AA4 – Energy and Markets



Stochastic Modeling of Intraday Electricity Markets

Project Heads

Dörte Kreher, Markus Reiß

Project Members

Cassandra Milbradt (HU) 

Project Duration

01.01.2019 – 31.12.2021

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 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 XBID 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 a one-sided discrete-time limit order book model, in which the order dynamics depend on the current best available price and the current volume density function, simultaneously. 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 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 zeros and the order arrival rate tends to infinity. It turns out that the limit order book dynamics are described by a one-dimensional jump diffusion coupled with an 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 XBID Market project (on-going):
A cross-border market model based on multiple limit order book models
In the XBID-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 local limit order books and a capacity process which may restrict cross-border trades in each direction. Assuming that the capacity process is non-restricted, we are able to derive heavy traffic approximations of the price, standing volumes and the capacity process. Of course, we further aim to derive limit results in the restricted setting.

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 one-sided limit order book (project 1). Left: the bid price dynamics. Right: the absolute volume density function of the standing volumes of the bid side.

a) Bid price dynamics

b) Volume dynamics of the bid side








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