Alexander Schied

Alexander Schied is Professor in the Department of Statistics and Actuarial Science at the University of Waterloo. He holds the Munich Re Chair in Stochastic Finance and a University Research Chair. His research focuses on quantitative finance, probability theory, and stochastic analysis, with recent work on risk measurement and management, financial modeling and optimization, robustness and model uncertainty, and market microstructure. He is co-author, with Hans Föllmer, of the book "Stochastic Finance: An Introduction in Discrete Time." He currently serves as Co-Editor of Finance and Stochastics and sits on several other editorial boards, including Applied Mathematics and Optimization, Mathematical Finance, the SIAM Book Series on Financial Mathematics, and Springer's Monographs in Mathematical Economics. Prior to joining Waterloo, he was a professor at the University of Mannheim and held positions as Associate Professor at TU Munich, Cornell University, and TU Berlin. He earned a doctorate in mathematics from the University of Bonn.

• Liu, Peng, and Alexander Schied. "Lambda Value-at-Risk under ambiguity and risk sharing." Submitted. arXiv:2511.00717.

• Han, Xiyue, and Alexander Schied. "Universal portfolios in continuous time: an approach in pathwise Itô calculus." Submitted. arXiv:2504.11881.

• Han, Xiyue, and Alexander Schied. "Estimating the roughness exponent of stochastic volatility from discrete observations of the realized variance." Submitted. arXiv:2307.02582.

• Schied, Alexander, and Zhenyuan Zhang. "Sample Path Properties of the Fractional Wiener–Weierstrass Bridge." To appear in Bernoulli.

• Han, Xiyue, and Alexander Schied. "Robust Faber–Schauder approximation based on discrete observations of an antiderivative." To appear in Mathematics of Operations Research.

• Han, Xiyue, and Alexander Schied. "On the rate of convergence of estimating the Hurst parameter of rough stochastic volatility models." SIAM Journal on Financial Mathematics 16, no. 4: 1336–1349. 2025.

• Föllmer, Hans, and Alexander Schied. Stochastic Finance. An Introduction in Discrete Time. 5th ed. De Gruyter. 2025.

• Han, Xiyue, and Alexander Schied. "The roughness exponent and its model-free estimation." Annals of Applied Probability 35, no. 2: 1049–1082. 2025.

• Hu, Hongda, Arthur Charpentier, Mario Ghossoub, and Alexander Schied. "Multiarmed bandits problem under the mean-variance setting." European Journal of Operational Research 324: 168–182. 2025.

• Han, Xiyue, and Alexander Schied. "A criterion for absolute continuity relative to the law of fractional Brownian motion." Electronic Communications in Probability 29, no. 80: 1–10. 2024.

• Luo, Peng, Alexander Schied, and Xiaole Xue. "The perturbation method applied to a robust optimization problem with constraint." Mathematics and Financial Economics 18: 95–112. 2024.

• Schied, Alexander, and Zhenyuan Zhang. "Weierstrass bridges." Transactions of the AMS 377, no. 4: 2947–2989. 2024.

• Han, Xiyue, Alexander Schied, and Zhenyuan Zhang. "A limit theorem for Bernoulli convolutions and the Φ-variation of functions in the Takagi class." Journal of Theoretical Probability 35: 2853–2878. 2022.

• Neuman, Eyal, and Alexander Schied. "Protecting Pegged Currency Markets from Speculative Investors." Mathematical Finance 32, no. 1: 405–420. 2022.

• Han, Xiyue, and Alexander Schied. "Step roots of Littlewood polynomials and the extrema of functions in the Takagi class." Mathematical Proceedings of the Cambridge Philosophical Society 173: 591–618. 2022.

• Embrechts, Paul, Alexander Schied, and Ruodu Wang. "Robustness in the Optimization of Risk Measures." Operations Research 70, no. 1: 95–110. 2022.

• Drapeau, Samuel, Peng Luo, Alexander Schied, and Dewen Xiong. "An FBSDE approach to market impact games with stochastic parameters." Probability, Uncertainty and Quantitative Risk 6, no. 3: 237–260. 2021.

