The research outcome of the stochastics group provides modeling tools for analysis, control, and numerical study of various stochastic systems that evolve in time and space, and are subject to randomness. Our study of structured dependence between stochastic processes helps to construct models of multivariate random dynamical systems with prescribed global structural features and prescribed marginal structural features. Random sequence comparison helps scientists to identify regions of similarity in the sequences of DNA, RNA, and proteins, or between strings in a natural language. Stochastic partial differential equations and stochastic dynamical systems serve as modeling tools for complex phenomena such as turbulent flows, climate change, and behavior of financial markets. Our research in the area of mathematical finance provides quantitative models of financial securities that allow pricing, hedging, and mitigating the risk of complex financial products.
Faculty with primary interests in stochastics
Faculty with secondary interests in stochastics
- Yue Cao
- Yu-Sin Chang
- Zhuan Cheng
- Ziteng Cheng
- Shibi Feng
- Xiao Huang
- Yicong Huang
- Senbao Jiang
- Kan Zhang
RECENT RESEARCH GRANTS
- NSF DMS-1620449 (PI X. Li and Co-PI J. Duan): Theoretical and Numerical Studies of Nonlocal Equations Derived from Stochastic Differential Equations with Lévy Noises, 2016-2020.
- NSF DMS-1642545 (PI J. Duan and Co-PI X. Li): CBMS Conference: Nonlocal Dynamics — Theory, Computation and Applications, 2017-2018.
- NSF DMS-1522687 (PI F. J. Hickernell and Co-PI G. E. Fasshauer): Stable, Efficient, Adaptive Algorithms for Approximation and Integration, 2015–2018.
- Fermilab (PI F. J. Hickernell): Modern Monte Carlo Methods for High Energy Event Simulation, Parts I, II, 2015.
- NSF DMS-1211256 (PI T. R. Bielecki and Co-PI I. Cialenco): Topics in Stochastic Processes and Mathematical Finance: Counterparty Risk Valuation and Hedging, Markov Consistency and Markov Copulas, and Dynamic Performance Assessment Indices, 2012-2015.
- CME Group (PIs I. Cialenco and T. R. Bielecki): CDS and CDX series pricing factor analysis, 2014.
- NSF DMS-1115392 (PI F. J. Hickernell and Co-PI G. E. Fasshauer): Kernel Methods for Numerical Computation, 2011–2014.
- NSF DMS-1025422 (PI J. Duan): CMG Collaborative Research: Mathematical Modeling by Bridging Primitive and Boussinesq Equations, 2010-2014.
- T. R. Bielecki, I. Cialenco, R. Gong, and Y. Huang. Wiener-Hopf Factorization for Time-Inhomogeneous Markov Chains and Its Application. Submitted, 2018. arXiv:1801.05553
- T. R. Bielecki, T. Chen, I. Cialenco, A. Cousin, and M. Jeanblanc. Adaptive Robust Control under Model Uncertainty. Submitted, 2017. arXiv:1706.02227
- T. R. Bielecki, I. Cialenco, and M. Rutkowski. Arbitrage-Free Pricing of Derivatives in Nonlinear Market Models. Submitted, 2017. arXiv:1701.08399
- Z. Chen, Z. Chen, L. X. Cai, Y. Cheng, and R. Gong. Performance Analysis of Energy Harvesting in Wireless Networks Using Stochastic Geometry. Submitted, 2017.
- I. Cialenco. Statistical Inference for SPDEs: an Overview. Submitted, 2017. arXiv:1712.05445
- I. Cialenco and Y. Huang. A Note on Parameter Estimation for Discretely Sampled SPDEs. Submitted, 2017. arXiv:1710.01649
- R. Gong, C. Mou, and A. Swiech. Stochastic Representations for Solutions to Nonlocal Bellman Equations. Submitted, 2017. arXiv:1709.00193
- R. Gong and C. Houdré. A Viscosity Solution to a Principal-Agent Problem. Submitted, 2016. arXiv:0911.0956
- R. Gong, C. Houdré, and Ü. Islak. A Central Limit Theorem for the Optimal Alignments Score in Multiple Random Words. Submitted, 2016. arXiv:1512.05699
- P. M.N. Feehan, R. Gong, and J. Song. Feynman-Kac Formulas for Solutions to Degenerate Elliptic and Parabolic Boundary-Value and Obstacle Problems with Dirichlet Boundary Conditions. Preprint, 2015. arXiv:1509.03864
- T. R. Bielecki, I. Cialenco, and M. Pitera. A Unified Approach to Time Consistency of Dynamic Risk Measures and Dynamic Performance Measures in Discrete Time. To appear in Mathematics of Operations Research, 2018.
