Preprints

• Gradient descent provably escapes saddle points in the training of shallow ReLU networks
  (with Arnulf Jentzen, and Florian Rossmannek)
  arXiv Preprint 2208.02083
  
• An efficient Monte Carlo scheme for Zakai equations
  (with Christian Beck, Sebastian Becker, Arnulf Jentzen and Ariel Neufeld )
  http://arxiv.org/abs/2210.13530
• Deep learning based numerical approximation algorithms for stochastic partial differential equations and high-dimensional nonlinear filtering problems
  (with Christian Beck, Sebastian Becker, Arnulf Jentzen and Ariel Neufeld )
  arXiv Preprint 2012.01194
• Variable annuities with high water mark withdrawal benefit
  (with Peiqi Wang) SSRN Preprint
• Existence of sequential competitive equilibrium in Krusell–Smith type economies
  (with Juan Sagredo) SSRN Preprint
• Comment on "Competitive equilibria of economies with a continuum of consumers and aggregate shocks" by J. Miao
  (with Juan Sagredo) SSRN Preprint
• Stochastic order-monotone uncertainty-averse preferences
  (with Freddy Delbaen, Samuel Drapeau and Michael Kupper) SSRN Preprint
• Portfolio execution with a dark pool under stochastic volatility and liquidity
  (with Tardu Sepin) SSRN Preprint
• Optimal trade execution with a dark pool and adverse selection
  (with Tardu Sepin) SSRN Preprint
• Pricing and hedging CoCos
  (with Zhikai Xu) SSRN Preprint

Publications

• Computation of conditional expectations with guarantees
  (with Balint Gersey) arXiv Preprint 2112.01804
  Forthcoming in Journal of Scientific Computing
• Landscape analysis for shallow neural networks: complete classification of critical points for affine target functions
  (with Arnulf Jentzen, and Florian Rossmannek)
  arXiv Preprint 2103.10922
  Journal of Nonlinear Science 32(64), 2022
• A proof of convergence for gradient descent in the training of artificial neural networks for constant target functions
  (with Arnulf Jentzen, Adrian Riekert and Florian Rossmannek)
  arXiv Preprint 2102.09924
  Journal of Complexity 72, 2022
• Deep splitting method for parabolic PDEs
  (with Christian Beck, Sebastian Becker, Arnulf Jentzen and Ariel Neufeld )
  arXiv Preprint 1907.03452
  SIAM Journal on Scientific Computing 43(5), 2021, p. A3135–A3154
• Representation of increasing convex functionals with countably additive measures
  (with Michael Kupper and Ludovic Tangpi) arXiv Preprint 1502.05763
  Studia Mathematica 260, 2021, p. 121–140
• Efficient approximation of high-dimensional functions with neural networks
  (with Arnulf Jentzen and Florian Rossmannek) arXiv Preprint 1912.04310
  Forthcoming in IEEE Transactions on Neural Networks and Learning Systems
• Non-convergence of stochastic gradient descent in the training of deep neural networks
  (with Arnulf Jentzen and Florian Rossmannek) arXiv Preprint 2006.07075
  Journal of Complexity 64, June 2021
• On non-local ergodic Jacobi semigroups: spectral theory, convergence-to-equilibrium and contractivity
  (with Pierre Patie, Anna Srapionyan and Aditya Vaidyanathan) arXiv Preprint 1905.07832
  Journal de l'Ecole Polytechnique – Mathématiques 8, 2021, p. 331–378
• Solving high-dimensional optimal stopping problems using deep learning
  (with Sebastian Becker, Arnulf Jentzen and Timo Welti) arXiv Preprint 1908.01602
  European Journal of Applied Mathematics 32(3), 2021, p. 470–514
• Pricing and hedging American-style options with deep learning
  (with Sebastian Becker and Arnulf Jentzen) arXiv Preprint 1912.11060
  Journal of Risk and Financial Management 13(7), 158, 2020, p. 1–12
• Assessing asset-liability risk with neural networks
  (with John Ery and Mario Wüthrich) arXiv Preprint 2105.12432
  Risks 8(1), 16, 2020, p. 1–17
• Martingale optimal transport duality
  (with Matti Kiiski, David Prömel and Mete Soner) arXiv Preprint 1904.04644
