On the dynamics of measure flows and multi-agent and mean-field financial games

Abstract

This thesis comprises two different collection of results. We present several Itô-Wentzell formulae on Wiener spaces for real-valued functional random field of Itô type that depends on measure flows. We distinguish the full- and the marginal-measure flow cases in the spirit of mean-field games without or with common noise respectively. Second part studies several portfolio management problems featuring many-player and mean-field competition and relative performance concerns under the forward performance processes (FPP) framework. In the first problem, we focus on agents using power (CRRA) type FPPs for their investment-consumption optimisation problem under a common noise Merton market model. We solve both the many-player and mean field game providing closed-form expressions for the solutions where the limit of the former yields the latter. In our case, the FPP framework yields a continuum of solutions for the consumption component as indexed to a market parameter we coin “market-risk relative consumption preference”. The parameter permits the agent to set a preference for their consumption going forward in time that, in the competition case, reflects a common market behaviour. We show the FPP framework, under both competition and no-competition, allows the agent to disentangle their risk-tolerance and elasticity of intertemporal substitution (EIS). This, in turn, allows a finer analysis on the agent’s consumption “income” and “substitution” regimes, and, of independent interest, motivates a new strand of economics research on EIS under the FPP framework. We find that competition rescales the agent’s perception of consumption in a non-trivial manner in addition to a time-dependent Elasticity of Conformity of the agent to the market-risk relative consumption preference. In the follow-up problem, we solve the forward-utility finite player and mean-field investment game for the agent following exponential (CARA) type FPPs. We explicitly derive best response and equilibrium strategies in the single common stock asset and the asset specialisation with common noise. As an application, we draw on the core features of the forward utility paradigm and discuss a problem of time-consistent mean-field dynamic model selection in sequential time-horizons.