Bergische Universität Wuppertal Fachbereich Mathematik und Naturwissenschaften Angewandte Mathematik - Numerische Analysis (AMNA)

People Research Publications Teaching

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Scientific goals

The project deals with qualitative and numerical analysis of nonlinear partial differential equations arising in mathematical finance. The main purpose is to develop new or to extend existing dynamic portfolio optimization models based on solving Hamilton-Jacobi-Bellman (HJB) equations, Black-Scholes equations for pricing financial instruments, as well as free boundary problems for advection-diffusion equations arising in financial mathematics. Specific research objectives include:

Main objective:

Specific research objectives

1. Comparison of different numerical methods for solving Hamilton-Jacobi-Bellman equations. Qualitative and quantitative evaluation of various numerical methods.
2. Investigating the viability of extending Hamilton-Jacobi-Bellman equation-based models from one-dimensional underlying process to a multi-dimensional one.
3. Analysing nonlinear option pricing extensions based on Black-Scholes partial differential equation, Frey-Stremme partial integro-differential equations and free-boundary problems.
4. Development and analysis of efficient numerical schemes for solving linear and nonlinear models with emphasis on models based on fully nonlinear parabolic PDEs including non-smooth diffusion terms.

The industrial partner of this project, the GEFA bank, has the main office located in Wuppertal and a branch in Bratislava. This setting will offer the unique opportunity for mutual exchange between academia and industry, transfer of knowledge and also with the option for jointly supervised theses.

German team:

• Anna Clevenhaus
• Matthias Ehrhardt (Leader)
• Mike Felpel
• Ali Ghafoor
• Michael Günther
• Lorenc Kapllani
• Tatiana Kossaczká
• Michelle Muniz
• Long Teng
• Sarah Treibert

Slovakian team:

• Maros Bobulský
• Terézia Fulová
• Ján Gašper
• Zuzana Girová
• Jakub Hrdina
• Cyril Izuchukwu
• Soňa Kilianová (Leader)
• Jan Komadel
• Radka Litvajová
• Igor Melichercik
• Daniel Ševčovič
• Beata Stehliková
• Magdaléna Zitnanská

German institutions:

• Department of Mathematics and Informatics, Faculty of Mathematics and Natural Sciences, University of Wuppertal.

Slovakian institutions:

• Department of Applied Mathematics and Statistics, Faculty of Mathematics, Physics and Informatics, Comenius University Bratislava

Publications related to the Project

2020

• L. Teng, X. Wu, M. Günther and M. Ehrhardt, A new methodology to create valid time-dependent correlation matrices via isospectral flows, ESAIM: Mathematical Modelling and Numerical Analysis (M2AN), Volume 54, Number 2, (March-April 2020), 361-371. DOI:10.1051/m2an/2019064
• M. Muniz, M. Ehrhardt and M. Günther, Approximating correlation matrices using stochastic Lie group methods, Mathematics 2021, 9(1), 94. DOI: 10.3390/math9010094
• A. Clevenhaus, M. Ehrhardt, M. Günther and D. Sevcovic, Pricing American Options with a Non-constant Penalty Parameter, J. Risk Financial Manag. 2020, 13(6), 124. DOI 10.3390/jrfm13060124
• L. Kapllani, L. Teng and M. Ehrhardt, A Multistep Scheme to solve Backward Stochastic Differential Equations for Option Pricing on GPUs, in: I. Dimov and S. Fidanova (eds.), Advances in High Performance Computing, Results of the International Conference on "High Performance Computing" Borovets, Bulgaria, 2019, Studies in Computational Intelligence 902, Springer, 2021, pp. 196-208. DOI: 10.1007/978-3-030-55347-0
• A. Clevenhaus, M. Ehrhardt and M. Günther, An ADI Sparse Grid method for pricing efficiently American Options under the Heston model, Preprint 20/45, October 2020.
• M. Muniz, M. Ehrhardt, M. Günther and R. Winkler, Higher Strong Order Methods for Itô SDEs on Matrix Lie Groups, Preprint 21/03, January 2021.

