Selected Publications<\/h4>\n\n\n\n Vieten, Martin G., and Stockbridge, Richard. “Convergence of Finite Element Methods for Singular Stochastic Control” SIAM Journal on Control and Optimization<\/em><\/a>56. (2018): 4336-4364. <\/div>\n Helmes, Kurt, Stockbridge, Richard, and Zhu, Chao. “Continuous Inventory Models of Diffusion Type: Long-term Average Cost Criterion” Annals of Applied Probability<\/em>27. (2017): 1831--1885. <\/div>\n Helmes, Kurt L., Stockbridge, Richard, and Zhu, Chao. “A Measure Approach for Continuous Inventory Models: Discounted Cost Criterion” SIAM Journal on Control and Optimization<\/em>53. (2015): 2100-2140. <\/div>\n Stockbridge, Richard. “Discussion of Dynamic Programming and Linear Programming Approaches to Stochastic Control and Optimal Stopping in Continuous Time” Metrika, Springer<\/em><\/a>77.1 (2013): 137-162. <\/div>\n Helmes, Kurt L., and Stockbridge, Richard. “Thinning and Harvesting of Stochastic Forest Models” Journal of Economic Dynamics and Control<\/em><\/a>35.1 (2011): 25-39. <\/div>\n Song, Qingshuo, Stockbridge, Richard, and Zhu, Chao. “On Optimal Harvesting Problems in Random Environments” SIAM Journal on Control and Optimization<\/em><\/a>49.2 (2011): 830--858. <\/div>\n 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<\/em><\/a>84.1 (2011): 55-84. <\/div>\n 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<\/em>42.1 (2010): 158-182. <\/div>\n <\/section>\n\n\n\n\nProf. Chao Zhu<\/h3>\n\n\n\n
Chao Zhu’s<\/a>\u00a0research 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 L\u00e9vy 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.<\/p>\n\n\n\nSelected Publications<\/h4>\n\n\n\n Xi, Fubao, and Zhu, Chao. “On Feller and Strong Feller Properties and Exponential Ergodicity of Regime-Switching Jump Diffusion Processes with Countable Regimes” SIAM J. Control and Optimization<\/em><\/a>55.3 (): 1789\u2013-1818. <\/div>\n Xi, Fubao, and Zhu, Chao. “Jump type stochastic differential equations with non-Lipschitz coefficients: Non-confluence, Feller and strong Feller properties, and exponential ergodicity” Journal of Differential Equations<\/em><\/a>(2018). <\/div>\n Xi, Fubao, and Zhu, Chao. “On the Martingale Problem and Feller and Strong Feller Properties for Weakly Coupled L\u00e9vy Type Operators” Stochastic Process. Appl<\/em><\/a>12.12 (2018). <\/div>\n Chao, Zhen, Wang, Kai, Zhu, Chao, and Zhu, Yanking. “Almost Sure and Moment Exponential Stability of Regime-Switching Jump Diffusions” SIAM J. Control and Optimization<\/em><\/a>55.6 (2017): 3458\u20133488. <\/div>\n Nguyen, Dang Hai, Yin, George, and Zhu, Chao. “Certain Properties Related to Well Posedness of Switching Diffusions” Stochastic Process. Appl<\/em><\/a>127.10 (2017): 3135-3158. <\/div>\n Chen, Xiaoshan, Huang, Yu-Jui, Song, Qingshuo, and Zhu, Chao. “The Stochastic Solution to a Cauchy Problem for Degenerate Parabolic Equations” Journal of Mathematical Analysis and Applications<\/em><\/a>451. (2017): 448-472. <\/div>\n Helmes, Kurt, Stockbridge, Richard, and Zhu, Chao. “Continuous Inventory Models of Diffusion Type: Long-term Average Cost Criterion” Annals of Applied Probability<\/em>27. (2017): 1831--1885. <\/div>\n Song, Qingshuo, and Zhu, Chao. “On Singular Control Problems with State Constraints and Regime-Switching: A Viscosity Solution Approach” Automatica<\/em><\/a>70. (2016): 66-73. <\/div>\n Weerasinghe, Ananda, and Zhu, Chao. “Optimal Inventory Control with Path-Dependent Cost Criteria” Stochastic Process. Appl<\/em><\/a>126. (2016): 1585-1621. <\/div>\n Zhu, Chao, Yin, George, and Baran, Nick. “Feynman-Kac formulas for regime-switching jump diffusions and their applications” Stochastics<\/em><\/a>87.6 (2015): 1000-1032. <\/div>\n Helmes, Kurt L., Stockbridge, Richard, and Zhu, Chao. “A Measure Approach for Continuous Inventory Models: Discounted Cost Criterion” SIAM Journal on Control and Optimization<\/em>53. (2015): 2100-2140. <\/div>\n Song, Qingshuo, Yin, George, and Zhu, Chao. “Optimal Switching with Constraints and Utility Maximization of an Indivisible Market” SIAM J. Control Optim<\/em><\/a>50.2 (2012): 629--651. <\/div>\n Zhu, Chao. “Optimal control of risk process in a regime-switching environment” Automatica<\/em><\/a>47.8 (2011): 1570--1579. <\/div>\n Song, Qingshuo, Stockbridge, Richard, and Zhu, Chao. “On Optimal Harvesting Problems in Random Environments” SIAM Journal on Control and Optimization<\/em><\/a>49.2 (2011): 830--858. <\/div>\n Yin, George, and Zhu, Chao. “Hybrid Switching Diffusions: Properties and Applications” Stochastic Modeling and Applied Probability<\/em><\/a> 63. Springer. (2010): xviii+395 pp. <\/div>\n <\/section>\n\n\n\n\nProbability at 51ÁÔÆæ<\/h2>\n\n\n\n
The Probability Faculty offer the courses Math 571 – Introduction to Probability Models<\/em><\/i>, Math 768 – Applied Stochastic Models<\/em><\/i>, Math 771 – Theory of Probability<\/em><\/i>, Math 873 – Advanced Topics in Probability<\/em><\/i> and participate in the teaching of MthStat 361 & 362 – Introduction to Mathematical Statistics I & II<\/em><\/i>.<\/p>\n\n\n\n\n- 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 \u201crandom experiments.\u201d<\/li>\n\n\n\n
- 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.<\/li>\n\n\n\n
- Math 768 examines basic Markov and other models from a more mathematically sophisticated point of view.<\/li>\n\n\n\n
- Math 771 develops the modern theory of probability using measure theory and provides the theoretical level appropriate for research in Probability.<\/li>\n\n\n\n
- 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.<\/li>\n\n\n\n
- 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:<\/li>\n<\/ul>\n\n\n\n
Prof. Chao Zhu<\/h3>\n\n\n\n
Chao Zhu’s<\/a>\u00a0research 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 L\u00e9vy 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.<\/p>\n\n\n\n The Probability Faculty offer the courses Math 571 – Introduction to Probability Models<\/em><\/i>, Math 768 – Applied Stochastic Models<\/em><\/i>, Math 771 – Theory of Probability<\/em><\/i>, Math 873 – Advanced Topics in Probability<\/em><\/i> and participate in the teaching of MthStat 361 & 362 – Introduction to Mathematical Statistics I & II<\/em><\/i>.<\/p>\n\n\n\nSelected Publications<\/h4>\n\n\n
Probability at 51ÁÔÆæ<\/h2>\n\n\n\n
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