• Han, Xiyue, Alexander Schied, and Zhenyuan Zhang. "A probabilistic approach to the Φ-variation of classical fractal functions with critical roughness." Statist. Probab. Lett. 168: 108920. 2021.

• Liu, Peng, Alexander Schied, and Ruodu Wang. "Distributional transforms, probability distortions, and their applications." Mathematics of Operations Research 46, no. 4: 1490–1512. 2021.

• Schied, Alexander, and Elias Strehle. "On the minimizers of energy forms with completely monotone kernel." Applied Mathematics and Optimization 83, no. 1: 177–205. 2021.

• Neuman, Eyal, Alexander Schied, Chengguo Weng, and Xiaole Xue. "A central bank strategy for defending a currency peg." Systems Control Lett. 144: 104761. 2020.

• Schied, Alexander, and Zhenyuan Zhang. "On the pth variation of a class of fractal functions." Proc. Amer. Math. Soc. 148: 5399–5412. 2020.

• Mishura, Yuliya, and Alexander Schied. "On (signed) Takagi-Landsberg functions: pth variation, maximum, and modulus of continuity." J. Math. Anal. Appl. 473: 258–272. 2019.

• Schied, Alexander, and Tao Zhang. "A market impact game under transient price impact." Math. Oper. Res. 44: 102–121. 2019.

• Luo, Xiangge, and Alexander Schied. "Nash equilibrium for risk-averse investors in a market impact game: finite and infinite time horizons." Market Microstructure and Liquidity 5, no. 01n04: 2050001. 2019.

• Schied, Alexander, Leo Speiser, and Iryna Voloshchenko. "Model-free portfolio theory and its functional master formula." SIAM J. Financial Math. 9: 1074–1101. 2018.

• Krätschmer, Volker, Alexander Schied, and Henryk Zähle. "Domains of weak continuity of statistical functionals with a view toward robust statistics." J. Multivariate Anal. 158: 1–19. 2017.

• Schied, Alexander, Elias Strehle, and Tao Zhang. "High-frequency limit of Nash equilibria in a market impact game with transient price impact." SIAM J. Financial Math. 8: 589–634. 2017.

• Klöck, Florian, Alexander Schied, and Yuemeng Sun. "Price manipulation in a market impact model with dark pool." Appl. Math. Finance 24: 417–450. 2017.

• Schied, Alexander, and Tao Zhang. "A state-constrained differential game arising in optimal portfolio liquidation." Math. Finance 27: 779–802. 2017.

• Schied, Alexander. "On a class of generalized Takagi functions with linear pathwise quadratic variation." J. Math. Anal. Appl. 433: 974–990. 2016.

• Neuman, Eyal, and Alexander Schied. "Optimal portfolio liquidation in target zone models and catalytic superprocesses." Finance Stoch. 20: 495–509. 2016.

• Mishura, Yuliya, and Alexander Schied. "Constructing functions with prescribed pathwise quadratic variation." J. Math. Anal. Appl. 442: 117–137. 2016.

• Alfonsi, Aurélien, Florian Klöck, and Alexander Schied. "Multivariate transient price impact and matrix-valued positive definite functions." Math. Oper. Res. 41: 914–934. 2016.

• Schied, Alexander, and Iryna Voloshchenko. "Pathwise no-arbitrage in a class of delta hedging strategies." Probab. Uncertain. Quant. Risk 1: Paper No. 3. 2016.

• Föllmer, Hans, and Alexander Schied. Stochastic Finance. De Gruyter. 2016.

• Schied, Alexander, and Iryna Voloshchenko. "The associativity rule in pathwise functional Itô calculus." arXiv:1605.0886. 2016.

• Krätschmer, Volker, Alexander Schied, and Henryk Zähle. "Quasi-Hadamard differentiability of general risk functionals and its application." Stat. Risk Model. 32: 25–47. 2015.

• Krätschmer, Volker, Alexander Schied, and Henryk Zähle. "Comparative and qualitative robustness for law-invariant risk measures." Finance Stoch. 18: 271–295. 2014.

