Translation invariant mean field games with common noise

This note highlights a special class of mean field games in which the coefficients satisfy a convolution-type structural condition. A mean field game of this type with common noise is related to a certain mean field game without common noise by a simple transformation, which permits a tractable construction of a solution of the problem with common noise from a solution of the problem without

The Self-Financing Equation in High Frequency Markets

High Frequency Trading (HFT) represents an ever growing proportion of all financial transactions as most markets have now switched to electronic order book systems. The main goal of the paper is to propose continuous time equations which generalize the self-financing relationships of frictionless markets to electronic markets with limit order books. We use NASDAQ ITCH data to identify significant empirical features such as price impact and recovery, rough paths of inventories and vanishing bid-ask spreads. Starting from these features, we identify microscopic identities holding on the trade clock, and through a diffusion limit argument, derive continuous time equations which provide a macroscopic description of properties of the order book. These equations naturally differentiate between trading via limit and market orders. We give several applications (including hedging European options with limit orders, market maker optimal spread choice, and toxicity indexes) to illustrate their impact and how they can be used to the benefit of Low Frequency Traders (LFTs)

A reinterpretation of set differential equations as differential equations in a Banach space

Set differential equations are usually formulated in terms of the Hukuhara differential, which implies heavy restrictions for the nature of a solution. We propose to reformulate set differential equations as ordinary differential equations in a Banach space by identifying the convex and compact subsets of $\R^d$ with their support functions. Using this representation, we demonstrate how existence and uniqueness results can be applied to set differential equations. We provide a simple example, which can be treated in support function representation, but not in the Hukuhara setting

Timing of Transients : Quantifying Reaching Times and Transient Behavior in Complex Systems

The authors thank the anonymous referees for their detailed and constructive feedback. This paper was developed within the scope of the IRTG 1740/TRP 2011/50151-0, funded by the DFG/FAPESP. This work was conducted in the framework of PIK’s flagship project on coevolutionary pathways (copan). The authors thank CoNDyNet (FKZ 03SF0472A) for their cooperation. The authors gratefully acknowledge the European Regional Development Fund (ERDF), the German Federal Ministry of Education and Research and the Land Brandenburg for supporting this project by providing resources on the high performance computer system at the Potsdam Institute for Climate Impact Research. The authors thank the developers of the used software: Python[47], Numerical Python[48] and Scientific Python[49]. The authors thank Sabine Auer, Karsten Bolts, Catrin Ciemer, Jonathan Donges, Reik Donner, Jasper Franke, Frank Hellmann, Jakob Kolb, Chiranjit Mitra, Finn Muller-Hansen, Jan Nitzbon, Anton Plietzsch Stefan Ruschel, Tiago Pereira da Silva, Francisco A. Rodrigues, Paul Schultz, and Lyubov Tupikina for helpful discussions and comments.Peer reviewedPublisher PD

Environmental complexity influences association network structure and network-based diffusion of foraging information in fish shoals

This project was funded by grants from the Natural Environment Research Council (NE/D010365/1) and the European Research Council (EVOCULTURE 232823) to K.N.L.Socially transmitted information can significantly affect the ways in which animals interact with their environments. We used network-based diffusion analysis, a novel and powerful tool for exploring information transmission, to model the rate at which sticklebacks (Gasterosteus aculeatus) discovered prey patches, comparing shoals foraging in open and structured environments. We found that for groups in the open environment, individuals tended to recruit to both the prey patch and empty comparison patches at similar times, suggesting that patch discovery was not greatly affected by direct social transmission. In contrast, in structured environments we found strong evidence that information about prey patch location was socially transmitted and moreover that the pathway of information transmission followed the shoals' association network structures. Our findings highlight the importance of considering habitat structure when investigating the diffusion of information through populations and imply that association networks take on greater ecological significance in structured than open environments.Publisher PDFPeer reviewe