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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

  • Y. Cao
  • Z. Chang
  • S. Feng
  • X. Huang
  • Y. Huang
  • S. Jiang
  • K. Zhang

Recent Publications

  • T.R. Bielecki, I. Cialenco, M. Rutkowski, Arbitrage-Free Pricing of Derivatives in Nonlinear Market Models (2017) submitted for publication (42 pages).
  • T.R. Bielecki, I. Cialenco, T. Chen, Recursive Construction of Confidence Regions (2016) submitted for publication (29 pages).
  • T. R. Bielecki, I. Cialenco, 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) 2:3, p.1-52.
  • T.R. Bielecki, I. Cialenco, T. Chen, Dynamic Conic Finance via Backward Stochastic Difference Equations (2015) SIAM Journal of Financial Mathematics, 6(1) p. 1068-1122.
  • T.R. Bielecki, I. Cialenco, M. Pitera, A unified approach to time consistency of dynamic risk measures and dynamic performance measures in discrete time (2017) Forthcoming in Mathematics of Operations Research.
  • I. Cialenco, R. Gong, Y. Huang, Trajectory Fitting Estimators for SPDEs Driven by Additive Noise (2016) Forthcoming in Statistical Inference for Stochastic Processes.
  • T.R. Bielecki, I. Cialenco, M. Pitera, Dynamic Limit Growth Indices in Discrete Times, Stochastic Models (2015) vol. 31, p. 494-523.
  • T.R. Bielecki, I. Cialenco, S. Drapeau, M. Karliczek, Dynamic Assessment Indices (2016) Stochastics: An International Journal of Probability and Stochastic Processes, vol. 88 No 1, p. 1-44.
  • I. Cialenco, L. Xu, A note on error estimation for hypothesis testing problems for some linear SPDEs (2015) Stochastic Partial Differential Equations: Analysis and Computations, vol. 2, No 3, p. 408-431.
  • I. Cialenco, L. Xu, Hypothesis testing for SPDE driven by additive noise, Stochastic Processes and Applications (2015) vol. 125, Issue 3, p. 819-866.
  • T.R. Bielecki, I. Cialenco and R. Rodriguez, No-Arbitrage Pricing for Dividend-Paying Securities in Discrete-Time Markets with Transaction Costs (2015) Mathematical Finance, vol. 25 (4), p. 673-701.
  • J. E. Figueroa-López, R. Gong, and C. Houdré, High-Order Short-Time Expansions for ATM Option Prices of Exponential Lévy Models (2016) Mathematical Finance, Vol. 26, Issue 3, p. 516-557.
  • 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 (2016) Submitted for Publication (28 pages).
  • R. Gong, C. Houdré, and Ü. Islak, A Central Limit Theorem for the Optimal Alignments Score in Multiple Random Words (2016) Submitted for Publication (30 pages).
  • R. Gong and C. Houdré, A Viscosity Solution to a Principal-Agent Problem (2016) Submitted for Publication (2016) Submitted for Publication (31 pages).
  • 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 (2015) Submitted for Publication (26 pages).
  • R. Gong, C. Houdré, and J. Lember, Lower Bounds on the Generalized Central Moments of the Optimal Alignments Score of Random Sequences (2015) Journal of Theoretical Probability, to appear.
  • T.R. Bielecki, S. Crepey, M. Jeanblanc and B. Zargari, “Valuation and Hedging of CDS Counterparty Exposure in a Markov Copula Model,” International Journal of Theoretical and Applied Finance, vol. 15, no. 1, 2012.
  • T.R. Bielecki and S. Crepey, “Dynamic Hedging of Counterparty Exposure,” The Musiela Festschrift, Zariphopoulou, T., Rutkowski, M. and Kabanov, Y., eds., Springer, 2013.
  • T.R. Bielecki, J. Jakubowski and M. Nieweglowski , “Intricacies of Dependence between Components of Multivariate Markov Chains: Weak Markov Consistency and Markov Copulae,” Electron. J. Probab. 18, no. 45, 2013
  • T.R. Bielecki, A. Cousin, S. Crepey and A. Herbertsson, “In search of grand unifying theory,” Creditflux Newsletter, July, 2013
  • T.R. Bielecki, A. Cousin, S. Crepey and A. Herbertsson, “Dynamic Hedging of Portfolio Credit Risk in a Markov Copula Model,” JOTA, Vol. 161, No. 1, 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, Volume 43, Issue 7, 2014
  • 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, 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, World Scientific, 2014
  • T.R. Bielecki and M. Rutkowski, “Valuation and hedging of contracts with funding costs and collateralization,” SIAM Journal on Financial Mathematics, 6-1 , 2015
  • T.R. Bielecki, J. Jakubowski and M. Nieweglowski, “Conditional Markov chains - construction and properties,” Stochastic Analysis and Control, Banach Center Publications, 2015
  • T.R. Bielecki, J. Jakubowski and M. Nieweglowski, “Conditional Markov Chains Revisited: Properties, Construction and Structured Dependence,” Stochastic Processes and Applications, Vol. 127, No. 4, 2017
  • T.R. Bielecki, J. Jakubowski and M. Nieweglowski. A note on independence copulae for conditional Markov chains, Fields Institute Communications: Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, Roderick Melnik, Roman Makarov, and Jacques Belair eds., to appear, 2017
  • T.R. Bielecki, M. Jeanblanc and D. Sezer. Joint Hitting-Time Densities for Finite State Markov Processes, Turkish Journal of Mathematics, to appear, 2017
  • 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, p. 108-114.
  • 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, p. 1-6.
  • T. Gao, J. Duan, X. Kan, and Z. Cheng. Dynamical Inference for Transitions in Stochastic Systems with Alpha-Stable Lévy Noise (2016). Journal of Physics A: Mathematical and Theoretical, Vol. 49, No. 19, 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, p. 1-20.
  • G. Lv, J. Duan, H. Gao, and J.-L.Wu. On a Stochastic Nonlocal Conservation Law in a Bounded Domain (2016). Bulletin des Sciences Mathématiques, Vol. 140, Issue 6, p. 718 -746.
  • H. Qiao and J. Duan. Stationary Measures for Stochastic Differential Equations with Jumps (2016). Stochastics, Vol. 88, Issue 6, p. 864-883.
  • L. Serdukova, Y. Zheng, J. Duan, and J. Kurths. Stochastic Basins of Attraction for Metastable States (2016). Chaos 26, Article number 073117.
  • T. Wang, J. Duan, and T. Liu. Competition promotes the persistence of populations in ecosystems. Nature - Scientific Reports (2016) 6, Article number 30477.
  • Y. Zheng, J. Duan, L. Serdukova, and J. Kurths. Transitions in a Genetic Transcriptional Regulatory System under Lévy Motion (2016). Nature - Scientific Reports 6, Article Number 29274.
  • G. Lv and J. Duan. Impacts of Noise on a Class of Partial Differential Equations (2015). Journal of Differential Equations. Vol. 258, Issue: 6, p. 2196-2220.
  • H. Qiao and J. Duan, Nonlinear Filtering of Stochastic Dynamical Systems with Lévy Noises (2015). Advances in Applied Probability, Vol. 47, No. 3, p. 902-918.
  • J. Ren, J. Duan, and C. K. R. T. Jones. Approximation of Random Slow Manifolds and Settling of Inertial Particles under Uncertainty (2015). Journal of Dynamics and Differential Equations, Vol. 27, Issue 3, p. 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, p. 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 (2015). Nature - Communications 6, Article Number 7709.
  • G. Chen, J. Duan, and J. Zhang, Slow Foliation of a Slow-Fast Stochastic Evolutionary System (2014). Journal of Functional Analysis, Vol. 267, Issue 8, p. 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, p. 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 (2014). Nonlinear Dynamics, Vol. 84, Issue 3, p. 1311-1325.
  • 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, Springer-Verlag, Berlin, 2016, pp. 367–383.
  • 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, Springer-Verlag, Berlin, 2016, pp. 407–422.

