Stochastic Graphon Games: II. The Linear-Quadratic Case

Abstract

In this paper, we analyze linear-quadratic stochastic differential games with a continuum of players interacting through graphon aggregates, each state being subject to idiosyncratic Brownian shocks. The major technical issue is the joint measurability of the player state trajectories with respect to samples and player labels, which is required to compute for example costs involving the graphon aggregate. To resolve this issue we set the game in a Fubini extension of a product probability space. We provide conditions under which the graphon aggregates are deterministic and the linear state equation is uniquely solvable for all players in the continuum. The Pontryagin maximum principle yields equilibrium conditions for the graphon game in the form of a forward-backward stochastic differential equation, for which we establish existence and uniqueness. We then study how graphon games approximate games with finitely many players over graphs with random weights. We illustrate some of the results with a numerical example.

A Proofs

1.1 A.1 Proof of Proposition 1

We drop the superscript ${\underline{\alpha }}$ since the strategy profile does not change throughout the proof. By Lemma 3, ${\mathbb {X}}^z\in L^2_\boxtimes (\varOmega \times I; E)$ so by Lemma 1, the mapping U is well defined. We turn to the contraction property.

To prove that U is a strict contraction, let $z, {\tilde{z}} \in L^2_{\boxtimes }(\varOmega \times I; E)$. By Gronwall’s inequality and the boundedness of the graphon,

$$\begin{aligned} \begin{aligned} {\mathbb {E}}^\boxtimes \left[ \Vert \left[ Uz\right] - \left[ U{\tilde{z}}\right] \Vert ^2_E\right] \le C\int _0^T{\mathbb {E}}^\boxtimes \left[ \sup _{s\in [0,t]}\big |z_s - \widetilde{z}_s\big |^2\right] dt, \end{aligned} \end{aligned}$$

(23)

where $C>0$ is a finite constant depending only on T, $\Vert W\Vert _2$, and the coefficient bound. Iterating the inequality (23) and making use of the fact that

$$\begin{aligned} {\mathbb {E}}^{\boxtimes }\Big [\sup _{s\in [0,t]}|z_s|^2\Big ] \le {\mathbb {E}}^{\boxtimes }\big [\Vert z_s\Vert ^2_E\big ],\quad t\in [0,T],\ z\in L^2_{\boxtimes }(\varOmega \times I; E), \end{aligned}$$

we have

$$\begin{aligned} {\mathbb {E}}^{\boxtimes }\Big [\big \Vert [(U^N)z] - [(U^N)\widetilde{z}]\big \Vert _E^2\Big ] \le \frac{(CT)^N}{N!}{\mathbb {E}}^{\boxtimes }\Big [\Vert z - {\tilde{z}}\Vert _E^2\Big ],\quad N\in {\mathbb {N}}. \end{aligned}$$

Hence, for some $N\in {\mathbb {N}}$, $(U^N)$ is a contraction mapping on from $L^2_{\boxtimes }(\varOmega \times I; E)$ to itself. The existence of a unique fixed point to U in the Banach space $L^2_{\boxtimes }(\varOmega \times I; E)$ then follows by the Banach fixed-point theorem for iterated mappings, see e.g. [5]. $\square$

1.2 A.2 Proof of Theorem 3

The coefficient matrices in (16) are given in terms of $a,b,c,C_f$ and $C_h$ as follows:

$$\begin{aligned}&\varGamma := \begin{bmatrix} a - \frac{b[C_f]_{12}}{[C_f]_{22}} &{} -\frac{b^2}{[C_f]_{22}} \\ \frac{[C_f]_{12}^2}{[C_f]_{22}}- [C_f]_{11} &{} \frac{b[C_f]_{12}}{[C_f]_{22}}- a \end{bmatrix},\ \varGamma _Z := \begin{bmatrix} c - \frac{b [C_f]_{32}}{[C_f]_{22}} \\ \frac{[C_f]_{12}[C_f]_{32}}{[C_f]_{22}}-[C_f]_{12} \end{bmatrix},\ \varGamma _T := \begin{bmatrix} [C_h]_{11} \\ [C_h]_{12} \end{bmatrix}. \end{aligned}$$

Step 0: Uniqueness in the Fubini extension

Assume that (X, p, q) and $({\widetilde{X}}, {\widetilde{p}}, \widetilde{q})$ are solutions to the FBSDE (16) in the sense that (16) is satisfied ${\mathbb {P}}\boxtimes \lambda$-a.s. and ${\mathbb {E}}^\boxtimes \big [ \Vert X\Vert _E^2 + \Vert p\Vert _E^2 + \int _0^T|q_t|^2dt\big ] < \infty$. Uniqueness, i.e., that

$$\begin{aligned} {\mathbb {E}}^\boxtimes \Big [\Vert p - {\widetilde{p}}\Vert _E^2 + \Vert X - \widetilde{X}\Vert ^2_E + \int _0^T |q_t - {\widetilde{q}}_t|^2dt \Big ] = 0 \end{aligned}$$

can be proven along standard lines of proof.