• Schied, Alexander. "Model-free CPPI." J. Econom. Dynam. Control 40: 84–94. 2014.

• Alfonsi, Aurélien, and Alexander Schied. "Capacitary measures for completely monotone kernels via singular control." SIAM J. Control Optim. 51: 1758–1780. 2013.

• Schied, Alexander. "A control problem with fuel constraint and Dawson-Watanabe superprocesses." Ann. Appl. Probab. 23: 2472–2499. 2013.

• Lorenz, Christopher, and Alexander Schied. "Drift dependence of optimal trade execution strategies under transient price impact." Finance Stoch. 17: 743–770. 2013.

• Föllmer, Hans, and Alexander Schied. "Probabilistic aspects of finance." Bernoulli 19: 1306–1326. 2013.

• Schied, Alexander. "Robust strategies for optimal order execution in the Almgren-Chriss framework." Appl. Math. Finance 20: 264–286. 2013.

• Gatheral, Jim, and Alexander Schied. "Dynamical models for market impact and algorithms for optimal order execution." In Handbook on Systemic Risk, 579–602. Cambridge University Press. 2013.

• Schied, Alexander. "Finanzmathematik." In Springer–Taschenbuch der Mathematik, 1015–1029. 3rd ed. Springer–Verlag. 2013.

• Alfonsi, Aurélien, Alexander Schied, and Alla Slynko. "Order book resilience, price manipulation, and the positive portfolio problem." SIAM J. Financial Math. 3: 511–533. 2012.

• Krätschmer, Volker, Alexander Schied, and Henryk Zähle. "Qualitative and infinitesimal robustness of tail-dependent statistical functionals." J. Multivariate Anal. 103: 35–47. 2012.

• Gatheral, Jim, Alexander Schied, and Alla Slynko. "Transient linear price impact and Fredholm integral equations." Math. Finance 22: 445–474. 2012.

• Gatheral, Jim, Alexander Schied, and Alla Slynko. "Exponential resilience and decay of market impact." In Econophysics of order-driven markets, 225–236. Springer. 2011.

• Gatheral, Jim, and Alexander Schied. "Optimal trade execution under geometric Brownian motion in the Almgren and Chriss framework." Int. J. Theor. Appl. Finance 14: 353–368. 2011.

• Schied, Alexander, and Alla Slynko. "Some mathematical aspects of market impact modeling." In Surveys in stochastic processes, 153–179. Eur. Math. Soc. 2011.

• Föllmer, Hans, and Alexander Schied. Stochastic Finance. Extended ed. Walter de Gruyter & Co. 2011.

• Schied, Alexander, Torsten Schöneborn, and Michael Tehranchi. "Optimal basket liquidation for CARA investors is deterministic." Appl. Math. Finance 17: 471–489. 2010.

• Alfonsi, Aurélien, Antje Fruth, and Alexander Schied. "Optimal execution strategies in limit order books with general shape functions." Quant. Finance 10: 143–157. 2010.

• Alfonsi, Aurélien, and Alexander Schied. "Optimal trade execution and absence of price manipulations in limit order book models." SIAM J. Financial Math. 1: 490–522. 2010.

• Föllmer, Hans, and Alexander Schied. "Coherent and convex risk measures." In Encyclopedia of quantitative finance, 355–363. John Wiley & Sons. 2010.

• Schied, Alexander, and Torsten Schöneborn. "Risk aversion and the dynamics of optimal liquidation strategies in illiquid markets." Finance Stoch. 13: 181–204. 2009.

• Schöneborn, Torsten, and Alexander Schied. "Liquidation in the face of adversity: stealth vs. sunshine trading." 2009.

• Föllmer, Hans, Alexander Schied, and Stefan Weber. "Robust preferences and robust portfolio choice." In Mathematical modelling and numerical methods in finance, 29–88. Elsevier. 2009.