- R. Gong, C. Houdré, and J. Lember. Lower Bounds on the Generalized Central Moments of the Optimal Alignments Score of Random Sequences. To appear in Journal of Theoretical Probability, 2017. DOI:10.1007/s10959-016-0730-4
- T. R. Bielecki, M. Jeanblanc, and D. Sezer. Joint Hitting-Time Densities for Finite State Markov Processes. Turkish Journal of Mathematics, Vol. 42, No. 2, pp. 586-608, 2018.
- I. Cialenco, R. Gong, Y. Huang. Trajectory Fitting Estimators for SPDEs Driven by Additive Noise. Statistical Inference for Stochastic Processes, Vol 21, Issue 1, pp. 1-19, 2018.
- J. E. Figueroa-López, R. Gong, and C. Houdré. Third-Order Short-Time Expansions for Close-to-the-Money Option Prices under the CGMY Model. Applied Mathematical Finance, Vol. 24, No. 6, pp. 547-574, 2018.
- J. E. Figueroa-López, R. Gong, and M. Lorig. Short-Time Expansions for Call Options on Leveraged ETFs under Exponential Lévy Models with Local Volatility. SIAM Journal on Financial Mathematics, Vol. 9, No. 1, pp. 347-380, 2018.
- T. R. Bielecki, T. Chen, and I. Cialenco. Recursive Construction of Confidence Regions. Electronic Journal of Statistics, Vol. 11, No. 2, pp. 4674-4700, 2017.
- T. R. Bielecki, I. Cialenco, and M. Pitera. A Survey of Time Consistency of Dynamic Risk Measures and Dynamic Performance Measures in Discrete Time: LM-Measure Perspective. Probability, Uncertainty and Quantitative Risk (2017), Vol. 2, Article 3, pp.1-52.
- T. R. Bielecki, J. Jakubowski, and M. Nieweglowski. Conditional Markov Chains: Properties, Construction and Structured Dependence. Stochastic Processes and Their Applications (2017), Vol. 127, Issue 4, pp. 1125-1170.
- T. R. Bielecki, I. Cialenco, S. Drapeau, and M. Karliczek. Dynamic Assessment Indices. Stochastics: An International Journal of Probability and Stochastic Processes (2016), Vol. 88, Issue 1, pp. 1-44.
- Z. Cheng, J. Duan, and L. Wang. Most Probable Dynamics of Some Non-Linear Systems under Noisy Fluctuations. Communications in Nonlinear Science and Numerical Simulation (2016), Vol. 30, Issue 1-3, pp. 108-114.
- J. E. Figueroa-López, R. Gong, and C. Houdré. High-Order Short-Time Expansions for ATM Option Prices of Exponential Lévy Models. Mathematical Finance (2016), Vol. 26, No. 3, pp. 516-557.
- T. Gao and J. Duan. Quantifying Model Uncertainty in Dynamical Systems Driven by Non-Gaussian Lévy Stable Noise with Observations on Mean Exit Time or Escape Probability. Communications in Nonlinear Science and Numerical Simulation (2016), Vol. 39, pp. 1-6.
- T. Gao, J. Duan, X. Kan, and Z. Cheng. Dynamical Inference for Transitions in Stochastic Systems with Alpha-Stable Lévy Noise. Journal of Physics A: Mathematical and Theoretical (2016), Vol. 49, No. 29, Article Number 294002.
- T. Gao, J. Duan, and X. Li. Fokker-Planck Equations for Stochastic Dynamical Systems with Symmetric Lévy Motions. Applied Mathematics and Computation (2016), Vol. 278, pp. 1-20.
- F. J. Hickernell and Ll. A. Jiménez Rugama. Reliable Adaptive Cubature Using Digital Sequences. Monte Carlo and Quasi-Monte Carlo Methods, MCQMC, Leuven, Belgium, April 2014 (R. Cools and D. Nuyens, eds.), Springer Proceedings in Mathematics and Statistics, Vol. 163, pp. 367–383, Springer, 2016.
