatica*47.8 (2011): 1570--1579.

*SIAM Journal on Control and Optimization*49.2 (2011): 830--858.

*Stochastic Modeling and Applied Probability*63. Springer, (2010): xviii+395 pp.

## Probability at UWM

The Probability Faculty offer the courses Math 571 – * Introduction to Probability Models*, Math 768 –

*, Math 771 –*

*Applied Stochastic Models**, Math 873 –*

*Theory of Probability**and participate in the teaching of MthStat 361 & 362 –*

*Advanced Topics in Probability**.*

*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 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.
- 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 771 – * Theory of Probability* is offered in the Spring Semester so that students may take Math 711 –

*in the preceding Fall semester so as to have the necessary prerequisites.*

*Real Analysis*