A Neural Network Approach to High-Dimensional Optimal Switching Problems with Jumps in Energy Markets

We develop a backward-in-time machine learning algorithm that uses a sequence of neural networks to solve optimal switching problems in energy production, where electricity and fossil fuel prices are subject to stochastic jumps. We then apply this algorithm to a variety of energy scheduling problems, including novel high-dimensional energy production problems. Our experimental results demonstrate that the algorithm performs with accuracy and experiences linear to sub-linear slowdowns as dimension increases, demonstrating the value of the algorithm for solving high-dimensional switching problems.

[1]  Alessandro Gnoatto,et al.  A deep solver for BSDEs with jumps , 2022, SSRN Electronic Journal.

[2]  Rüdiger Frey,et al.  Deep Neural Network Algorithms for Parabolic PIDEs and Applications in Insurance Mathematics , 2021, ArXiv.

[3]  Niklas L. P. Lundström,et al.  Management strategies for run-of-river hydropower plants: an optimal switching approach , 2020, Optimization and Engineering.

[4]  Maxime Sangnier,et al.  Approximating Lipschitz continuous functions with GroupSort neural networks , 2020, AISTATS.

[5]  Huyên Pham,et al.  Approximation Error Analysis of Some Deep Backward Schemes for Nonlinear PDEs , 2020, SIAM J. Sci. Comput..

[6]  Arnulf Jentzen,et al.  Pricing and Hedging American-Style Options with Deep Learning , 2019, Journal of Risk and Financial Management.

[7]  H. Pham,et al.  Deep backward schemes for high-dimensional nonlinear PDEs , 2019, Math. Comput..

[8]  Huyên Pham,et al.  Some machine learning schemes for high-dimensional nonlinear PDEs , 2019, ArXiv.

[9]  Xavier Warin,et al.  Nesting Monte Carlo for high-dimensional non-linear PDEs , 2018, Monte Carlo Methods Appl..

[10]  Justin A. Sirignano,et al.  DGM: A deep learning algorithm for solving partial differential equations , 2017, J. Comput. Phys..

[11]  Arnulf Jentzen,et al.  Solving high-dimensional partial differential equations using deep learning , 2017, Proceedings of the National Academy of Sciences.

[12]  Theodore B. Trafalis,et al.  Natural gas storage valuation via least squares Monte Carlo and support vector regression , 2017 .

[13]  Dmitry Yarotsky,et al.  Error bounds for approximations with deep ReLU networks , 2016, Neural Networks.

[14]  Matt Thompson.  Natural gas storage valuation, optimization, market and credit risk management , 2016 .

[15]  J. Chassagneux,et al.  Rate of convergence for the discrete-time approximation of reflected BSDEs arising in switching problems , 2016, Stochastic Processes and their Applications.

[16]  Lingfei Li,et al.  An Efficient Algorithm Based on Eigenfunction Expansions for Some Optimal Timing Problems in Finance , 2015, J. Comput. Appl. Math..

[17]  Erhan Bayraktar,et al.  Ergodicity of Robust Switching Control and Nonlinear System of Quasi-Variational Inequalities , 2015, SIAM J. Control. Optim..

[18]  Jimmy Ba,et al.  Adam: A Method for Stochastic Optimization , 2014, ICLR.

[19]  Erhan Bayraktar,et al.  Robust Feedback Switching Control: Dynamic Programming and Viscosity Solutions , 2014, SIAM J. Control. Optim..

[20]  Romuald Elie,et al.  BSDE representations for optimal switching problems with controlled volatility , 2014 .

[21]  Eugene A. Feinberg,et al.  Optimizing cloud utilization via switching decisions , 2014, PERV.

[22]  Huyên Pham,et al.  A Probabilistic Numerical Method for Optimal Multiple Switching Problems in High Dimension , 2014, SIAM J. Financial Math..

[23]  H. Pham,et al.  A probabilistic numerical method for optimal multiple switching problem and application to investments in electricity generation , 2012, 1210.8175.

[24]  Romuald Elie,et al.  Discrete-time Approximation of Multidimensional BSDEs with oblique reflections , 2012, 1210.1407.

[25]  Nicola Bruti-Liberati.  Numerical Solution of Stochastic Differential Equations with Jumps in Finance , 2010 .

[26]  Espen R. Jakobsen,et al.  Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes , 2010 .

[27]  Jianfeng Zhang,et al.  Switching problem and related system of reflected backward SDEs , 2010 .

[28]  Arnaud Porchet,et al.  Valuation of power plants by utility indifference and numerical computation , 2009, Math. Methods Oper. Res..

[29]  Yun Li,et al.  Natural Gas Storage Valuation , 2009 .

[30]  R. Carmona,et al.  Pricing Asset Scheduling Flexibility using Optimal Switching , 2008 .

[31]  Nicola Bruti-Liberati,et al.  Strong approximations of stochastic differential equations with jumps , 2007 .

[32]  Erhan Bayraktar,et al.  On the One-Dimensional Optimal Switching Problem , 2007, Math. Oper. Res..

[33]  Monique Jeanblanc,et al.  On the Starting and Stopping Problem: Application in Reversible Investments , 2007, Math. Oper. Res..

[34]  Rémi Munos,et al.  Numerical Methods for the Pricing of Swing Options: A Stochastic Control Approach , 2006 .

[35]  B. Bouchard,et al.  Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations , 2004 .

[36]  S. Peng,et al.  Reflected solutions of backward SDE's, and related obstacle problems for PDE's , 1997 .

[37]  Xunjing Li,et al.  Necessary Conditions for Optimal Control of Stochastic Systems with Random Jumps , 1994 .

[38]  Kurt Hornik,et al.  Multilayer feedforward networks are universal approximators , 1989, Neural Networks.

[39]  T. Faniran.  Numerical Solution of Stochastic Differential Equations , 2015 .

[40]  Christoph Weber,et al.  Gas storage valuation applying numerically constructed recombining trees , 2012, Eur. J. Oper. Res..

[41]  B. Bouchard,et al.  Discrete time approximation of decoupled Forward-Backward SDE with jumps , 2008 .

[42]  Y. Maghsoodi,et al.  Exact solutions and doubly efficient approximations of jump-diffusion itô equations , 1998 .