  Mathematische Annalen 2020
• Deep optimal stopping
  (with Sebastian Becker and Arnulf Jentzen) arXiv Preprint 1804.05394
  Journal of Machine Learning Research 20(74), 2019, p. 1–25
• Measuring and allocating systemic risk
  (with Markus Brunnermeier) SSRN Preprint
  Risks 7(2), 46, 2019, p. 1–19
• Robust expected utility maximization with medial limits
  (with Daniel Bartl and Michael Kupper) arXiv Preprint 1712.07699
  Journal of Mathematical Analysis and Applications 471(1–2), 2019, p. 752–775
• Duality formulas for robust pricing and hedging in discrete time
  (with Michael Kupper and Ludovic Tangpi) arXiv Preprint 1602.06177
  SIAM Journal on Financial Mathematics 8(1), 2017, p. 738–765
• BSEs, BSDEs and fixed point problems
  (with Kihun Nam) arXiv Preprint 1410.1247
  Annals of Probability 45(6), 2017, p. 3795–3828
• Duality for increasing convex functionals with countably many marginal constraints
  (with Daniel Bartl, Michael Kupper and Ludovic Tangpi) arXiv Preprint 1509.08988
  Banach Journal of Mathematical Analysis 11(1), 2017, p. 72–89
• Equilibrium pricing in incomplete markets under translation invariant preferences
  (with Ulrich Horst, Michael Kupper, Traian Pirvu) SSRN Preprint
  Mathematics of Operations Research 41(1), 2016, p. 174–195
• Multidimensional quadratic and subquadratic BSDEs with special structure
  (with Kihun Nam) arXiv Preprint 1309.6716
  Stochastics 87(5), 2015, p. 871–884
• Conditional Analysis on ℝ^d
  (with Michael Kupper, Nicolas Vogelpoth) arXiv Preprint 1211.0747
  Set Optimization and Applications – The State of the Art. 2015, p. 179–212
• A reduced form CoCo model with deterministic conversion intensity
  (with Zhikai Xu) SSRN Preprint
  The Journal of Risk 17(3), 2015, p. 1–18
• Optimal trade execution under stochastic volatility and liquidity
  (with Tardu Sepin) SSRN Preprint
  Applied Mathematical Finance 21(4), 2014, p. 342–362
• BSDEs with terminal conditions that have bounded Malliavin derivative
  (with Kihun Nam) arXiv Preprint 1309.6716
  Journal of Functional Analysis 266(3), 2014, p. 1257–1285
• Controlling portfolio skewness and kurtosis without directly optimizing third and fourth moments (with Frank Fabozzi, Charles Fox, Woo Chang Kim) PDF
  Economic Letters 122(2), 2014, p. 154–158
• How to value a coco (with Zhikai Xu) PDF
  Creditflux Newsletter October 4, 2013
• Reward-risk ratios (with Eduard Kromer) SSRN Preprint
  Journal of Investment Strategies 3(1), 2013, p. 1–16
• Weak closedness of monotone sets of lotteries and robust representation of risk preferences
  (with Samuel Drapeau, Michael Kupper) PDF
  Risk Measures and Attitudes. European Actuarial Academy Series 2013, p. 3–9
• BSΔEs and BSDEs with non-Lipschitz drivers: comparison, convergence and robustness (with Mitja Stadje) PDF
  Bernoulli 19(3), 2013, p. 1047–1085
• Existence, minimality and approximation of solutions to BSDEs with convex drivers
  (with Mitja Stadje) PDF
  Stochastic Processes and their Applications 122(4), 2012, p. 1540–1565
• Pricing and hedging in affine models with possibility of default
  (with Alexander Wugalter) PDF
  SIAM Journal on Financial Mathematics 3(1), 2012, p. 328–350
• Processes of class Sigma, last passage times and drawdowns
  (with Ashkan Nikeghbali and Eckhard Platen) PDF
  SIAM Journal on Financial Mathematics 3(1), 2012, p. 208–303
• Ordered contribution allocations: theoretical properties and applications
  (with Eduard Kromer) PDF
  The Journal of Risk 14(1), 2011
• Optimal consumption and investment in incomplete markets with general constraints
  (with Ying Hu) arXiv Preprint 1010.0080
  Stochastics and Dynamics 11(2), 2011, p. 283–299
• Composition of time-consistent dynamic monetary risk measures in discrete time
  (with Michael Kupper) PDF
