Mean Field Portfolio Games with Consumption

As a game-theoretic extension of classical optimal investment problems in [16], portfolio games have received substantial considerations in the financial mathematics literature in recent years. In a portfolio game, each player chooses her investment and/or consumption to maximize her utility induced by some risk preference criterion, by taking her competitors’ decisions into consideration. The goal of the portfolio game is to search for a Nash equilibrium (NE), such that no one would like to change her strategy unilaterally. One way to model the interaction among players in the portfolio game is through the price equilibrium; however, it typically leads to tractability issue. Another way to model the interaction is through the relative performance: each player’s utility is driven by her own wealth as well as the relative wealth to her competitors.

The study of portfolio games with relative performance concerns dates back to [7], where many player portfolio games with common stocks and trading constraint were studied: in the context of complete markets, the unique NE was obtained for general utility functions; in the context of incomplete markets, the unique NE was obtained for games with exponential utility functions, where the uniqueness result was proved by establishing an equivalent relation between each NE for the game and each solution to a multidimensional BSDE. [8] examined similar games as [7] with a different focus: [8] constructed counterexamples where no NE exists, by proving that the corresponding multidimensional BSDE has no solution. In contrast to [7, 8] where all players traded common stocks, [14] studied portfolio games where each player traded a different but correlated stock. Assuming all market parameters to be constant, [14] obtained the unique constant NE by solving coupled HJB equations. In constrast to classical utility functions, [1, 5] studied portfolio games with forward utilities. Recently, [9] studied portfolio games with general market parameters. A one-to-one correspondence between each NE and each solution to some FBSDE was established by dynamic programming principle (DPP) and martingale optimality principle (MOP), so that the portfolio game was solved by solving the FBSDE. In [9], we also obtained an asymptotic expansion result in powers of the competition parameter.

All the aforementioned results do not incorporate consumption. The only results on portfolio games with consumption, to the best of our knowledge, are [13] and [6]. Assuming all market parameters to be constant, [13] obtained a unique NE, which was called strong equilibrium²²2By [13, Definition 2.1 and Definition 3.1], a strong equilibrium is the one with time-independent investment rate and continuous consumption rate. Moreover, both the investment rate and the consumption rate are adapted to the initial filtration.. [6] examined a portfolio game with both investment and consumption under the framework of forward performance processes; the market parameters were also assumed to be constant.

In this paper, we will study portfolio games with consumption under classical utility criteria, where market parameters are allowed to be time-dependent. Assume there are $N$ risky assets in the market, with price dynamics of asset $i\in\{1,\cdots,N\}$ following

where $h^{i}$ is the return rate, $\sigma^{i}$ is the volatility corresponding to the idiosyncratic noise $W^{i}$, and $\sigma^{i0}$ is the volatility corresponding to the common noise $W^{0}$. We further assume that player $i$ specializes in asset $i$. Let $X^{i}$ be the wealth process of player $i$, whose dynamics is given by

where $x^{i}$ is the initial wealth, $\pi^{i}$ is the investment rate and $c^{i}$ is the consumption rate. The risk preference for each player is described by a power utility function, i.e. given other players’ strategies, player $i$ chooses the pair of investment rate and consumption rate $(\pi^{i},c^{i})$ to maximize the expected power utility induced by her terminal wealth and intermediate consumption:

Here, $\overline{X}^{-i}=\left(\Pi_{j\neq i}X^{j}\right)^{\frac{1}{N-1}}$ and $\overline{cX}^{-i}=\left(\Pi_{j\neq i}c^{j}X^{j}\right)^{\frac{1}{N-1}}$ are the performance indices of player $i$, and the constant $\alpha^{i}$ describes the relative importance of the utility induced by consumption and terminal wealth.

Although our $N$-player game (1.2)-(1.3) is solvable, in this paper we will focus on the corresponding mean field game (MFG), in order to make the statement more concise. According to e.g. [3, 11, 15], the corresponding MFG is:

Our contributions. Our paper has two contributions. First, under general market parameters, we provide a one-to-one correspondence between each NE of (1.4) and each solution to some FBSDE. The FBSDE is further proved to be equivalent to some BSDE, which completely characterizes the NE of the MFG (1.4). Specifically, the optimal consumption rate and the optimal investment rate for the representative player are characterized by the $Y$-component and the $Z$-component of the BSDE, respectively. In order to establish the equivalence, we need to prove two sides. On the one hand, for each NE of the MFG (1.4), we will prove that there exists an FBSDE such that the NE can be characterized by this FBSDE. On the other hand, for each solution to this FBSDE, we will prove that the solution corresponds to an NE of the MFG (1.4). The former claim can be proved by DPP, and the latter claim can be proved by MOP. Second, when market parameters do not depend on the Brownian paths, we explicitly solve the BSDE characterizing the NE. In particular, we obtain the unique NE in closed form. The assumption on market parameters implies that the BSDE reduces to an ODE; the $Z$-component is zero. Thus, the optimal investment rate can be obtained, since it is completely characterized by the $Z$-component following our first contribution. The optimal consumption rate is the unique solution to a Riccati equation, which is derived from the above ODE. We emphasize that the uniqueness result strongly relies on the one-to-one correspondence established in the first main contribution.

