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Stochastics

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.

» Laboratory for Stochastics and Dynamics

Faculty with primary interests in stochastics

» A. Adler » T. R. Bielecki » I. Cialenco » J. Duan » R. Gong » F. Hickernell

Faculty with secondary interests in stochastics

» X. li 

Related Seminar

» Mathematical Finance & Stochastic Analysis Seminar

Ph.D. Students

  • Yue Cao
  • Yu-Sin Chang
  • Zhuan Cheng
  • Ziteng Cheng
  • Shibi Feng
  • Xiao Huang
  • Yicong Huang
  • Senbao Jiang
  • Kan Zhang

Recent Publications

  • 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.
  • 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. Submitted, 2017. arXiv:1608.07863
  • T. R. Bielecki, T. Chen, and I. Cialenco. Recursive Construction of Confidence Regions. Submitted, 2016. arXiv:1605.08010
  • 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
  • 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. Submitted, 2015. arXiv:1305.4719
  • 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, 2017.
  • T. R. Bielecki, J. Jakubowski, and M. Nieweglowski. A Note on Independence Copulae for Conditional Markov Chains. To appear in Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science (R. Melnik, R. Makarov, and J. Belair, eds.), Fields Institute Communications, Vol. 79, Springer, 2017.
  • T. R. Bielecki, M. Jeanblanc, and D. Sezer. Joint Hitting-Time Densities for Finite State Markov Processes. To appear in Turkish Journal of Mathematics, 2017. DOI: 10.3906/mat-1608-29
  • I. Cialenco, R. Gong, Y. Huang. Trajectory Fitting Estimators for SPDEs Driven by Additive Noise. To appear in Statistical Inference for Stochastic Processes, 2017. DOI:10.1007/s11203-016-9152-2
  • 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, I. Cialenco, and M. Pitera. A Survey of Time Consistency of Dynamic Risk Measures and Dynamic Performance Measures in Discrete Time: LM-Measure PerspectiveProbability, 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 NoiseStochastic 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.
  • T. R. Bielecki, I. Cialenco, and Z. Zhang. Dynamic Coherent Acceptability Indices and Their applications to Finance. Mathematical Finance (2014), Vol. 24, No. 3, pp. 411-441.
  • T. R. Bielecki, A. Cousin, S. Crepey, and A. Herbertsson. A Bottom-Up Dynamic Model of Portfolio Credit Risk - Part I: Markov Copula Perspective. Recent Advances in Financial Engineering 2012: Proceedings of the International Workshop on Finance 2012, Tokyo, Japan, October 30-31, 2012 (A. Takahashi, Y. Muromachi, and T. Shibata, eds.), pp. 25-50, World Scientific, 2014.
  • T. R. Bielecki, A. Cousin, S. Crepey, and A. Herbertsson. A Bottom-Up Dynamic Model of Portfolio Credit Risk - Part II: Common-Shock Interpretation, Calibration and Hedging Issues. Recent Advances in Financial Engineering 2012: Proceedings of the International Workshop on Finance 2012, Tokyo, Japan, October 30-31, 2012 (A. Takahashi, Y. Muromachi, and T. Shibata, eds.), pp. 51-74, World Scientific, 2014.
  • T. R. Bielecki, A. Cousin, S. Crepey, and A. Herbertsson. A Bottom-Up Dynamic Model of Portfolio Credit Risk with Stochastic Intensities and Random Recoveries. Communications in Statistics - Theory and Methods (2014), Vol. 43, Issue 7, pp. 1362-1389.
  • T. R. Bielecki, A. Cousin, S. Crepey, and A. Herbertsson. Dynamic Hedging of Portfolio Credit Risk in a Markov Copula Model. Journal of Optimization Theory and Applications (2014), Vol. 161, Issue 1, pp. 90-102.
  • G. Chen, J. Duan, and J. Zhang. Slow Foliation of a Slow-Fast Stochastic Evolutionary System. Journal of Functional Analysis (2014), Vol. 267, Issue 8, pp. 2663-2697.
  • T. Gao, J. Duan, X. Li, and R. Song. Mean Exit Time and Escape Probability for Dynamical Systems Driven by Lévy Noise. SIAM Journal on Scientific Computing (2014), Vol. 36, No. 3, pp. A887-A906.
  • X. Sun, J. Duan, and X. Li. Modeling Nonlinear Oscillators under Excitation of Combined Gaussian and Poisson White Noise: A Viewpoint Based on Energy Conservation Law. Nonlinear Dynamics (2014), Vol. 84, Issue 3, pp. 1311-1325.

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.
  • NSF DMS-0938235 (PI J. Duan and CO-PIs I. Cialenco and F. J. Hickernell): NSF/CBMS Regional Conference in the Mathematical Sciences — Recent Advances in the Numerical Approximation of Stochastic Partial Differential Equations, 2010.
  • NSF CBET-0731201 (Co-PI J. Duan): Stochastic Agent-Based Modeling of Angiogenesis and Tissue Growth, 2007-2010.