I am considering to take a position as a phd student in a computer science department. I am a mathematician with a master degree in finance and my research interests are mainly focused in game theory. I would like to study the field of mechanism and information design with applications in financial markets. To name but a few papers in the field, you can take a look here Dynamic Information Provision (2019), Dynamic Mechanism Design: An Introduction (2017) and Bayes correlated equilibrium andthe comparison of information structures in games (2016). This is a small part of the literature that I am interested, but I do not know if I can work on a phd in this field as a doctoral student in a computer sience department. Could you please tell me, how this literature can be combined with computer science or maybe if it does not matter and I can do whatever kind of reasearch I want no matter the department?
This depends on whether the CS department you are studying at has somebody working in this field. Some of them (at least three of the top ten in the U.S.) do, and some of them don't, and some of them may let you work with a PhD advisor in a different department. You are going to have to do some research on your own.
Note: it's possible in many computer science departments to do a PhD in an area that your advisor isn't working in. I strongly recommend that you do not do this.