Recent Research Grants

  • CME Group (Pi I. Cialenco and T.R. Bielecki): CDS and CDX series pricing factor analysis, 2014.
  • NSF Grant DMS-1211256 (PI T.R. Bielecki, Co-PI I. Cialenco): “Topics in stochastic processes and mathematical finance: counterparty risk valuation and hedging, Markov consistency and Markov copulae, and dynamic performance assessment indices, 2012-2015.
  • NSF DMS-1620449 (Co-PI J. Duan): Theoretical and numerical studies of nonlocal equations derived from stochastic differential equations with Levy noises, 2016-2020.
  • NSF DMS-1642545 (PI J. Duan), CBMS Conference: Nonlocal Dynamics Theory, Computation and Applications, 2016-2017.
  • NSF DMS-1025422 (Lead PI J. Duan). Collaborative Research: Mathematical Modeling by Bridging Primitive and Boussinesq Equations, 2010-2015.
  • NSF-0938235 (PI J. Duan): Recent Advances in the Numerical Approximation of Stochastic Partial Differential Equations, NSF-CBMS Regional Conference in the Mathematical Sciences, August 2010, Chicago.
  • NSF-0731201 (Co-PI J. Duan): Stochastic Agent-based Modeling of Angiogenesis and Tissue Growth, 2007-2010.
  • NSF-DMS-1522687 (G. E. Fasshauer and F. J. Hickernell (PI)) Stable, Efficient, Adaptive Algorithms for Approximation and Integration, 2015–2018.
  • Fermilab (F. J. Hickernell), Modern Monte Carlo Methods for High Energy Event Simulation, Parts I, II, 2015.
  • NSF-DMS-1115392 (G. E. Fasshauer and F. J. Hickernell (PI)) Kernel Methods for Numerical Computation, 2011–2014.
  • NSF-DMS-0938235 (I. Cialenco, J. Duan (PI), and F. J. Hickernell) NSF/CBMS Regional Conference in the Mathematical Sciences — Recent Advances in the Numerical Approximation of Stochastic Partial Differential Equations, 2010.