# Decisions in energy markets via deep learning and optimal control

The change in energy has boosted the share of renewable energies in Germany’s total energy production from a meager 3% in 1990 to over 25% last year. This change is accompanied by a massive and ongoing restructuration of the energy sector. The geographic locations of energy production and energy consumption have become more and more disconnected, and as a result, the interdependencies of the market players and the complexity of their behavior have increased dramatically. Unlike energy production from fossil fuels, the input in the grid from renewable sources such as wind and solar energy comes along with a high uncertainty and is difficult to predict, resulting in an increased volatility of the energy prices and in the emergence of energy markets to facilitate matching offer and demand. So, like actors on the classical markets, both producers and large scale consumers of energy face risks and seek protection from these risks by structured contracts, so-called energy derivatives. In the gas and electricity markets for example, so called “swing options” have become very popular. A typical swing option gives the holder the right to buy or sell a certain amount of gas, electricity or storage capacity at a certain prescribed number of trading dates. As a consequence, market players are faced with optimal decision problems involving, for example, buying, storing, or selling energy over time, exercising certain energy contracts and so on. Therefore, on the one hand, design of optimal strategies for complexly structured decision problems is called for, and on the other hand, a high demand of adequate statistical prediction algorithms based on adequate statistical modeling of market prices is arisen. The distinctive properties of energy markets make the existent repertoire of statistical methods and stochastic algorithms developed in the framework of classical financial mathematics insufficient and call for the development of new tailored methods. For example, finding correct prices for energy derivatives is typically difficult due to their complex structured exercise features and their highly path-dependent structure. When developing strategies for energy producers and/or traders, both the particularities of the energy market and the constraints posed by the limited storage and production resources of the actor have to be adequately modeled. As a general consequence, the rising complexity of the markets poses challenging mathematical problems that may be categorized into the following main streams:

1. Advanced methods for solving complex decision problems

2. Adequate modeling of underlying dynamics, such as energy price processes

Generally, the aim of solving an optimal decision problem, that is an optimal stopping or optimal control problem, is twofold. On the one hand, one aims at bounding its “true” value from below and above, and on the other hand one tries to find a “good” decision policy consistent with these bounds. In fact, a “good” (primal) decision policy yields a lower bound, and a “good” system of (dual) martingales yields an upper bound to the “true” value, respectively. Thus, naturally, solution methods for optimal decision problems can be classified in primal and dual approaches. For the standard optimal stopping problem, [BBS09] succeeded to avoid the time consuming sub-simulations in the Andersen-Broadie algorithm by constructing the dual martingale via a discrete Clark-Ocone derivative of some approximation to the Snell-envelope, obtained by regression on a suitable set of basis functions. Later on in [SZH13] a related regression method was developed that also avoids sub-simulations and, even more, does not require any input approximation to the solution of the problem (i.e. the Snell envelope). Particularly the later approach looked promising for generalization to quite general control problems. As a first non-trivial application, this method was successfully applied in the con- text of a hydro electricity storage model [HSZ16]. One of the main goals in this project proposal is a systematic numerical treatment of generic optimal decision problems in “real-life” applications by incorporating recent ideas of a relatively new concept of data analysis and prediction: Deep Learning. On the other hand, it is intended to include principles of Deep Learning in methods for forecasting and estimating price distribution processes in a systematic way.

**Responsible persons:** Denis Belomestny, Vladimir Spokoiny