# 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.

### Ph.D. Students

• Angel J. Santores Almenara
• Ziteng Cheng
• Senbao Jiang
• Kan Zhang

### RECENT RESEARCH GRANTS

• NSF DMS-1907568 (PI T. R. Bielecki and Co-PI I. Cialenco): Collaborative Research: Risk-Averse Control of Markov Systems with Model Uncertainty, 2019-2022.
• NSF EEC-1840433 (Co-PI M. Dixon): Planning Grant: Engineering Research Center for Infrastructure Finance through Intelligent Design and Operations (Joint with University of Michigan), 2017-2018.
• NSF CAREER DMS-1651294 (PI S. Nadtochiy): Quantitative Approach to Large-population Stochastic Dynamic Games, 2017-2022.
• 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 EEC-1840433 (Co-PI M. Dixon): Planning Grant: Engineering Research Center for Infrastructure Finance through Intelligent Design and Operations (InFinIDO), 2018-2019.
• Intel Corporation (PI M. Dixon): Scalable Uncertainty Quantification Methodology for Counterparty Credit Risk, 2018-2019.
• NSF DMS-1642545 (PI J. Duan and Co-PI X. Li): CBMS Conference: Nonlocal Dynamics — Theory, Computation and Applications, 2017-2018.
• Intel Corporation (PI M. Dixon): High Performance Algorithms for Computational Finance, 2015-2018.
• NSF DMS-1522687 (PI F. J. Hickernell and Co-PI G. E. Fasshauer): Stable, Efficient, Adaptive Algorithms for Approximation and Integration, 2015–2018.
• Intel Funded Research in Computational Finance (PI M. Dixon), 2014-2019.
• NSF DMS-1411824 (PI S. Nadtochiy): Mean-field Games for Market Microstructure and Liquidity Risk, 2014-2017.
• 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.