  International Journal of Theoretical and Applied Finance 14(1), 2011, p. 137–162
• A note on the Dai–Singleton canonical representation of affine term structure models
  (with Damir Filipovic and Bob Kimmel) PDF
  Mathematical Finance 20(3), 2010, p. 509–519
• Recursiveness of indifference prices and translation-invariant preferences
  (with Michael Kupper) PDF
  Mathematics and Financial Economics, 2(3), 2009, p. 173–188
• Time-inconsistency of VaR and time-consistent alternatives
  (with Mitja Stadje) PDF
  Finance Research Letters, 6(1), 2009, p. 40–46
• Dual characterization of properties of risk measures on Orlicz hearts
  (with Tianhui Michael Li) PDF
  Mathematics and Financial Economics 2(1), 2008, p. 29–55
• Risk measures on Orlicz hearts
  (with Tianhui Michael Li) PDF (more general version of an earlier paper called Monetary risk measures on maximal subspaces of Orlicz classes)
  Mathematical Finance 19(2), 2009, p. 189–214
• Second order backward stochastic differential equations and fully non-linear parabolic PDEs
  (with H. Mete Soner, Nizar Touzi and Nicolas Victoir) PDF
  Communications on Pure and Applied Mathematics 60(7), 2007, p. 1081–1110
• Market price of risk specifications for affine models: theory and evidence
  (with Damir Filipovic and Bob Kimmel) PDF
  Journal of Financial Economics 83(1), 2007, p. 123–170
• Dynamic monetary risk measures for bounded discrete-time processes
  (with Freddy Delbaen and Michael Kupper) PDF
  Electronic Journal of Probability, 11, 2006, p. 57–106
• Coherent and convex monetary risk measures for unbounded càdlàg processes
  (with Freddy Delbaen and Michael Kupper) PDF
  Finance and Stochastics, 10(3), 2006, p. 427–448
• Utility maximization under increasing risk aversion in one-period models
  (with Christopher Summer) PDF
  Finance and Stochastics, 10(1), 2006, p. 147–158
• Equivalent and absolutely continuous measure changes for jump-diffusion processes
  (with Damir Filipovic and Marc Yor) PDF
  Annals of Applied Probability, 15(3), 2005, p. 1713–1732
• The multi-dimensional super-replication problem under gamma constraints
  (with H. Mete Soner and Nizar Touzi) PDF
  Annales de l'Institute Henri Poincaré (C) Non Linear Analysis, 22(5), 2005, p. 633–666
• Small time path behavior of double stochastic integrals and applications to stochastic control
  (with H. Mete Soner and Nizar Touzi) PDF
  Annals of Applied Probability, 15(4), 2005, p. 2472–2495
• Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter H in (0,1/2) (with David Nualart) PDF
  Annales de l'Institute Henri Poincaré (B) Probability and Statistics, 41(6), 2005, p. 1049–1081
• Coherent and convex monetary risk measures for bounded càdlàg processes
  (with Freddy Delbaen and Michael Kupper) PDF
  Stochastic Processes and their Applications, 112(1), 2004, p. 1–22
• Gaussian moving averages, semimartingales and option pricing PDF
  Stochastic Processes and their Applications, 109(1), 2003, p. 47–68
• Fractional Ornstein–Uhlenbeck processes
  (with Hideyuki Kawaguchi and Makoto Maejima) PDF
  Electronic Journal of Probability, 8(3), 2003, p. 1–14
• Arbitrage in fractional Brownian motion models PDF
  Finance and Stochastics, 7(4), 2003, p. 533–553
• Representations of Gaussian measures that are equivalent to Wiener measure
  PDF Séminaire de Probabilités , Vol. XXXVII, 2003, p. 81-89. Springer Lecture Notes in Mathematics, Vol. 1832
• Sensitivity of the Black–Scholes option price to the local path behavior of the stochastic process modeling the underlying asset PDF
  Proceedings of the Steklov Institute of Mathematics, Vol. 237, 2002, p. 225–239
• Mixed fractional Brownian motion PDF
  Bernoulli, 7(6), 2001, p. 913–934
• Regularizing fractional Brownian motion with a view towards stock price modelling PDF
  Diss. ETH No. 14051, 2001