A mean field game approach to relative investment–consumption games with habit formation

Abstract

This paper studies an optimal investment–consumption problem for competitive agents with exponential or power utilities and a common finite time horizon. Each agent regards the average of habit formation and wealth from all peers as benchmarks to evaluate the performance of her decision. We formulate the n-agent game problems and the corresponding mean field game problems under the two utilities. One mean field equilibrium is derived in a closed form in each problem. In each problem with n agents, an approximate Nash equilibrium is then constructed using the obtained mean field equilibrium when n is sufficiently large. The explicit convergence order in each problem can also be obtained. In addition, we provide some numerical illustrations of our results.

Appendices

Some estimations for the fixed point system (3.42)

In this section, we make some prior estimations on $\left( {\overline{X}}_{T},{\hat{Z}}\right)$ in the system (3.42).

Proposition A.1

For any given ${\hat{Z}}\in {\mathcal {C}}_{T,+}$, there exists a unique solution ${\overline{X}}^{{\hat{Z}}}_{T}$ to the following equation

$$\begin{aligned} {\overline{X}}_{T}=C_{2}\exp \left\{ -\int _{0}^{T}\sum _{k=1}^{K}\exp \left( \frac{\theta _{k} p_{k}\delta }{1-p_{k}}t\right) \left( {\hat{Z}}_{t}\right) ^{\frac{\theta _{k}p_{k}}{p_{k}-1}}{\hat{G}}_{k}(t,{\overline{X}}_{T},{\hat{Z}})dt\right\} , \end{aligned}$$

(A.1)

where $C_{2}$ is defined in (3.43). Moreover, we have the following estimation:

$$\begin{aligned} C_{1}\le {\overline{X}}^{{\hat{Z}}}_{T}\le C_{2}, \end{aligned}$$

(A.2)

where $C_{1}$ and $C_{2}$ are two positive constants only depending on $\left\{ o_{k}\right\} _{k=1}^{K}, z_{0}, \delta$ and T.

Proof

As the LHS of (A.1) is strictly increasing with respect to ${\overline{X}}_{T}$ and the RHS of (A.1) is strictly decreasing with respect to ${\overline{X}}_{T}$, the equation (A.1) has at most one solution.

When ${\overline{X}}_{T}=0$, we get $LHS<RHS$; When ${\overline{X}}_{T}$ tends to infinity, we get $LHS>RHS$. Hence, for any given ${\hat{Z}}\in {\mathcal {C}}_{T,+}$, there exists a unique solution ${\overline{X}}^{{\hat{Z}}}_{T}$ to (A.1).

It is obviously to see that ${\overline{X}}^{{\hat{Z}}}_{T}\le C_{2}$. Note that

$$\begin{aligned} {\hat{G}}_{k}(s,{\overline{X}}_{T},{\hat{Z}})\le e^{ a_{k}(s-T)}\left( {\overline{X}}_{T}\right) ^{\frac{p_{k}\theta _{k}}{1-p_{k}}}, \end{aligned}$$

(A.3)

we have from (A.1)

$$\begin{aligned} \begin{aligned} {\overline{X}}_{T}&\ge C_{2}\exp \left\{ -\int _{0}^{T}\sum _{k=1}^{K}\exp \left( \frac{\theta _{k} p_{k}\delta }{1-p_{k}}s\right) z_{0}^{\frac{\theta _{k}p_{k}}{p_{k}-1}}e^{ a_{k}(s-T)}\left( {\overline{X}}_{T}\right) ^{\frac{p_{k}\theta _{k}}{1-p_{k}}}ds\right\} \\&= C_{2}\exp \left\{ -\sum _{k=1}^{K}D_{k}\left( {\overline{X}}_{T}\right) ^{\frac{p_{k}\theta _{k}}{1-p_{k}}}\right\} , \end{aligned} \end{aligned}$$

(A.4)

where $D_{k}:=z_{0}^{\frac{\theta _{k}p_{k}}{p_{k}-1}}e^{-a_{k}T}\int _{0}^{T}\exp \left[ (\frac{\theta _{k}p_{k}\delta }{1-p_{k}}+a_{k})s\right] ds$. Then it is straightforward to get ${\overline{X}}_{T}>C_{1}$, where $C_{1}$ is the root of the following equation

$$\begin{aligned} C_{1} = C_{2}\exp \left( -\sum _{k=1}^{K}D_{k}C_{1}^{\frac{p_{k}\theta _{k}}{1-p_{k}}}\right) , \end{aligned}$$

