Groupe d’études et de recherche en analyse des décisions

# An optimal execution problem in finance with acquisition and liquidation objectives: An MFG formulation

## Dena Firoozi et Peter E. Caines

Partially observed major minor LQG mean field game theory is applied to an optimal execution problem in finance; following standard financial models, controlled linear system dynamics are postulated where an institutional investor (interpreted as a major agent) in the market aims to liquidate a specific amount of shares and has partial observations of its own state (which includes its inventory). Furthermore, the market is assumed to have two populations of high frequency traders (interpreted as minor agents) who wish to liquidate or acquire a certain number of shares within a specific time, and each one of them has partial observations of its own state and the major agent's state (which include the corresponding 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. The existence of $\epsilon$-Nash equilibria together with the individual agents' trading strategies yielding the equilibria, were established. A simulation example is provided.

, 17 pages