• A. Clevenhaus, C. Totzeck and M. Ehrhardt, A Gradient Descent Algorithm for the Heston Model, Preprint 21/32, October 2021.
• D. Ševčovič and C. Izuchukwu Udeani, Application of maximal monotone operator method for solving Hamilton-Jacobi-Bellman equation arising from optimal portfolio selection problem, Japan Journal of Industrial and Applied Mathematics, 38(3), 2021, 693-713.
• D. Ševčovič and C. Izuchukwu Udeani, Multidimensional linear and nonlinear partial integro-differential equation in Bessel potential spaces with applications in option pricing, Mathematics 2021, 9(13), 1463.
• T. Kossaczká, M. Ehrhardt and M. Günther, A Deep Smoothness WENO Method with Applications in Option Pricing, Preprint 21/26, August 2021.
• M. Muniz, M. Ehrhardt, M. Günther and R. Winkler, Stochastic Runge-Kutta-Munthe-Kaas methods in the modelling of perturbed rigid bodies, accepted: Advances in Applied Mathematics and Mechanics, 2021.
• T. Kossaczká, M. Ehrhardt and M. Günther, Enhanced fifth order WENO Shock-Capturing Schemes with Deep Learning, in press: Results in Applied Mathematics (2021). DOI: 10.1016/j.rinam.2021.100201 (open access).
• A. Clevenhaus, M. Ehrhardt and M. Günther, An ADI Sparse Grid method for pricing efficiently American Options under the Heston model, Adv. Appl. Math. Mech., Volume 13 (2021), 1384-1397. DOI: 10.4208/aamm.OA-2020-0317
• M. Muniz, M. Ehrhardt and M. Günther, Approximating correlation matrices using stochastic Lie group methods, Mathematics 2021, 9(1), 94. DOI: 10.3390/math9010094
• M. Trnovská and J. Hrdina, Duality aspects in convex conic programming, submitted in Elsevier, Linear Algebra and its Application, 2021.
• S. Kilianová and D. Ševčovič, Riccati transformation for Hamilton-Jacobi-Bellman equations with multidimensional underlying processes, Working paper, 2021, FMFI UK.

• LAURA DRAHT (Kombi BA)
  Sensitivitätsanalyse und Optimale Steuerung eines Mathematischen Modells für COVID-19 und numerische Simulation mit Nicht-Standard Finite Differenzen Verfahren,
  Bachelorarbeit Bergische Universität Wuppertal, März 2022.
  (Joint Supervision with Dr. Soňa Kilianová, Comenius U Bratislava)
• LEON HOFFE (Wirtschaftsmathematik)
  Die numerische Lösung eines Problems der optimalen Steuerung mit der Bellman Gleichung anhand des Beispiels der Besteuerung von CO₂-Emissionen,
  Bachelorarbeit Bergische Universität Wuppertal, Mai 2021.
  (Joint Supervision with Dr. Soňa Kilianová, Comenius U Bratislava)
• BENJAMIN LEONHARDT (Wirtschaftsmathematik)
  Optimale Steuerung erschöpfbarer Resourcen am Beispiel der globalen CO₂-Emissionen,
  Bachelorarbeit Bergische Universität Wuppertal, Mai 2021.
  (Joint Supervision with Dr. Soňa Kilianová, Comenius U Bratislava)
• PATRICK NGAPGOU DONFACK (Wirtschaftsmathematik)
  Eine Schätzung der optimalen CO₂-Steuer mit dem DICE-Modell,
  Bachelorarbeit Bergische Universität Wuppertal, November 2020.
  (Joint Supervision with Dr. Soňa Kilianová, Comenius U Bratislava)