(A.5)

which completes the proof. $\square$

Proposition A.2

If there exists a solution $\left( {\overline{X}}_{T},{\hat{Z}}\right)$ of (3.42), it holds that

$$\begin{aligned} z_{0}\le {\hat{Z}}_{t}\le M(t)\le M(T),\quad \, \forall t \in [0,T], \end{aligned}$$

where $M(\cdot )$ is a positive increasing function only depending on $\left( x_{0},z_{0}, \delta ,T\right)$ and $\left\{ o_{k}\right\} _{k=1}^{K}$.

Proof

It is enough to find a function $M(\cdot )$ and prove ${\hat{Z}}_{t}\le M(t)$ for all $t\in [0,T]$.

We have from (3.42)

$$\begin{aligned} \begin{aligned} d{\hat{Z}}_{t}&=\delta \sum _{k=1}^{K}e^{\beta _{k}t}\left( {\hat{Z}}_{t}\right) ^{\frac{\theta _{k}p_{k}}{p_{k}-1}}{\hat{G}}_{k}(t,{\overline{X}}_{T},{\hat{Z}}){\hat{f}}_{k}(t,{\overline{X}}_{T},{\hat{Z}})F(\left\{ k\right\} )dt\\&\le \delta x_{0}\sum _{k=1}^{K}e^{\beta _{k}t}z_{0}^{\frac{\theta _{k}p_{k}}{p_{k}-1}}e^{a_{k}(t-T)}{\overline{X}}_{T}^{\frac{p_{k}\theta _{k}}{1-p_{k}}}\exp \left\{ \frac{\mu _{k}^{2}}{(1-p_{k})\sigma _{k}^{2}}t\right\} dt\\&\le \delta x_{0}\sum _{k=1}^{K}e^{-a_{k}T}z_{0}^{\frac{\theta _{k}p_{k}}{p_{k}-1}}C_{2}^{\frac{p_{k}\theta _{k}}{1-p_{k}}}\exp \left\{ \left( \beta _{k}+a_{k}+\frac{\mu _{k}^{2}}{(1-p_{k})\sigma _{k}^{2}}\right) t\right\} dt\\&\le EKdt,\quad \forall t\in [0,T], \end{aligned} \end{aligned}$$

(A.6)

where

$$\begin{aligned} \begin{aligned}&\beta _{k}:= \frac{\delta }{1-p_{k}}\left( 1+\theta _{k}p_{k}-p_{k}\right) ,\\&E:=\delta x_{0}\max _{1\le k\le K}e^{-a_{k}T}z_{0}^{\frac{\theta _{k}p_{k}}{p_{k}-1}}C_{2}^{\frac{p_{k}\theta _{k}}{1-p_{k}}}\left( \max _{t\in [0,T]}\exp \left\{ \left( \beta _{k}+a_{k}+\frac{\mu _{k}^{2}}{(1-p_{k})\sigma _{k}^{2}}\right) t\right\} \right) . \end{aligned} \end{aligned}$$

(A.7)

For the first inequality, we have just used

$$\begin{aligned} \begin{aligned}&{\hat{f}}_{k}(t,{\overline{X}}_{T},{\hat{Z}})\le x_{0}\exp \left\{ \frac{\mu _{k}^{2}}{(1-p_{k})\sigma _{k}^{2}}t\right\} ,\\&{\hat{G}}_{k}(t,{\overline{X}}_{T},{\hat{Z}})\le e^{ a_{k}(t-T)}\left( {\overline{X}}_{T}\right) ^{\frac{p_{k}\theta _{k}}{1-p_{k}}},\\&{\hat{Z}}_{t}\ge z_{0}. \end{aligned} \end{aligned}$$

(A.8)

And for the second inequality, we have just used ${\overline{X}}_{T}\le C_{2}$.

