**Project Heads**

*Christian Bayer, Peter Friz, John Schoenmakers, Vladimir Spokoiny*

**Project Members**

**Former Members**

*Paul Hager, Paolo Pigato, Sebastian Riedel, Wiliam **Salkeld, Nikolas Tapia*

**Project Duration**

First funding period: 01.01.2019 – 31.12.2021; second funding period: 01.01.2022 – 31.03.2025

**Located at**

TU Berlin / WIAS

In this project, we will develop efficient methods for modeling energy price processes and methods for solving related control or decision problems. Following (Bennedsen, M. 2017, ‘A rough multi-factor model of electricity spot prices’, *Energy Economics*, bind 63, s. 301-313), we explore the use of rough pricing models, which have been very successful for modeling equity markets. Due to the lacking Markov property, rough models pose new mathematical challenges for stochastic control. In this area deep learning is playing an increasingly important role. In this respect, a big challenge is the incorporation of deep learning architectures in new methods for optimal stopping, multiple stopping and control problems.

We compute solutions to these control problems by combining new methods from machine learning (reinforcement learning) with classical tools from optimal control (dynamic programming, regression methods, duality formulas).

Further, we employ methods that are based on the signature, that is the sequence consisting of iterated integrals of the underlying path – giving an efficient compression of the signal, particularly promising for optimal stopping problems in non Markovian frameworks.

**Selected Publications
**

. Quantitative Finance, 20:11, 1749-1760, 2020 (journal, arXiv).**Pricing American options by exercise rate optimization**- P. K. Friz, P. Gassiat, P. Pigato.
. The Annals of Applied Probability, 31(2):896–940, 2021 (journal, arXiv).**Precise asymptotics: Robust stochastic volatility models** - C. Bayer, D. Belomestny, P. Hager, P. Pigato, J. Schoenmakers.
SIAM Journal on Financial Mathematics, 12(3), 1201–1225, 2021, (journal, arXiv)*Randomized optimal stopping algorithms and their convergence analysis.* - P. Hager, E. Neuman.
The Annals of Applied Probability, 32(3):2139-2179, 2022 (journal, arXiv).*The Multiplicative Chaos of H=0 Fractional Brownian Fields.* - C. Bayer, D. Belomestny, P. Hager, P. Pigato, J. Schoenmakers, V. Spokoiny.
Comm. in Math. Sci., 20:1951-1978, 2022 (journal, arXiv)*Reinforced optimal control.* - P. K. Friz, P. Hager, N. Tapia.
Forum of Mathematics, Sigma. Vol. 10. Cambridge University Press, 2022 (journal, arXiv)*Unified Signature Cumulants and Generalized Magnus Expansions.* - C. Bayer, J. Qiu, Y. Yao.
**Pricing Options Under Rough Volatility with Backward SPDEs.**SIAM Journal on Financial Mathematics 13 (1), 179-212, 2022 (journal, arXiv) - C. Bayer, P. Hager, S. Riedel, J. Schoenmakers.
The Annals of Applied Probability, 33(1):238-273, 2023 (journal, arXiv).*Optimal stopping with signatures.* - D. Belomestny, J. Schoenmakers.
**From optimal martingales to randomized dual optimal stopping**. Quantitative Finance 23.7-8: 1099-1113, 2023 (journal, arXiv) - D. Belomestny, C. Bender, J. Schoenmakers.
**Solving optimal stopping problems via randomization and empirical dual optimization**. Mathematics of Operations Research 48.3: 1454-1480, 2023 (journal, WIAS preprint) - C. Bayer, M. Eigel, L. Sallandt, P. Trunschke.
**Pricing high-dimensional Bermudan options with hierarchical tensor formats.**SIAM Journal on Financial Mathematics, 14(2), 383-406, 2023 (journal, arXiv) - P.K. Friz, T. Wagenhofer.
**Reconstructing volatility: Pricing of index options under rough volatility**. Mathematical Finance 33.1: 19-40, 2023 (journal, arXiv) - F. Bourgey, S. De Marco, P.K. Friz, P. Pigato.
**Local volatility under rough volatility**. Mathematical Finance 33.4:1119-1145, 2023 (journal, arXiv) - C. Bayer, D. Belomestny, O. Butkovsky, J. Schoenmakers.
**RKHS regularization of singular local stochastic volatility model McKean-Vlasov models**. Finance & Stochastic, to appear in 2024 (arXiv)

**Books**

- C. Bayer, P.K. Friz, M. Fukasawa, J. Gatheral, A. Jacquier, M. Rosenbaum.
**Rough volatility.**SIAM, 2023. (SIAM)

**Preprints**

- P. Bank, C. Bayer, P. K. Friz, L. Pelizzari.
**Rough PDEs for local stochastic volatility models,**18. Jul 2023,*arXiv:2307.09216* - C. Bayer, L. Pelizzari, J. Schoenmakers.
**Primal and dual optimal stopping with signatures**, 06. Dez 2023, arXiv:2312.03444

**Selected Pictures
**

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