### Recent Publications

• T. R. Bielecki, Z. Cheng, I. Cialenco, and R. Gong. Wiener-Hopf Factorization for Time-Inhomogeneous Markov Chains. Submitted, 2019. arXiv:1902.10850
• T. R. Bielecki, I. Cialenco, M. Pitera, and T. Schmidt. Fair Capital Risk Allocation. Submitted, 2019. arXiv:1902.10044
• I. Cialenco, F. Delgado-Vences, and H.-J. Kim. Drift Estimation for Discretely Sampled SPDEs. Submitted, 2019. arXiv:1904.10884
• F. Delarue, S. Nadtochiy, and M. Shkolnikov. Global Solution to Super-cooled Stefan Problem with Blow-ups: Regularity and Uniqueness. Submitted, 2019. arXiv:1902.05174
• R. Gong and C. Houdré. A Viscosity Solution to a Principal-Agent Problem. Submitted, 2018. arXiv:0911.0956
• S. Nadtochiy and M. Shkolnikov. Mean Field Systems on Networks, with Singular Interaction Through Hitting Times. Submitted, 2018. arXiv:1807.02015
• C. G. Akcroa, M. F. Dixon, Y. R. Gel, and M. Kantarcioglu. Blockchain Analytics for Intraday Financial Risk Modeling. Forthcoming in Digital Finance, 2019+. DOI:10.1007/s42521-019-00009-8
• T. R. Bielecki, I. Cialenco, R. Gong, and Y. Huang. Wiener-Hopf Factorization for Time-Inhomogeneous Markov Chains and Its Application. Forthcoming in Probability and Mathematical Statistics, 2019+. arXiv:1801.05553
• Z. Cheng, I. Cialenco, and R. Gong. Bayesian Estimations for Diagonalizable Bilinear SPDEsForthcoming in Stochastic Processes and Their Applications, 2019+. DOI:10.1016/j.spa.2019.03.020
• I. Cialenco and Y. Huang. A Note on Parameter Estimation for Discretely Sampled SPDEs. Forthcoming in Stochastics and Dynamics, 2019+. DOI:10.1142/S02194937205001
• I. Cialenco, H.-J. Kim, and S. Lototsky. Statistical Analysis of Some Evolution Equations Driven by Space-Only Noise. Forcoming in Statistical Inference for Stochastic Processes, 2019+. DOI: 10.1007/s11203-019-09205-0
• R. Gayduk and S. Nadtochiy. Control-Stopping Games for Market Microstructure and Beyond. Forthcoming in Mathematics of Operations Research, 2019+. arXiv:1708.00506
• R. Gong, C. Mou, and A. Swiech. Stochastic Representations for Solutions to Nonlocal Bellman Equations. Forthcoming in The Annals of Applied Probability, 2019+. arXiv:1709.00193
• I. Halperin and M.F. Dixon. Quantum Equilibrium-Disequilibrium": Asset Price Dynamics, Symmetry Breaking and Defaults as Dissipative Instantons. Forthcoming in Physica A: Statistical Mechanics and its Applications, 2019+. DOI:10.1016/j.physa.2019.122187
• Y. Liu, X. Chen, L. X. Cai, Q. Chen, R. Gong, and D. Tang. On the Fairness Performance of NOMA-Based Wireless Powered Communication Networks. Forthcoming in the Proceedings of IEEE International Conference on Communications, 2019. DOI:10.1109/ICC.2019.8761702
• S. Nadtochiy and T. Zariphopoulou. Optimal Contract for a Fund Manager, with Capital Injections and Endogenous Trading Constraints. Forthcoming in SIAM Journal on Financial Mathematics, 2019+. arXiv:1802.09165
• T. R. Bielecki, T. Chen, I. Cialenco, A. Cousin, and M. Jeanblanc. Adaptive Robust Control under Model Uncertainty. SIAM Journal on Control and Optimization (2019), Vol. 57, No. 2, pp. 925-946.
• M. F. Dixon, N. G. Polson, and V. Sokolov. Deep Learning for Spatio‐Temporal Modeling: Dynamic Traffic Flows and High Frequency Trading. Applied Stochastic Models in Business and Industry (2019), Vol. 35, No. 3, pp. 788-807
• R. Gayduk and S. Nadtochiy. Endogenous Formation of Limit Order Books: Dynamics Between Trades. SIAM Journal on Control and Optimization (2019), Vol. 56, No. 3, pp. 1577-1619.
• S. Nadtochiy and M. Shkolnikov. Particle Systems with Singular Interaction Through Hitting Times: Application in Systemic Risk Modeling. The Annals of Applied Probability (2019), Vol. 29, No. 1, pp. 89-129.
• C. G. Akcora, M. F. Dixon, Y. R. Gel, and M. Kantarcioglu. Bitcoin Risk Modeling with Blockchain Graphs. Economic Letters (2018), Vol. 173, pp. 138-142.
• C. G. Akcroa, M. F. Dixon, Y. R. Gel, and M. Kantarcioglu. Blockchain Data Analytics. The IEEE Intelligent Informatics Bulletin (2018), Vol. 19, No. 2, Article 1, pp. 1-6.
• T. R. Bielecki, I. Cialenco, and S. Feng. A Dynamic Model of Central Counterparty RiskInternational Journal of Theoretical and Applied Finance (2018), Vol. 21, No. 08, 1850050.
• T. R. Bielecki, I. Cialenco, and M. Rutkowski. Arbitrage-Free Pricing of Derivatives in Nonlinear Market Models. Probability, Uncertainty and Quantitative Risk (2018), Vol. 3, Article 2, pp. 1-56.
• 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. Mathematics of Operations Research (2018), Vol. 43, No. 1, pp. 204-221.
• T. R. Bielecki, M. Jeanblanc, and D. Sezer. Joint Hitting-Time Densities for Finite State Markov Processes. Turkish Journal of Mathematics (2018), Vol. 42, No. 2, pp. 586-608.
• Z. Chen, Z. Chen, L. X. Cai, Y. Cheng, and R. Gong. Performance Analysis of Energy Harvesting in Wireless Networks Using Stochastic Geometry. Proceedings of IEEE International Conference on Green Computing and Communications (2018), pp. 280-286.
• I. Cialenco. Statistical Inference for SPDEs: an Overview. Statistical Inference for Stochastic Processes (2018), Vol. 21, Issue 2, pp. 309-329.
• I. Cialenco, R. Gong, Y. Huang. Trajectory Fitting Estimators for SPDEs Driven by Additive Noise. Statistical Inference for Stochastic Processes (2018), Vol 21, Issue 1, pp. 1-19.
• M. F. Dixon. A High Frequency Trade Execution Model for Supervised Learning. High Frequency (2018), Vol. 1, No. 1, pp. 32-52.
• 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 (2018), Vol. 24, No. 6, pp. 547-574.
• 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 (2018), Vol. 9, No. 1, pp. 347-380.
• R. Gayduk and S. Nadtochiy. Liquidity Effects of Trading Frequency. Mathematical Finance (2018), Vol. 28, No. 3, pp. 839-876.
• R. Gong, C. Houdré, and J. Lember. Lower Bounds on the Generalized Central Moments of the Optimal Alignments Score of Random Sequences. Journal of Theoretical Probability (2018), Vol. 31, No. 2 pp. 643-683.
• T. R. Bielecki, T. Chen, and I. Cialenco. Recursive Construction of Confidence Regions. Electronic Journal of Statistics (2017), Vol. 11, No. 2, pp. 4674-4700.
• 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.
• R. Carmona, Y. Ma, and S. Nadtochiy. Simulation of Implied Volatility Surfaces via Tangent Levy Models. SIAM Journal on Financial Mathematics (2017), Vol. 8, No. 1, pp. 171-213.
• P. Carr and S. Nadtochiy. Local Variance Gamma and Explicit Calibration to Option Prices. Mathematical Finance (2017), Vol. 27, No. 1, pp. 151-193.
• M. F. Dixon. Sequence Classification of the Limit Order Book Using Recurrent Neural Networks. Journal of Computational Science (2017), Vol. 24, pp. 277-286.
• M. F. Dixon, D. Klabjan, and J. H. Bang. Classification-Based Financial Markets Prediction Using Deep Neural Networks. Algorithmic Finance (2017), Vol. 6, No. 3-4, pp. 66-99.
• S. Nadtochiy and J. Obloj. Robust Trading of Implied Skew. International Journal of Theoretical and Applied Finance (2017), Vol. 20, No. 2, 1750008.
• S. Nadtochiy and M. Tehranchi. Optimal Investment for All Time Horizons and Martin Boundary of Space-Time Diffusions. Mathematical Finance, Vol. 27, No. 2, pp. 438-470.
• 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.
• M. F. Dixon, J. Lotze, and M. Zubair. A Portable, Extensible and Fast Stochastic Volatility Model Calibration Using Multi- and Many-Core Processors. Concurrency and Computation: Practice and Experience (2016), Vol. 28, No. 3, pp. 866–877.
• 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.
• R. Jordan and C. Tier. Asymptotic Approximations for Pricing Derivatives Under Mean-Reverting Processes. International Journal of Theoretical and Applied Finance (2016), Vol. 19, No. 5, 1650030.
• 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.
• E. Bayraktar and S. Nadtochiy. Weak Reflection Principle for Levy Processes. The Annals of Applied Probability (2015), Vol. 25, No. 6, pp. 3251-3294.
• 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. 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.