Probability

Probability & Stochastic Analysis Research Group

Richard Stockbridge
Professor
Mathematical Sciences - Generalstockbri@uwm.edu(414) 229-3947Eng & Math Sciences W447
Wei Wei
Assistant Professor
Mathematical Sciences - Generalweiw@uwm.edu(414) 229-4600Eng & Math Sciences E455
Chao Zhu
Associate Professor
Mathematical Sciences - Generalzhu@uwm.edu(414) 229-3528Eng & Math Sciences E489

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

Helmes, Kurt, Stockbridge, Richard, and Zhu, Chao. “Continuous Inventory Models of Diffusion Type: Long-term Average Cost Criterion.” Annals of Applied Probability. 46 pages.
Helmes, Kurt L., Stockbridge, Richard, and Zhu, Chao. “A Measure Approach for Continuous Inventory Models: Discounted Cost Criterion.” SIAM Journal on Control and Optimization 53. (2015): 2100-2140.
Dufour, Francois, and Stockbridge, Richard. “On the Existence of Strict Optimal Controls for Constrained, Controlled Markov Processes in Continuous-Time.” Stochastics: An International Journal of Probability and Stochastic Processes 84.1 (2011): 55-84.
Song, Qingshuo, Stockbridge, Richard, and Zhu, Chao. “On Optimal Harvesting Problems in Random Environments.” SIAM Journal on Control and Optimization 49.2 (2011): 830--858.
Helmes, Kurt L., and Stockbridge, Richard. “Thinning and Harvesting of Stochastic Forest Models.” Journal of Economic Dynamics and Control 35.1 (2011): 25-39.
Helmes, Kurt L., and Stockbridge, Richard. “Construction of the Value Function and Stopping Rules for Optimal Stopping of One-Dimensional Diffusions.” 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

Samanthi, Ranadeera G., Brazauskas, Vytaras, and Wei, Wei. “Comparing the Riskiness of Dependent Portfolios via Nested L-Statistics.” Annals of Actuarial Science.
Cai, Jun, Landriault, David, Shi, Tianxiang, and Wei, Wei. “Joint insolvency analysis of a shared MAP risk process: a capital allocation application.” North American Actuarial Journal.
Samanthi, Ranadeera G., Wei, Wei, and Brazauskas, Vytaras. “Ordering Gini indexes of multivariate elliptical risks.” Insurance: Mathematics and Economics 68. (2016): 84-91.
Cai, Jun, and Wei, Wei. “Notions of multivariate dependence with applications in optimal portfolio selections.” Journal of Multivariate Analysis 138. (2015): 156-169.
Cai, Jun, and Wei, Wei. “Some new notions of dependence with applications in optimal allocation problems.” Insurance: Mathematics and Economics 55.1 (2014): 200-209.
Cai, Jun, and Wei, Wei. “Optimal reinsurance with positively dependent risks.” Insurance: Mathematics and Economics 50.1 (2012): 57-63.
Cai, Jun, and Wei, Wei. “On the invariant properties of notions of positive dependence and copulas under increasing transformations.” Insurance: Mathematics and Economics 50.1 (2012): 43-49.

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

Helmes, Kurt, Stockbridge, Richard, and Zhu, Chao. “Continuous Inventory Models of Diffusion Type: Long-term Average Cost Criterion.” Annals of Applied Probability. 46 pages.
Weerasinghe, Ananda, and Zhu, Chao. “Optimal Inventory Control with Path-Dependent Cost Criteria.” Stochastic Process. Appl. 126. (2016): 1585-1621.
Helmes, Kurt L., Stockbridge, Richard, and Zhu, Chao. “A Measure Approach for Continuous Inventory Models: Discounted Cost Criterion.” SIAM Journal on Control and Optimization 53. (2015): 2100-2140.
Zhu, Chao, Yin, George, and Baran, Nick. “Feynman-Kac formulas for regime-switching jump diffusions and their applications.” Stochastics 87.6 (2015): 1000-1032.
Feng, Runhuan, Zhang, Shuaiqi, Volkmer, Hans W., and Zhu, Chao. “Optimal Dividend Payments for the Piecewise Deterministic Poisson Risk Model.” Scandinavian Actuarial Journal 5. (2015): 423--454.
Song, Qingshuo, Yin, George, and Zhu, Chao. “Optimal Switching with Constraints and Utility Maximization of an Indivisible Market.” SIAM J. Control Optim. 50.2 (2012): 629--651.
Song, Qingshuo, Stockbridge, Richard, and Zhu, Chao. “On Optimal Harvesting Problems in Random Environments.” SIAM Journal on Control and Optimization 49.2 (2011): 830--858.
Yin, George, and Zhu, Chao. “Hybrid Switching Diffusions: Properties and Applications.” Stochastic Modeling and Applied Probability 63. Springer, (2010): xviii+395 pp.

Graduate Students

Current Graduate Students

Probability Graduate Students

Charles Beer
Teaching Assistant
Mathematical Sciences - Generalcbeer@uwm.eduEng & Math Sciences E480
Nyles Breecher
Teaching Assistant
Mathematical Sciences - Generalbreecher@uwm.eduEng & Math Sciences W423
Samuel Nehls
Teaching Assistant
Mathematical Sciences - Generalsmnehls@uwm.eduEng & Math Sciences E499
Martin Vieten
Teaching Assistant
Mathematical Sciences - Generalmgvieten@uwm.eduEng & Math Sciences W423
Sheng Wang
Teaching Assistant
Mathematical Sciences - Generalwang297@uwm.eduEng & Math Sciences E450

Past Students

Student Archive

Probability at UWM

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.”
  • 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 – Real Analysis in the preceding Fall semester so as to have the necessary prerequisites.