**Project Heads**

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

**Project Members**

William Salkeld

**Former Members**

**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 in non Markovian situations.

**Selected Publications
**

- C. Bayer, P. K. Friz, and J. Gatheral.
*Pricing under rough volatility. Quantitative Finance* - D. Belomestny and J. Schoenmakers.
. Palgrave Macmillan, London, 2018.*Advanced simulation-based methods for optimal stopping and control* - D. Belomestny, J. Schoenmakers, V. Spokoiny, and B. Zharkynbay.
. Comm. in Math. Sci., 18(1):109–121, 2020.*Optimal stopping via reinforced regression* . 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.*

**Preprints**

- P. Hager, E. Neuman.
, 4 Aug 2020, arxiv:2008.01385 (accepted at AAP).*The Multiplicative Chaos of H=0 Fractional Brownian Fields* - C. Bayer, D. Belomestny, P. Hager, P. Pigato, J. Schoenmakers, V. Spokoiny.
, 24 Nov 2020, arxiv:2011.1238.*Reinforced optimal control* - C. Bayer, P. Hager, S. Riedel, J. Schoenmakers.
, 26 Nov 2020, WIAS.PREPRINT.2790.*Optimal stopping with signatures* - P. K. Friz, P. Hager, N. Tapia,
, 8 Feb 2021, arxiv:2102.03345.*Unified Signature Cumulants and Generalized Magnus Expansions*

**Selected Pictures
**

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