Integrating (A.6), we obtain

$$\begin{aligned} {\hat{Z}}_{t}\le M(t):= EKt+z_{0}\le M(T),\quad \forall t\in [0,T]. \end{aligned}$$

(A.9)

$\square$

Combining (A.1) with Proposition A.2, we get a new lower bound for ${\overline{X}}_{T}$:

$$\begin{aligned} \begin{aligned} {\overline{X}}_{T}\ge C_{2}\exp&\left\{ -\int _{0}^{T}\sum _{k=1}^{K}\exp \left( \frac{\theta _{k}p_{k}\delta }{1-p_{k}}s\right) z_{0}^{\frac{\theta _{k}p_{k}}{p_{k}-1}}\right. \\&\left. \left( \int _{s}^{T}e^{a_{k}(v-s)}\exp \left( \frac{\theta _{k}p_{k}\delta }{1-p_{k}}v\right) \left( {\hat{Z}}_{v}\right) ^{\frac{\theta _{k}p_{k}}{p_{k}-1}}dv\right) ^{-1}ds\right\} \\ \ge C_{0}\\:= C_{2}\exp&\left\{ -\int _{0}^{T}\sum _{k=1}^{K}\exp \left( \frac{\theta _{k}p_{k}\delta }{1-p_{k}}s\right) z_{0}^{\frac{\theta _{k}p_{k}}{p_{k}-1}}\right. \\&\left. \left( \int _{s}^{T}e^{a_{k}(v-s)}\exp \left( \frac{\theta _{k}p_{k}\delta }{1-p_{k}}v\right) \left( M(T)\right) ^{\frac{\theta _{k}p_{k}}{p_{k}-1}}dv\right) ^{-1}ds\right\} , \end{aligned} \end{aligned}$$

(A.10)

where the first inequality follows from

$$\begin{aligned} {\hat{G}}_{k}(s,{\overline{X}}_{T},{\hat{Z}})<\left( \int _{s}^{T}e^{a_{k}(v-s)}\exp \left( \frac{\theta _{k} p_{k}\delta }{1-p_{k}}v\right) \left( {\hat{Z}}_{v}\right) ^{\frac{\theta _{k}p_{k}}{p_{k}-1}}dv\right) ^{-1} \end{aligned}$$

and the second inequality follows from ${\hat{Z}}_{t}\le M(T)$ for any $t\in [0,T]$.

Additional explanation to Assumption 5

Let $i\in \left\{ 1,\ldots ,n\right\}$ and $x_{0}\in {\mathbb {R}}$ be given. Assume that $\Pi ^{i}:[0,T]\rightarrow {\mathbb {R}}$ is a continuous function and $C^{i}:[0,T]\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is of an affine form:

$$\begin{aligned} C^{i}(t,x):=p^{i}(t)x+q^{i}(t),\quad (t,x)\in [0,T]\times {\mathbb {R}}, \end{aligned}$$

for some continuous functions $p^{i},q^{i}:[0,T]\rightarrow {\mathbb {R}}$. Then, solving the following closed-loop system for $\left( X^{*,1},\ldots ,X^{*,n},X^{i}\right)$:

$$\begin{aligned} {\left\{ \begin{array}{ll} dX^{*,j}_{t}=\left( \Pi ^{*,j}(t)\mu _{\alpha (j)}-C^{*,j}(t,X^{*,j}_{t})\right) dt+\Pi ^{*,j}(t)\sigma _{\alpha (j)}dW^{j}_{t},\quad j=1,\ldots , n,\\ dX^{i}_{t}=\left( \Pi ^{i}(t)\mu _{\alpha (i)}-C^{i}(t,X^{i}_{t})\right) dt+\Pi ^{i}(t)\sigma _{\alpha (i)}dW^{i}_{t},\\ X^{*,j}_{0}=x_{0},\quad j=1,\ldots ,n,\\ X^{i}_{0}=x_{0}, \end{array}\right. } \end{aligned}$$