Step 1: An ansatz for $p^x$

We will look for a solution defined with the following ansatz: for each $x\in I$ there exists differentiable and deterministic mappings $t\mapsto \eta ^x_t$ and $t\mapsto \zeta ^x_t$ such that

$$\begin{aligned} p^x_t = \eta ^x_t {\hat{X}}^x_t + \zeta ^x_t. \end{aligned}$$

(24)

Plugging (24) into (16) and matching terms we obtain $\eta ^x = q^x$ and the following system for $({\hat{X}}^x,\zeta ^x,\eta ^x; x\in I)$:

$$\begin{aligned} \begin{aligned}&{\dot{\eta }}^x_t = - (\eta ^x_t)^2\varGamma _{12}(x) - \eta ^x_t(\varGamma _{11}(x) + \varGamma _{21}(x) - \varGamma _{22}(x)) + \varGamma _{21}(x),&\eta ^x_T = \varGamma _{T,1}(x), \\&d\begin{bmatrix} {\hat{X}}_t^x \\ \zeta ^x_t \end{bmatrix} = \begin{bmatrix} \varGamma _{11}(x) + \eta ^x_t\varGamma _{12}(x) &{} \varGamma _{12}(x) \\ 0 &{} \varGamma _{22}(x) - \eta ^x_t\varGamma _{12}(x) \end{bmatrix} \begin{bmatrix} {\hat{X}}_t^x \\ \zeta ^x_t \end{bmatrix} dt \\&\quad + \begin{bmatrix} \varGamma _{Z,1}(x) \\ \varGamma _{Z,2}(x) - \eta ^x_t\varGamma _{Z,1}(x) \end{bmatrix}{\hat{Z}}^x_t dt + \begin{bmatrix} 1 \\ 0 \end{bmatrix} dB^x_t,&x\in I,\ t\in [0,T], \\&{\hat{X}}^x_0 = \xi ^x,\quad \zeta ^x_T = \varGamma _{T,2}(x){\hat{Z}}^x_T,\quad {\hat{Z}}^x_t = \int _I w(x,y){\mathbb {E}}[{\hat{X}}^y_t]\lambda (dy),&x\in I,\ t\in [0,T]. \end{aligned} \end{aligned}$$

(25)

The Riccati equation for $\eta ^x$ in (25) does not depend on the other variables and can be solved independently. Furthermore, under Assumption 1 and 2 it has a unique solution $(\eta ^x_t)_{t\in [0,T]}$ for all $x\in I$ and $\sup _{(t,x)\in [0,T]\times I}|\eta ^x_t| < \infty$, see for example [9, Sec. 2.4.1]. Thus, to prove existence of a solution to (25) it is sufficient to study the forward-backward system for $({\hat{X}}, \zeta )$, which is the subject matter of the next steps.

Step 2: Unique solvability of (25) for short time horizons

If we fix a collection of aggregates ${\hat{Z}}^x\in E$, $x\in I$, then $\zeta ^x$ and subsequently ${\hat{X}}^x$ can be solved explicitly for all $x\in I$. This "decoupling" property of the aggregate provides us with a simple proof of short time existence and uniqueness. By a fixed-point argument there exists a unique solution $({\hat{X}}^x,\zeta ^x)$ to (25) in $L^2_\boxtimes (\varOmega \times I; E)\times L^2(I;E)$ when T is small enough.

Step 3: Setting the stage for the induction approach

Inspired by the induction approach, described in detail in [9, Sec. 4.1.2.], we now extend existence and uniqueness from the previous step to any finite time horizon.