Talks related to the Project

2020

• Bulgarian-German-Slovakian MATTHIAS Workshop, Wuppertal, March 4, 2020
  • Jakub Hrdina, Duality in Conic Programming
  • Terézia Fulová, Finding the nearest low-rank Correlation Matrix
  • Michele Muniz, Correlation Flows - and how not to compute them
  • Ján Gašper, Gillespie Algorithm for non-Markovian Transitions
  • Anna Clevenhaus, Pricing American Options with an affine-linear penalty function
  • Lorenc Kapllani, Machine learning and GPU computing for solving high dimensional BSDEs with applications in finance
  (postponed to September 3, 2020)
• Minisymposium "Novel methods in computational finance" by Matthias Ehrhardt and Jan ter Maten at ECMI 2020 conference, Limerick, Ireland, June 21-26, 2020.
  (ECMI 2020 has been postponed due to Covid-19. Provisional rescheduled dates are June 21-25, 2021).
• German-Slovakian MATTHIAS Workshop, Wuppertal, September 3, 2020
  • Ján Gašper, Gillespie algorithm for non-Markovian processes
  • Terézia Fulová, Finding low-rank solutions in financial factor models
  • Cyril Izuchukwu, Application of maximal monotone operator method for solving Hamilton-Jacobi-Bellman equation arising in optimal portfolio selection problem
  • Michelle Muniz, Structure-preserving schemes for SDEs on Matrix Lie groups: A comparison between the exponential and the Cayley map
  • Anna Clevenhaus, An ADI Sparse Grid method for American Options under the Heston model
  • Mike Felpel, Effective Stochastic Volatility: Applications to ZABR-type Models
  • Lorenc Kapllani, Deep learning for high dimensional nonlinear BSDEs
• Minisymposium "Novel methods in computational finance" by Daniel Sevcovic (Bratislava) and Matthias Ehrhardt (Wuppertal) at ALGORITMY 2020 - Conference on Scientific Computing, Vysoke Tatry, Podbanske, Slovakia, September 10-15, 2020.
  • Matthias Ehrhardt, High-order methods for parabolic equations in multiple space dimensions for option pricing
  • Zuzana Buckova, Dynamic correlation in a convergence model of interest rates
  • Tatiana Jasurkova, Calibration of the Vasicek model of interest rates using bicriteria optimization
  • Beatriz Salvador Mancho, XVA for a stochastic volatility model
  • Soňa Kilianová, Transformation techniques of Hamilton-Jacobi-Bellman equations
  • Cyril Izuchukwu, Application of maximal monotone operator method for solving Hamilton-Jacobi-Bellman equations
  • Igor Melichercik, Different ways of using second pillar savings in Slovakia
  • Terézia Fulová, Finding low-rank solutions in financial factor models
  

• 4th International Conference on Computational Finance 2021, Wuppertal, May 24-28, 2021, postponed to June 6-10, 2022.
• Minisymposium "Novel methods in computational finance" by Matthias Ehrhardt and Jan ter Maten at virtual ECMI 2021 conference, hosted by Wuppertal, Germany, April 13-15, 2021.
• 20th Joint Czech-German-Slovak Conference on Mathematical Methods in Economy and Industry, Smolenice Castle, Slovakia, September 15-19, 2021.
• Matthias Ehrhardt, Deep Smoothness WENO method with applications in finance
• Soňa Kilianová, Hamilton-Jacobi-Bellman equation in stochastic dynamic portfolio optimization
• Daniel Ševčovič, Nonlinearities in financial modelling
• Terézia Fulová, Searching for low-rank solutions to semidefinite problems with a specific structure
• Jakub Hrdina, Cone of Non-negative Polynomials
• Ján Gašper, Maximum likelihood parameter estimation for discrete state space and continuous time stochastic processes
• Cyril Izuchukwu Udeani, Application of maximal monotone operator method for solving Hamilton-Jacobi-Bellman equation arising from optimal portfolio selection problem.

Staff Exchange

2019/2020

• Soňa Kilianová, Lecture Numerics of ODEs and Lecture Advanced Topics Transformation Methods in Computational Finance, University of Wuppertal, Winter term 2019/20.

Joint Supervision of Theses

2020/2021

• Patrick Ngapgou Donfack, Bachelor Thesis, Wuppertal, 2020.
• Benjamin Leonhardt, Bachelor Thesis, Wuppertal, 2021.
• Leon Hoffe, Bachelor Thesis, Wuppertal, 2021.
• Laura Draht, Bachelor Thesis, Wuppertal, 2022.

Habilitation Thesis

2021

• Soňa Kilianová, Optimization and Differential Equations in Mathematical Modeling, habilitačná práca, Comenius University, 2021.

Pre-Dissertation Theses

2021

• Jakub Hrdina, Cone of Non-negative Polynomials, 2021-2022, FMFI UK
• Ján Gašper, Modelovanie a numerické riešenie modelov šírenia infekčných ochorení, 2021-2022, FMFI UK