Group for Research in Decision Analysis

# Mean Field Game $$\epsilon$$-Nash equilibria for partially observed optimal execution problems in finance

## Dena Firoozi and Peter E. Caines

Partially observed Mean Field Game (PO MFG) theory was introduced and developed in (Caines and Kizilkale, 2013, 2014, Şen and Caines 2014, 2015), where it is assumed the major agent's state is partially observed by each minor agent, and the major agent completely observes its own state. Accordingly, each minor agent can recursively estimate the major agent's state, compute the system's mean field and thence generate the feedback control which yields the $$\epsilon$$-Nash property. This PO MM LQG MFG theory was further extended in recent work (Firoozi and Caines, 2015) to major-minor LQG systems in which both the major agent and the minor agents partially observe the major agent's state. The existence of $$\epsilon$$-Nash equilibria, together with the individual agents' control laws yielding the equilibria, were established wherein each minor agent recursively generates (i) an estimate of the major agent's state, and (ii) an estimate of the major agent's estimate of its own state (in order to estimate the major agent's control feedback), and hence generates a version of the system's mean field. In the current work, PO MM LQG MFG theory is applied to the optimal execution problem in the financial sector where an institutional investor, interpreted as a major agent, has partial observations of its own inventories, and high frequency traders (HFTs), interpreted as minor agents, have partial observations of the major agent's inventories. The objective for each agent is to maximize its own wealth and to avoid the occurrence of large execution prices, large rates of trading and large trading accelerations which are appropriately weighted in the agent's performance function. PO LQG MFG theory is utilized to establish the existence of $\epsilon$-Nash equilibria and a simulation example is provided.

, 17 pages