Propagation of Chaos of Forward–Backward Stochastic Differential Equations with Graphon Interactions

Abstract

In this paper, we study graphon mean field games using a system of forward–backward stochastic differential equations. We establish the existence and uniqueness of solutions under two different assumptions and prove the stability with respect to the interacting graphons which are necessary to show propagation of chaos results. As an application of propagation of chaos, we prove the convergence of n-player game Nash equilibrium for a general model, which is new in the theory of graphon mean field games.

Appendices

Appendix A: Measurability

Lemma A.1

$\varvec{\mu }: [0,1] \rightarrow \mathcal {C}\left( [0,T]; \mathcal {P}_p(\mathbb {R}) \right)$ is measurable if and only if for any $t \in [0,T]$,

$$\begin{aligned} \lambda \mapsto \varvec{\mu }^{\lambda }(t) \in \mathcal {P}_p(\mathbb {R}) \text { is measurable}. \end{aligned}$$

(A.1)

Proof

The proof of ‘only if’ is trivial. For the proof of ‘if’ part, we note that with sup norm, $\mathcal {C}\left( [0,T];\mathcal {P}_p(\mathbb {R})\right)$ is a topological subspace of $\mathcal {C}^0 \left( [0,T];\mathcal {P}_p(\mathbb {R})\right)$. For any $n \in \mathbb {N}$, due to (A.1) we know that

$$\begin{aligned} \lambda \mapsto (\varvec{\mu }^{\lambda }(T/n), \cdots , \varvec{\mu }^{\lambda }(T) ) \text { is measurable}. \end{aligned}$$

We construct $\varvec{\mu }_n \in \mathcal {M}\left( [0,T];\mathcal {P}_p(\mathbb {R})\right)$

$$\begin{aligned} \varvec{\mu }_n^{\lambda }(t):= \varvec{\mu }^{\lambda }\left( \frac{i}{n}\right) , \, t \in \left( \frac{(i-1)T}{n},\frac{iT}{n} \right] , \, i=1, \cdots , n. \end{aligned}$$

Then it can be easily verified that $\lambda \rightarrow \varvec{\mu }_n^{\lambda }$ is measurable. By the continuity of $\varvec{\mu }^{\lambda }(\cdot )$, $\lim \limits _{n \rightarrow \infty } \varvec{\mu }_n^{\lambda }= \varvec{\mu }^{\lambda }$ is measurable in $\lambda$. $\square$

Lemma A.2

A function $x: \lambda \mapsto x^{\lambda } \in L^{p,c}_{\mathcal {F}}$ belongs to $\mathcal {M}L^{p,c}_{\mathcal {F}}$ if and only if $\lambda \mapsto x^{\lambda }_t \in L^p_{\mathcal {F}_t}$ is measurable for any $t \in [0,T]$.

Proof

Note that $L^{p,c}_{\mathcal {F}} \ni \varvec{\eta } \mapsto \varvec{\eta }_t \in L^p_{\mathcal {F}_t}$ is continuous. Therefore it can be readily seen that the measurability of $\lambda \mapsto x^{\lambda }$ implies the measurability of $\lambda \mapsto x^{\lambda }_t$ for any $t \in [0,T]$.

Conversely, define $x^{N} \in \mathcal {M}L^{p,2}_{\mathcal {F}}$ for $N \in \mathbb {N}$ as follows,

$$\begin{aligned} x^{N,\lambda }_t&:=x^{\lambda }_{nT/N}, \quad \forall t \in [nT/N, (n+1)T/N), \, n =0,\cdots , N-2, \, \lambda \in [0,1], \\ x^{N,\lambda }_t&:=x^{\lambda }_{(N-1)T/N}, \quad \forall t \in [(N-1)T/N,T], \, \lambda \in [0,1]. \end{aligned}$$

According to our hypothesis, it can be easily seen that

$$\begin{aligned} \lambda \mapsto x^{N, \lambda } \in \left( L^{p,2}_{\mathcal {F}}, \widetilde{ \Vert \cdot \Vert }_S^p\right) \end{aligned}$$

is measurable, and also the limit

$$\begin{aligned} \lambda \mapsto x^{\lambda } \in \left( L^{p,c}_{\mathcal {F}}, \widetilde{ \Vert \cdot \Vert }_S^p \right) . \end{aligned}$$

$\square$

Lemma A.3

Take a polish space $\Omega$ and a Borel probability measure $(\mathcal {F},P)$ over $\Omega$. Take another measure space $(E, \Sigma , m)$. Suppose $\rho : E \times \mathbb {R}\rightarrow \mathbb {R}$ is a real-valued function such that $x \mapsto \rho (e,x)$ is continuous for any $e \in E$, $e \mapsto \rho (e,x)$ is measurable for any $x \in \mathbb {R}$, and $|\rho (e,x)| \le C(1+|x|),\, \forall (e,x) \in E \times \mathbb {R}$ for some positive constant C. Then given any measurable mapping $e \mapsto X(e) \in L^p(\Omega , \mathcal {F},P)$, the Banach-valued function

$$\begin{aligned} e \mapsto \rho (e, X(e)) \in L^p(\Omega , \mathcal {F},P) \end{aligned}$$

is also measurable.