(B.1)

where $\left( \Pi ^{*,i},C^{*,i}\right) _{i=1}^{n}$ is given by (4.5), we obtain that for $t\in [0,T]$,

$$\begin{aligned} \begin{aligned}&X^{*,j}_{t}=f^{j}(t)+\left( T+1-t\right) \frac{\mu _{\alpha (j)}}{\sigma _{\alpha (j)}}\beta _{\alpha (j)}W^{j}_{t},\quad j=1,\ldots ,n,\\&X^{i}_{t}=g^{i}(t)+e^{-\int _{0}^{t}p^{i}(s)ds}\int _{0}^{t}e^{\int _{0}^{s}p^{i}(v)dv}\Pi ^{i}(s)\sigma _{\alpha (i)}dW^{i}_{s}, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned}&f^{j}(t):=x_{0}\frac{T+1-t}{T+1}+\left( T+1-t\right) \int _{0}^{t}\left[ \frac{\theta _{\alpha (j)}}{\left( T+1-s\right) ^{2}}\left( \int _{s}^{T}{\overline{Z}}_{v}dv+{\overline{X}}_{T}\right) -\frac{\theta _{\alpha (j)}}{T+1-s}{\overline{Z}}_{s}\right. \\&\left. \quad +\frac{1}{4}\left( \frac{\mu _{\alpha (j)}}{\sigma _{\alpha (j)}}\right) ^{2}\beta _{\alpha (j)}\frac{1}{\left( T+1-s\right) ^{2}}+\frac{3}{4}\left( \frac{\mu _{\alpha (j)}}{\sigma _{\alpha (j)}}\right) ^{2}\beta _{\alpha (j)}\right] ds,\quad j=1,\ldots ,n,\\&g^{i}(t):=x_{0}e^{-\int _{0}^{t}p^{i}(s)ds}+e^{-\int _{0}^{t}p^{i}(s)ds}\int _{0}^{t}e^{\int _{0}^{s}p^{i}(v)dv}\left( \Pi ^{i}(s)\mu _{\alpha (i)}-q^{i}(s)\right) ds. \end{aligned} \end{aligned}$$

(B.2)

It is obvious that there exists a constant M independent of j and n, such that for all $j\in \left\{ 1,\ldots ,n\right\}$,

$$\begin{aligned} \sup _{t\in [0,T]}|f^{j}(t)|\le M. \end{aligned}$$

(B.3)

Note that

$$\begin{aligned} \begin{aligned}&\sup _{t\in [0,T]}|\Pi ^{*,i}(t)|\le \sup _{1\le k\le K}\left( \beta _{k}\frac{\mu _{k}}{(\sigma _{k})^{2}}\right) (T+1),\quad i=1,\ldots ,n,\\&C^{*,i}(t,x)=\frac{1}{T+1-t}x+h^{i}(t),\quad i=1,\ldots ,n, \end{aligned} \end{aligned}$$

(B.4)

where $h^{i}:[0,T]\rightarrow {\mathbb {R}}$ is continuous and has a uniform bound M, to abuse the notation, which is independent of i and n, i.e., $\sup _{t\in [0,T]}|h^{i}(t)|\le M$ for all $i\in \left\{ 1,\ldots ,n\right\}$.

In this section, we only show

$$\begin{aligned} {\mathbb {E}}\left[ \int _{0}^{T}\left| U_{\alpha (i)}\left( C^{i}_{t}-\theta _{\alpha (i)}{\overline{Z}}^{*,n}_{t}\right) \right| ^{2}dt+\left| U_{\alpha (i)}\left( X^{i}_{T}-\theta _{\alpha (i)}{\overline{X}}_{T}^{*,n}\right) \right| ^{2}\right]<M<\infty , \end{aligned}$$

(B.5)

where M is a constant independent of n, as the other condition in Assumption 5 is much easier to verify.

