Deep learning-based schemes for solving high-dimensional nonlinear BSDEs and their uncertainty quantification / submitted by Lorenc Kapllani, M. Sc. Wuppertal, February 25, 2025
Content
- List of Figures
- List of Tables
- List of Publications
- List of Presentations
- Abbreviations and List of Symbols
- Summary
- Deep Learning BSDE Schemes
- Introduction
- Preliminaries
- Forward deep learning schemes
- Backward deep learning schemes
- Numerical results
- The simple bounded BSDE
- BSDE with quadratic control
- The Black-Scholes-Barenblatt BSDE
- Option pricing with different interest rates
- BSDE with non-additive diffusion
- Conclusions
- Differential Deep Learning BSDE Schemes
- Introduction
- Malliavin differentiable BSDEs and differential deep learning
- Backward differential deep learning schemes
- Forward differential deep learning schemes
- Numerical results
- The Black-Scholes BSDE
- Option pricing with different interest rates
- The Hamilton-Jacobi-Bellman equation
- The Black-Scholes extended with local volatility
- BSDE with non-additive diffusion
- The Black-Scholes BSDE with correlated noise
- Conclusions
- UQ for Deep Learning BSDE Schemes
- Introduction
- Uncertainty decomposition in the DBSDE scheme
- A suitable stochastic control problem to represent the DBSDE scheme
- Sources of uncertainty in the DBSDE scheme
- UQ model
- Numerical results
- Experimental setup
- The impact of the sources of uncertainty in the DBSDE scheme
- Performance of the UQ model
- The DBSDE scheme for the Black-Scholes example
- The DBSDE scheme for the Burgers type example
- The LaDBSDE scheme for the Burgers type example
- Practical implications of the UQ model
- Conclusions
- Conclusions and outlook
- Impact of the sources of uncertainty for the Burgers type BSDE
- Normality assumption of the error distribution
- Bibliography