- Ll. A. Jiménez Rugama and F. J. Hickernell. Adaptive Multidimensional Integration Based on Rank-1 Lattices. Monte Carlo and Quasi-Monte Carlo Methods, MCQMC, Leuven, Belgium, April 2014 (R. Cools and D. Nuyens, eds.), Springer Proceedings in Mathematics and Statistics, Vol. 163, pp. 407–422, Springer, 2016.
- G. Lv, J. Duan, H. Gao, and J.-L.Wu. On a Stochastic Nonlocal Conservation Law in a Bounded Domain. Bulletin des Sciences Mathématiques (2016), Vol. 140, Issue 6, pp. 718-746.
- H. Qiao and J. Duan. Stationary Measures for Stochastic Differential Equations with Jumps. Stochastics: An International Journal of Probability and Stochastic Processes (2016), Vol. 88, Issue 6, pp. 864-883.
- L. Serdukova, Y. Zheng, J. Duan, and J. Kurths. Stochastic Basins of Attraction for Metastable States. Chaos: An Interdisciplinary Journal of Nonlinear Science (2016), Vol. 26, Issue 7, Article Number 073117.
- T. Wang, J. Duan, and T. Liu. Competition Promotes the Persistence of Populations in Ecosystems. Nature - Scientific Reports (2016), Vol. 6, Article Number 30477.
- Y. Zheng, J. Duan, L. Serdukova, and J. Kurths. Transitions in a Genetic Transcriptional Regulatory System under Lévy Motion. Nature - Scientific Reports (2016), Vol. 6, Article Number 29274.
- T. R. Bielecki, I. Cialenco, and T. Chen. Dynamic Conic Finance via Backward Stochastic Difference Equations. SIAM Journal of Financial Mathematics (2015), Vol. 6, Issue 1, pp. 1068-1122.
- T. R. Bielecki, I. Cialenco, and M. Pitera. Dynamic Limit Growth Indices in Discrete Times. Stochastic Models (2015), Vol. 31, Issue 3, pp. 494-523.
- T. R. Bielecki, I. Cialenco, and R. Rodriguez. No-Arbitrage Pricing for Dividend-Paying Securities in Discrete-Time Markets with Transaction Costs. Mathematical Finance (2015), Vol. 25, No. 4, pp. 673-701.
- T. R. Bielecki, J. Jakubowski, and M. Nieweglowski. Conditional Markov chains - Construction and Properties. Stochastic Analysis: Special Volume in Honour of Jerzy Zabczyk, the Banach Center Conference on Stochastic Analysis and Control, Bedlewo, Poland, May 2013 (A. Chojnowska-Michalik, S. Peszat, and Ł.Stettner, eds.), Banach Center Publications, Vol. 105, Issue 1, pp. 33-42, Warszawa, 2015.
- T. R. Bielecki and M. Rutkowski. Valuation and Hedging of Contracts with Funding Costs and Collateralization. SIAM Journal on Financial Mathematics (2015), Vol. 6, Issue 1, pp. 594-655.
- I. Cialenco and L. Xu. A Note on Error Estimation for Hypothesis Testing Problems for Some Linear SPDEs. Stochastic Partial Differential Equations: Analysis and Computations (2015), Vol. 2, Issue 3, pp. 408-431.
- I. Cialenco and L. Xu. Hypothesis Testing for SPDE Driven by Additive Noise. Stochastic Processes and Their Applications(2015), Vol. 125, Issue 3, pp. 819-866.
- G. Lv and J. Duan. Impacts of Noise on a Class of Partial Differential Equations. Journal of Differential Equations (2015). Vol. 258, Issue 6, pp. 2196-2220.
- H. Qiao and J. Duan. Nonlinear Filtering of Stochastic Dynamical Systems with Lévy Noises. Advances in Applied Probability (2015), Vol. 47, No. 3, pp. 902-918.
- J. Ren, J. Duan, and C. K. R. T. Jones. Approximation of Random Slow Manifolds and Settling of Inertial Particles under Uncertainty. Journal of Dynamics and Differential Equations (2015), Vol. 27, Issue 3-4, pp. 961-979.
- J. Ren, J. Duan, and X. Wang. A Parameter Estimation Method Based on Random Slow Manifolds. Applied Mathematical Modelling (2015), Vol. 39, Issue 13, pp. 3721-3732.
- W. Zou, D. V. Senthilkumar, R. Nagao, I. Z. Kiss, Y. Tang, A. Koseska, J. Duan, and J. Kurths. Restoration of Rhythmicity in Diffusively Coupled Dynamical Networks. Nature Communications (2015), Vol. 6, Article Number 7709.