Without loss of generality, we assume $\alpha (i)=1$. Then,

$$\begin{aligned} \begin{aligned}&\left| U_{1}\left( C^{i}_{t}-\theta _{1}{\overline{Z}}^{*,n}_{t}\right) \right| ^{2}\\&=\exp \left\{ -\frac{2}{\beta _{1}}\left( p^{i}(t)X^{i}_{t}+q^{i}(t)\right) +2\frac{\theta _{1}}{\beta _{1}}{\overline{Z}}^{*,n}_{t}\right\} \\&=\exp \left\{ -\frac{2}{\beta _{1}}\left( p^{i}(t)X^{i}_{t}+q^{i}(t)\right) +2\frac{\theta _{1}}{\beta _{1}}e^{-\delta t}z_{0}+\frac{\theta _{1}}{\beta _{1}}\frac{2}{n}\sum _{j=1}^{n}\int _{0}^{t}\delta e^{\delta (s-t)}C^{*,j}(s,X^{*,j}_{s})ds\right\} \\&\le C\exp \left\{ -\frac{2}{\beta _{1}}p^{i}(t)X^{i}_{t}+\delta \frac{\theta _{1}}{\beta _{1}}\frac{2}{n}\sum _{j=1}^{n}\int _{0}^{t}\frac{e^{\delta (s-t)}}{T+1-s}X^{*,j}_{s}ds\right\} \\&\le C\exp \left\{ -\frac{2}{\beta _{1}}p^{i}(t)e^{-\int _{0}^{t}p^{i}(s)ds}\int _{0}^{t}e^{\int _{0}^{s}p^{i}(v)dv}\Pi ^{i}(s)\sigma _{1}dW^{i}_{s}+\delta \frac{\theta _{1}}{\beta _{1}}\frac{2}{n}\sum _{j=1}^{n}\int _{0}^{t}e^{\delta (s-t)}\frac{\mu _{\alpha (j)}}{\sigma _{\alpha (j)}}\beta _{\alpha (j)}W^{j}_{s}ds\right\} . \end{aligned} \end{aligned}$$

By Itô’s formula, we get

$$\begin{aligned} \int _{0}^{t}e^{\delta s}\frac{\mu _{\alpha (j)}}{\sigma _{\alpha (j)}}\beta _{\alpha (j)}W^{j}_{s}ds=\int _{0}^{t}\left( F^{j}(t)-F^{j}(s)\right) dW^{j}_{s}, \end{aligned}$$

(B.6)

where

$$\begin{aligned} F^{j}(t):=\frac{\mu _{\alpha (j)}\beta _{\alpha (j)}}{\delta \sigma _{\alpha (j)}}\left( e^{\delta t}-1\right) . \end{aligned}$$

Therefore, we have

$$\begin{aligned} \begin{aligned} \left| U_{1}\left( C^{i}_{t}-\theta _{1}{\overline{Z}}^{*,n}_{t}\right) \right| ^{2} \le C\exp \left\{ \int _{0}^{t}G(s,t)dW^{i}_{s}+\frac{1}{n}\sum _{j=1}^{n}\int _{0}^{t}H^{j}(s,t)dW^{j}_{s}\right\} , \end{aligned} \end{aligned}$$

for some bounded functions $G,H^{j}:[0,T]\times [0,T]\rightarrow {\mathbb {R}}$. It is worth noting that $H^{j}$ has a uniform bound, which is still denoted by M.

Hence,

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\left[ \left| U_{1}\left( C^{i}_{t}-\theta _{1}{\overline{Z}}^{*,n}_{t}\right) \right| ^{2}\right]&\le C{\mathbb {E}}\left[ \exp \left\{ \int _{0}^{t}G(s,t)dW^{i}_{s}+\frac{1}{n}\sum _{j=1}^{n}\int _{0}^{t}H^{j}(s,t)dW^{j}_{s}\right\} \right] \\&=\exp \left\{ \int _{0}^{t}\left( G(s,t)+\frac{1}{n}H^{i}(s,t)\right) ^{2}ds+\frac{1}{n}\sum _{j\ne i}^{n}\int _{0}^{t}\left( H^{j}(s,t)\right) ^{2}ds\right\} . \end{aligned}\nonumber \\ \end{aligned}$$

(B.7)

Then, it is straightforward to get

$$\begin{aligned} {\mathbb {E}}\left[ \int _{0}^{T}\left| U_{\alpha (i)}\left( C^{i}_{t}-\theta _{\alpha (i)}{\overline{Z}}^{*,n}_{t}\right) \right| ^{2}dt\right]<M<\infty , \end{aligned}$$

(B.8)

where M is a constant independent of n. The estimation on term ${\mathbb {E}}\left[ \left| U_{1}\left( X^{i}_{T}-\theta _{1}{\overline{X}}_{T}^{*,n}\right) \right| ^{2}\right]$ is similar and we omit it here.