# Game theory and stochastic calculus

Does anybody know any details of game theory literature combined with stochastic calculus in finance? If yes, please recommend some papers of any authors who are doing exceptional work on the filed. Thank you in advance!

• Does this answer your question? Recommendations for a text on game theory in the context of market making? Oct 27, 2021 at 16:43
• @DimitriVulis no way. This does not answer my question and I can not understand how a book for which I can find little or no details about it can answer to such a special topic. Oct 27, 2021 at 17:18
• Perhaps the literature on mean field games is a good place to start? Oct 27, 2021 at 19:55
• @JoseAvilez do you have to suggest any specific paper, book or author in your mind? Oct 27, 2021 at 20:42
• Oct 27, 2021 at 21:47

## 1 Answer

The field you have in mind is covered with differential game theory, and it game birth to Mean Field Games (MFG), the book posted in a comment is certainly the reference: Probabilistic Theory of Mean Field Games with Applications volume 1 and 2 by Carmona and Delarue.

MFG started with two independent trends of research:

It is now applied to finance, the first paper doing it has been Efficiency of the Price Formation Process in Presence of High Frequency Participants: a Mean Field Game analysis by Lachapelle, Lasry, L and Pierre-Louis Lions.

Later on, different authors applied it to trading flows, see

A simple way to understand MFG. Say you have a stochastic control problem on each agent of a game, and that the value function of all agents is a function of the "positions" of all other agents. For instance each agent could desire to be away from this other, but to have an incentive that the standard deviation of the distribution of all agents is not too large.

(step 0) You can express this problem as a standard stochastic problem involving the density $$m$$ of all the agents. You can solve it as a function of $$m_0$$ that is an arbitrary density to start with.

Now you have to consider that each agent is implementing its optimal strategies:

• whereas the solution of the stochastic control problem is backward
• now the controls of the agents act as a "push forward" of their positions; it is a forward PDE.

(step 1) It is possible to "run" this PDE on the distribution of the agents: it gives you a new position $$m_1$$ of these agents.

And you iterate steps (0-1), ultimately is may converge to a fixed point (as usual in game theory) that will indeed be the optimal positions of the agents.