Proof

According to [42, Proposition 3.4.5], the Banach space $L^p(\Omega , \mathcal {F}, P)$ is separable. Therefore as a result of Pettis measurability theorem, any measurable function $X: E \rightarrow L^p(\Omega , \mathcal {F}, P)$ is also strongly measurable, i.e., X can be written as a pointwise limit of simple functions

$$\begin{aligned} X^n= \sum _{i=1}^{m_n}\mathbbm {1}_{S_{m_n}} x_{m_n}, \end{aligned}$$

where $m_n \in \mathbb {N}$, $S_1, \cdots , S_{m_n}$ is a finite collection of disjoint subsets of E, and $x_1, \cdots , x_{m_n} \in L^p(\Omega , \mathcal {F}, P)$. It is then readily seen that

$$\begin{aligned} e \mapsto \rho (e, X^n(e)) \end{aligned}$$

is measurable, and thus $\rho (\cdot , X(\cdot ))= \lim \limits _n \rho (\cdot , X^n(\cdot ))$ is measurable. $\square$

Lemma A.4

Suppose $\psi : [0,1] \times [0,T] \times \mathbb {R}\rightarrow \mathbb {R}$ is a measurable function such that $x \mapsto \psi (\lambda ,t,x)$ is continuous and grows at most linearly uniformly for $(\lambda ,t) \in [0,1] \times [0,T]$. Given any measurable $\lambda \mapsto X^{\lambda } \in L^{p,c}_{\mathcal {F}}$, we have that

$$\begin{aligned} \lambda \mapsto \int _0^{\cdot } \psi (\lambda , s ,X^{\lambda }(s)) \, ds \in L^{p,c}_{\mathcal {F}} \end{aligned}$$

(5.1)

is measurable.

Proof

By our assumption, it is clear that $(\lambda ,s) \mapsto X^{\lambda }(s)$ is measurable. Applying Lemma A.3 with $E=[0,1] \times [0,T]$, it is readily seen that

$$\begin{aligned} (\lambda ,s) \mapsto \psi (\lambda , s ,X^{\lambda }(s)) \end{aligned}$$

(5.2)

is measurable. The function (5.2) is also Bochner integrable due to our linear growth assumption in x. Thus by the Fubini theorem of Bochner theorem,

$$\begin{aligned} \lambda \mapsto \int _0^{t} \psi (\lambda , s,X^{\lambda }(s)) \, ds \end{aligned}$$

is measurable for any $t \in [0,T]$. Now the measurability of (5.1) follows from Lemma A.2. $\square$

Remark A.1

Using approximation of simple functions, one can easily verify that Bochner integral coincides with Lebesgue integral.

Appendix B: Weak Uniqueness of FBSDE

The notion of weak existence and uniqueness for FBSDEs are almost the same to the ones considered for classical SDEs, see e.g. [43,44,45].

Definition B.1

A five-tuple $(\Omega , \mathcal {F}, \mathbb {F}, P, W)$ is said to be a standard set-up if W is a Brownian motion over the probability space $(\Omega , \mathcal {F},\mathbb {F},P)$ and $\mathbb {F}:=\{ \mathcal {F}\}_{t \ge 0}$ is complete and right continuous.

Consider an FBSDE

$$\begin{aligned} {\left\{ \begin{array}{ll} X_t=x+\int _0^t B(s,X_s, Y_s) \, ds + \sigma \, W_t, \\ Y_t= Q(X_T)+\int _t^T F(s,X_s,Y_s) \, ds- \int _t^T Z_s \, d W_s, \end{array}\right. } \end{aligned}$$

(B.1)

where B, F, Q are progressively measurable functions.

Definition B.2

A triple of processes (X, Y, Z) is said to be a weak solution of (B.1) if there exists a standard set-up $(\Omega , \mathcal {F}, \mathbb {F}, P, W)$ such that (X, Y, Z) are adapted to the filtration $\mathbb {F}$ and satisfy (B.1) a.s. If (X, Y, Z) and $(\widetilde{X}, \widetilde{Y},\widetilde{Z})$ are two weak solutions of (B.1) on the same set-up, we say that pathwise uniqueness holds if

$$\begin{aligned} \mathbb {P} \left[ (X_t,Y_t)=(\widetilde{X}_t,\widetilde{Y}_t), \, \forall t \in [0,T] \right] =1. \end{aligned}$$

By Yamada-Watanabe Theorem for SDEs, pathwise uniqueness implies uniqueness in law. We have the same result for FBSDEs.

Lemma B.1

Suppose the pathwise uniqueness property holds for FBSDE (B.1). Then for any two weak solutions (X, Y, Z) on $(\Omega , \mathcal {F}, \mathbb {F}, P, W)$ and $(\widetilde{X},\widetilde{Y},\widetilde{Z})$ on $(\widetilde{\Omega }, \widetilde{\mathcal {F}}, \widetilde{\mathbb {F}}, \widetilde{P}, \widetilde{W})$, their distributions coincide.

Proof

See [43, Theorem 5.1]. $\square$