AA4 – Energy and Markets



Stochastic Modeling of Intraday Electricity Markets

Project Heads

Dörte Kreher, Markus Reiß

Project Members

Cassandra Milbradt (HU) 

Project Duration

15.04.2019 – 14.04.2022

Located at

HU Berlin


This project aims at developing infinite dimensional, stochastic models for intraday electricity markets with continuous trading, with a special focus on the newly launched integrated European intraday electricity market “Singe Intraday Coupling” (short SIDC, formerly known as XBID). Starting from a microscopic description of the market model, we will derive heavy traffic approximations of the system dynamics yielding a tractable description of the volumes, prices, and available transmission capacities via stochastic (partial) differential equations.


In a first step, we study the dynamics for the volumes and prices of a single electricity contract traded on national limit order market. Second, we will look at a single contract being traded in an integrated market network, exhibiting limited transmission capacities. Third, we will look at several contracts simultaneously, constructing an underlying term structure model. Furthermore, we shall develop statistical methods to estimate the model’s parameters and to provide uncertainty quantification via confidence statements, for which explicit limit distributions of the estimators and test statistics will be derived.


As opposed to the existing structural energy market models, we develop analytically tractable, mathematical models for intraday electricity markets based on their observable market microstructure and therefore on statistically measurable input factors. This may presumably be useful to gain not only qualitative, but also quantitative insights into the German Intraday Electricity Market as well as the SIDC market project.


Project 1 – Approximation for the German Intraday Electricity Market (finished Nov 20):
Jump diffusion approximation for the price dynamics of a fully state dependent limit order book model
We study the microscopic dynamics of limit order book, in which the order dynamics depend on the current best bid and ask prices as well as the standing volumes of the bid and sell side. Based on the fact that the German Intraday Electricity Market is not as liquid as classical financial markets, we include price changes to our model which do not scale with the tick size. Besides, we derive a functional convergence result, which states that the microscopic limit order book model converges to a continuous-time limit when the size of an individual order as well as the tick size tend to zero and the order arrival rate tends to infinity. It turns out that the limit order book dynamics are described by two one-dimensional jump diffusions coupled with two infinite-dimensional fluid process in the high-frequency limit. Our scaling limit provides a tractable approximation for the discrete-time order book dynamics.

Preprint available at arXiv.


Project 2 – Approximation for the European SIDC Market project (on-going):
A cross-border market model based on multiple limit order book models
In the SIDC-market, multiple European countries can trade electricity domestically as well as cross-border on their foreign markets. However, cross-border trading is limited via the total amount of available transmission capacities during a trading session. We analyse a cross-border market model between two countries. Therefore, we introduce reduced-form representations (see Cont and de Larrard) of the national limit order books in which a single limit order book is approximated by the best bid and ask prices and the size of the queues at the best bid and ask price. Moreover, we introduce a so-called capacity process which may restrict cross-border trades in each direction. With help of so-called active and inactive regimes, we are able to describe the microscopic dynamics of the cross-border market if cross-border trading is allowed or not, latter due to a full occupation of transmission capacity. It turns out that our microscopic dynamics can be defined as a regime switching discrete-time process. Moreover, we are able to approximate the rather complex discrete-time dynamics via a continuous-time regime switching process if the size of an individual order converges to zero while the order arrival rate tends to inifinity. In the limit, the queue sizes are approaximed by non-negative semimartingales with jumps at hitting times of transformations of themselves at zero, the price processes are approximated by pure jump processes with jump times equal to those of the queue length process, and the capacity process is approximated by a bounded process of finite variation. 


Change point detection at a random point in time

Motivated by our derived cross-border market model between two countries, we are interested to estimate the possible random time of a regime switch in real-world order book data. The model suggests that the time of a regime switch is a stopping time depending on the evolution of the net order flow process. Hence, we study existing literature on the detection of change points and extend it in such a way to allow also random occurences of change points. Note that this only needs to develop new results under the alternative that a change point indeed occurs. In particular, this needs to study the convergence of stochastic process with two time parameters (estimated and true change point).


Project Webpages

Selected Publications

  • D. Kreher and C. Milbradt. Jump diffusion approximation for the price dynamics of a fully state dependent limit order book model. Preprint. Available at arXiv. 2020.
  • U. Horst and D. Kreher. Second order approximations for limit order book. Finance and Stochastics, 22(4):827–877, 2018.
  • U. Horst and D. Kreher. A diffusion approximation for limit order book models. Stochastic Processes and their Applications, 129(11):4431-4479, 2019.
  • U. Horst and D. Kreher. A weak law of large numbers for a limit order book model with fully state dependent order dynamics. SIAM J. Financial Mathematics, 8:314–343, 2017.
  • M. Bibinger, M. Jirak, and M. Reiß. Volatility estimation under one-sided errors with applications to limit order books. Annals of Applied Probability, 26:2754–2790, 2016.

Selected Pictures

Simulation of the cross-border market model between two countries with limited transmission capacities (project 2). Upper Left: Best bid queues. Upper Right: Best ask queues. Lower Left: price dynamics. Lower Right: Dynamics of the capacity process. Dependent of the state of the capacity process, we observe three different regimes (white, gray, colored) in which the dynamics evolve differently.

Cross border market dynamics with limited transmission capacities.

Simulation the limit order book dynamics (project 1). Left: the bid and ask price dynamics. Middle: the standing volumes at the bid side. Right: the standing volumes at the ask side.







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