Common Noise by Random Measures: Constructing Mean-Field Equilibria for Competitive Investment and Hedging

TL;DR

The paper addresses Nash equilibria in mean-field portfolio games with relative performance using CARA utility, incorporating common noise via integer-valued random measures and Brownian motion. It proves existence and uniqueness of mean-field equilibria driven by McKean–Vlasov BSDEs with jumps without a weak-interaction restriction and provides a constructive representation of the MFE from a single-agent optimization through a projection, along with a three-component decomposition of the equilibrium strategy. In Markov-switching settings, the MFE is characterized by a system of PDEs, and the authors demonstrate a PDE-based computational pathway with explicit examples illustrating the impact of common noise. In the limit of vanishing risk aversion, the model converges to a quadratic-hedging MFG under relative performance, revealing a bridge between utility-based and quadratic hedging frameworks in a jump-driven mean-field context.

Abstract

We construct Nash-equilibria in mean-field portfolio games of optimal investment and hedging under relative performance concerns with exponential (CARA) utility preferences. Common noise dynamics are modeled by integer-valued random measures, for instance Poisson random measures, in addition to Brownian motions. Agents differ in individual risk aversions, competition weights, and initial capital endowments, while their contingent claim liabilities depend on both common and idiosyncratic risk factors. Mean-field equilibria are characterized by solutions to McKean-Vlasov backward stochastic differential equations with jumps, for which we prove existence and uniqueness of solutions, without assuming mean field interaction to be small. Moreover, we show how the equilibrium can be constructed from the optimal strategy of a single-agent optimization problem (without mean-field interaction) via an appropriate projection. Using successive changes of measure, our derivation provides a decomposition of the equilibrium strategy into three components with clear interpretations. Finally, we show how a limiting mean-field game of quadratic (instead of utility-based) hedging with relative performance concerns arises for vanishing risk aversion.

Figures (2)

• Figure 1: MFE strategy $\widetilde{\vartheta}$ (recalling relation \ref{['parametrizationVarthetaTheta']}) for the MFG \ref{['MFG']} with claim $B$ from \ref{['DefFinancialStopLossContractForPDE']}. The MFE is expressed as the number of shares held at the corresponding time and given by a function $\widetilde{\vartheta}_t=\widetilde{\vartheta}(t,S_t,L_{t-})$ of time $t$, the asset price $S_t=s$, and the state $L_{t-}^B$ of the Markov chain defined in \ref{['DefLtB']}. Parameters: $S_0=1,\sigma=0.3, \varphi = 0, C=1, K_1=0.5, K_2=0.5, \alpha=2.5, \rho=0.9, T=3, \lambda^0=0.9$ and $\lambda^1=1-\lambda^0$
• Figure 2: Illustration of how the MFE strategy $\widetilde{\vartheta}$ (recalling relation \ref{['parametrizationVarthetaTheta']}) of the MFG \ref{['MFG']} with claim $B$ from \ref{['DefFinancialStopLossContractForPDE']} varies with the intensity $\lambda^0$ of the common Poisson process. The plots compare for $\lambda^1=1-\lambda^0$ the MFE strategies $\widetilde{\vartheta}$ for $\lambda^0\in \{0.3,0.5,0.7,0.9\}$ against the benchmark MFE strategy with $\lambda^0=0.1$ in the state $L_{t-}^B=(0,1)$ of the Markov chain from \ref{['DefLtB']}. All other parameters are as in \ref{['fig:MFE']}.

Theorems & Definitions (44)

• Example 2.2
• Remark 2.3
• Remark 2.5
• Example 2.7
• Remark 2.8
• Definition 3.2: mean-field equilibrium
• Example 3.3
• Remark 3.4
• Remark 3.5
• Remark 3.6