Connections with existing literature. From a methodology perspective, our paper shares similarities with [4, 7, 8]. Specifically, [4] used MOP to solve utility maximization problems with both investment and consumption. We apply a similar argument to our portfolio game and prove that each solution to some mean field FBSDE yields an NE of (1.4). A key difference of the MOP in [4] and in our paper is the choice of strategies. We claim that the strategies used in [4] is not suitable to portfolio games; refer to [9, Remark 2.2]. The DPP we use is adapted from [7, 8], where all players traded common stocks and there was no consumption. This is a key step to prove the uniqueness result. Note that [6] also obtained a uniqueness result for portfolio games with consumption under forward utilities, using MOP implicitly. However, the uniqueness result in [6] was implied by the definition of, especially the (super)martingale properties of forward utilities. Thus, it does not imply the uniqueness result for games with classical utilities functions. Admittedly, the most similar paper to the current one is [9], where we studied mean field portfolio games with only investment in a general framework. We also used DPP and MOP to establish the one-to-one correspondence. The current paper can be considered as a continuation of [9]. From a modeling perspective, our paper is similar to [13]; when all market parameters become time-independent, our model reduces to the one in [13]. Using a PDE approach, [13] obtained a strong equilibrium, which was proved to be unique among all strong ones. By our more probabilistic approach, we conclude that the strong equilibrium obtained in [13] is unique in the essentially bounded space. Thus, we answered the question proposed in [13].

The rest of the paper is organized as follows. After the introduction of notation, in Section 2, we establish an equivalent relationship between each solution to some mean field (F)BSDE and each NE of the MFG (1.4) with general market parameters. In Section 3, assuming that the market parameters do not depend on the Brownian paths, we construct the unique NE in closed form.

Notation. Let $(W,W^{0})$ be a two dimensional Brownian motion, defined on a probability space $(\Omega,\mathbb{P})$. Here, $W$ denotes the idiosyncratic noise for the representative player, and $W^{0}$ denotes the common noise for all players. Moreover, $\mathbb{G}=\{\mathcal{G}_{t},t\in[0,T]\}$ is assumed to be the augmented natural filtration of $(W,W^{0})$. The augmented natural filtration of $W^{0}$ is denoted by $\mathbb{F}^{0}=\{\mathcal{F}^{0}_{t},t\in[0,T]\}$. To allow for additional heterogeneity across players, we let $\mathcal{A}$ be a $\sigma$-algebra that is independent of $\mathbb{G}$. Let $\mathbb{F}=\{\mathcal{F}_{t},t\in[0,T]\}$ be the $\sigma$-algebra generated by $\mathcal{A}$ and $\mathbb{G}$.

Denote $\textrm{Prog}(\Omega\times[0,T])$ the space of all stochastic processes that are $\mathcal{F}$-progressively measurable. For each $\eta\in\textrm{Prog}(\Omega\times[0,T])$, define $\|\eta\|_{\infty}=\operatorname*{ess\,sup}_{\omega\in\Omega,t\in[0,T]}|\eta_{t}(\omega)|$. Let $L^{\infty}$ be the space of all essentially bounded stochastic processes, i.e.,

Define the BMO space under $\mathbb{P}$ as

For each positive random variable $\xi$, denote $\widehat{\xi}:=\log(\xi)$.

Let $C$ be a generic positive constant, which may vary from line to line.

We assume the space of admissible strategies for the representative player is $L^{\infty}\times L^{\infty}$.³³3The space $L^{\infty}$ to accommodate investment rates is smaller than $H^{2}_{BMO}$, which is commonly used in the literature. We use $L^{\infty}$ as our admissible space for technical purpose; see the estimates in the proof of Theorem 2.2. However, we do not lose much generality because the closed form investment rate constructed in Section 3 stays in $L^{\infty}$. Moreover, we say that the tuple $(\mu^{*},\nu^{*},\pi^{*},c^{*})$ is an NE of the MFG (1.4), if $(\pi^{*},c^{*})$ is admissible, with $X^{*}$ being the corresponding wealth, $\mu^{*}_{t}=\exp\left(\mathbb{E}[\log X^{*}_{t}|\mathcal{F}^{0}_{t}]\right)$ and $\nu^{*}_{t}=\exp\left(\mathbb{E}[\log(c^{*}_{t}X^{*}_{t})|\mathcal{F}^{0}_{t}]\right)$, $t\in[0,T]$, and if the optimality condition holds for each admissible strategy $(\pi,c)$:

Throughout the paper, the following assumptions are in force.

Assumption 1. The initial wealth $x$, risk aversion parameter $\gamma$, competition parameter $\theta$, and weight parameter $\alpha$ of the population are assumed to be bounded $\mathcal{A}$-random variables. Moreover, $x$ and $\alpha$ are $\mathbb{R}_{+}$-valued, $\gamma$ is valued in $(-\infty,1)/\{0\}$ and $\theta$ is valued in $[0,1]$.

Assume the return rate $h\in L^{\infty}$ and the volatilities $(\sigma,\sigma^{0})\in L^{\infty}\times L^{\infty}$. Moreover, $|\gamma|$ and $|\sigma|+|\sigma^{0}|$ are bounded away from $0$, i.e., there exist positive constants $\underline{\gamma}$ and $\underline{\sigma}$ such that $|\gamma|\geq\underline{\gamma}>0$ a.s. and $|\sigma|+|\sigma^{0}|\geq\underline{\sigma}>0$ a.s. a.e..