• Alfonsi, Aurélien, Antje Fruth, and Alexander Schied. "Constrained portfolio liquidation in a limit order book model." In Advances in mathematics of finance, 9–25. Polish Acad. Sci. Inst. Math. 2008.

• Schied, Alexander. "Robust optimal control for a consumption-investment problem." Math. Methods Oper. Res. 67: 1–20. 2008.

• Schied, Alexander, and Torsten Schöneborn. "Optimal portfolio liquidation: market impact models and optimal control." In MFO Oberwolfach Reports, 192–194. 2008.

• Hernández-Hernández, Daniel, and Alexander Schied. "A control approach to robust utility maximization with logarithmic utility and time-consistent penalties." Stochastic Process. Appl. 117: 980–1000. 2007.

• Schied, Alexander. "Optimal investments for risk- and ambiguity-averse preferences: a duality approach." Finance Stoch. 11: 107–129. 2007.

• Schied, Alexander, and Mitja Stadje. "Robustness of delta hedging for path-dependent options in local volatility models." J. Appl. Probab. 44: 865–879. 2007.

• Hernández-Hernández, Daniel, and Alexander Schied. "Robust maximization of consumption with logarithmic utility." In Proceedings of the 2007 American Control Conference, 1120–1123. 2007.

• Schied, Alexander. "Risk measures and robust optimization problems." Stoch. Models 22: 753–831. 2006.

• Hernández-Hernández, Daniel, and Alexander Schied. "Robust utility maximization in a stochastic factor model." Statist. Decisions 24: 109–125. 2006.

• Schied, Alexander, and Ching-Tang Wu. "Duality theory for optimal investments under model uncertainty." Statist. Decisions 23: 199–217. 2005.

• Schied, Alexander. "Optimal investments for robust utility functionals in complete market models." Math. Oper. Res. 30: 750–764. 2005.

• Schied, Alexander. "On the Neyman-Pearson problem for law-invariant risk measures and robust utility functionals." Ann. Appl. Probab. 14: 1398–1423. 2004.

• Föllmer, Hans, and Alexander Schied. Stochastic Finance. De Gruyter Studies in Mathematics, vol. 27. Extended ed. Walter de Gruyter & Co. 2004.

• Föllmer, Hans, and Alexander Schied. "Convex measures of risk and trading constraints." Finance Stoch. 6: 429–447. 2002.

• Schied, Alexander. "Geometric analysis for symmetric Fleming-Viot operators: Rademacher's theorem and exponential families." Potential Anal. 17: 351–374. 2002.

• Föllmer, Hans, and Alexander Schied. "Robust preferences and convex measures of risk." In Advances in finance and stochastics, 39–56. Springer. 2002.

• Föllmer, Hans, and Alexander Schied. Stochastic Finance. De Gruyter Studies in Mathematics, vol. 27. Walter de Gruyter & Co. 2002.

• Schied, Alexander. "Existence and regularity for a class of infinite-measure (ξ,ψ,K)-superprocesses." J. Theoret. Probab. 12: 1011–1035. 1999.

• Röckner, Michael, and Alexander Schied. "Rademacher's theorem on configuration spaces and applications." J. Funct. Anal. 169: 325–356. 1999.

• Schied, Alexander. "Cramer's condition and Sanov's theorem." Statist. Probab. Lett. 39: 55–60. 1998.

• Djehiche, Boualem, and Alexander Schied. "Large deviations for hierarchical systems of interacting jump processes." J. Theoret. Probab. 11: 1–24. 1998.

• Schied, Alexander. "Geometric aspects of Fleming-Viot and Dawson-Watanabe processes." Ann. Probab. 25: 1160–1179. 1997.

• Schied, Alexander. "Moderate deviations and functional LIL for super-Brownian motion." Stochastic Process. Appl. 72: 11–25. 1997.

• Schied, A. "Sample path large deviations for super-Brownian motion." Probab. Theory Related Fields 104: 319–347. 1996.

• Schied, Alexander. Grosse Abweichungen für die Pfade der Super-Brownschen Bewegung. Bonner Mathematische Schriften, vol. 277. Universität Bonn. 1995.