AA4 – Energy Transition



Optimal Control in Energy Markets Using Rough Analysis and Deep Networks

Project Heads

Christian Bayer, Peter Friz, John Schoenmakers, Vladimir Spokoiny

Project Members

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 for optimal stopping problems in non Markovian frameworks.

Selected Publications


  • C. Bayer, R. Tempone, S. Wolfers. Pricing American options by exercise rate optimization. Quantitative Finance, 20:11, 1749-1760, 2020 (journal, arXiv).
  • P. K. Friz, P. Gassiat, P. Pigato. Precise asymptotics: Robust stochastic volatility models. The Annals of Applied Probability, 31(2):896–940, 2021 (journal, arXiv).
  • C. Bayer, D. Belomestny, P. Hager, P. Pigato, J. Schoenmakers. Randomized optimal stopping algorithms and their convergence analysis. SIAM Journal on Financial Mathematics, 12(3), 1201–1225, 2021, (journal, arXiv)
  • P. Hager, E. Neuman. The Multiplicative Chaos of H=0 Fractional Brownian Fields. The Annals of Applied Probability, 32(3):2139-2179, 2022 (journal, arXiv).
  • C. Bayer, D. Belomestny, P. Hager, P. Pigato, J. Schoenmakers, V. Spokoiny. Reinforced optimal control. Comm. in Math. Sci., 20:1951-1978, 2022 (journal, arXiv)
  • P. K. Friz, P. Hager, N. Tapia. Unified Signature Cumulants and Generalized Magnus Expansions. Forum of Mathematics, Sigma. Vol. 10. Cambridge University Press, 2022 (journal, arXiv)
  • 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. Optimal stopping with signatures. The Annals of Applied Probability, 33(1):238-273, 2023 (journal, arXiv).
  • 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)


  • P. Bank, C. Bayer, P. K. Friz, L. Pelizzari. Rough PDEs for local stochastic volatility models, 18. Jul 2023, arXiv:2307.09216

Selected Pictures

Deep Signature Stopping – Optimal Stopping of fractional Brownian motion

Deep Signature Stopping - Optimal Stopping of fractional Brownian motion

Optimal stopping of a fractional Brownian motion for different Hurst parameters H and time discretization with J=100 and J=1000 steps using the Deep signature stopping methodology.

Hierarchical Reinforced Regression

Hierarchical Reinforced Regression

The hierarchical reinforced regression algorithm gradually improves the quality of approximation of the value function by reinforcing the regression basis in the backward induction by using the value function of the lower hierarchy.

Hierarchical Reinforced Regression – Multiple Stopping Example

Multiple Stopping / Swing Option Problem

Multiple stopping/swing option problem. Lower bounds (color) and upper bounds (grey) calculated using the hierarchical reinforced regression methods with different recursion depths I. Ψ_i is a polynomial regression basis of degree i.

Rough volatility

Rough Volatility

Exercise rate for max-call option

exercise rate for max-call option

Asymptotic formulas

Implied smile for rough volatility

Implied smile for rough volatility

Moderate deviations - Rough Bergomi model

Moderate deviations – Rough Bergomi model

Please insert any kind of pictures (photos, diagramms, simulations, graphics) related to the project in the above right field (Image with Text), by choosing the green plus image on top of the text editor. (You will be directed to the media library where you can add new files.)
(We need pictures for a lot of purposes in different contexts, like posters, scientific reports, flyers, website,…
Please upload pictures that might be just nice to look at, illustrate, explain or summarize your work.)

As Title in the above form please add a copyright.

And please give a short description of the picture and the context in the above textbox.

Don’t forget to press the “Save changes” button at the bottom of the box.

If you want to add more pictures, please use the “clone”-button at the right top of the above grey box.