For any $\tau \in [T_0,T]$, where $T_0 := T-c_0$ and $c_0>0$, let $\xi _\tau$ be such that $(\xi ^x_\tau )_{x\in I}$ are e.p.i. and $\xi ^x_\tau$ is ${\mathcal {F}}^x_\tau$-measurable for all $x\in I$. Assume that $c_0>0$ is small enough so that

$$\begin{aligned} \begin{aligned}&d\begin{bmatrix} {\hat{X}}_t^x \\ \zeta ^x_t \end{bmatrix} = \begin{bmatrix} \varGamma _{11}(x) + \eta ^x_t\varGamma _{12}(x) &{} \varGamma _{12}(x) \\ 0 &{} \varGamma _{22}(x) - \eta ^x_t\varGamma _{12}(x) \end{bmatrix} \begin{bmatrix} {\hat{X}}_t^x \\ \zeta ^x_t \end{bmatrix} dt \\&\quad + \begin{bmatrix} \varGamma _{Z,1}(x) \\ \varGamma _{Z,2}(x) - \eta ^x_t\varGamma _{Z,1}(x) \end{bmatrix}{\hat{Z}}^x_t dt + \begin{bmatrix} 1 \\ 0 \end{bmatrix} dB^x_t,&x\in I,\ t\in [\tau , T], \\&{\hat{X}}^x_\tau = \xi ^x_\tau ,\quad \zeta ^x_T = \varGamma _{T,2}(x){\hat{Z}}^x_T, \quad {\hat{Z}}^x_t = \int _I w(x,y){\mathbb {E}}[{\hat{X}}^y_t]\lambda (dy),&x\in I,\ t\in [\tau ,T], \end{aligned} \end{aligned}$$

(26)

has a unique solution as found in Step 2. Denote the solution $({\hat{X}}^{0:x,\tau ,\xi _\tau ^\cdot }_t,\zeta ^{0:x,\tau ,\xi _\tau ^\cdot }_t; t\in [\tau ,T])$.

Assume now that the forward-backward system (25) has a solution over the full time horizon: $({\hat{X}}^x_t,\zeta ^x_t; t\in [0,T])$. It is also a solution to (26) on the subinterval $[T_0,T]$ with ${\hat{X}}_{T_0}^x$ as initial condition at $T_0 = T-c_0$. By the unique solvability of (26),

$$\begin{aligned} {\mathbb {P}}\boxtimes \lambda \Big ((\omega ,x) : ({\hat{X}}^x_t(\omega ),\zeta ^x_t) = ({\hat{X}}^{0:x,T_0,{\hat{X}}^\cdot _{T_0}}_t(\omega ), \zeta ^{0:x,T_0,{\hat{X}}^\cdot _{T_0}}_t),\ T_0 \le t\le T \Big ) = 1. \end{aligned}$$

Now consider for some $\tau \in [0,T_0]$ the forward-backward system

$$\begin{aligned} \begin{aligned}&d\begin{bmatrix} {\hat{X}}_t^x \\ \zeta ^x_t \end{bmatrix} = \begin{bmatrix} \varGamma _{11}(x) + \eta ^x_t\varGamma _{12}(x) &{} \varGamma _{12}(x) \\ 0 &{} \varGamma _{22}(x) - \eta ^x_t\varGamma _{12}(x) \end{bmatrix} \begin{bmatrix} {\hat{X}}_t^x \\ \zeta ^x_t \end{bmatrix} dt \\&\quad + \begin{bmatrix} \varGamma _{Z,1}(x) \\ \varGamma _{Z,2}(x) - \eta ^x_t\varGamma _{Z,1}(x) \end{bmatrix}{\hat{Z}}^x_t dt + \begin{bmatrix} 1 \\ 0 \end{bmatrix} dB^x_t,&x\in I,\ t\in [\tau , T_0], \\&{\hat{X}}^x_{\tau } = \xi ^x_{\tau },\quad \zeta ^x_{T_0} = \zeta ^{0:x,T_0,{\hat{X}}^\cdot _{T_0}}_{T_0}, \quad {\hat{Z}}^x_t = \int _I w(x,y){\mathbb {E}}[{\hat{X}}^y_t]\lambda (dy),&x\in I,\ t\in [\tau ,T_0], \end{aligned} \end{aligned}$$

(27)

with $\xi ^x_\tau$ satisfying the same assumptions as above but with the new $\tau \in [0, T_0]$ replacing the old $\tau \in [T_0,T]$. System (27) differs from (25) in the terminal condition for the backward equation. In the next steps we deduce small-time existence and uniqueness of solutions of (27), we patch the solution with $({\hat{X}}^{0:x,T_0,{\hat{X}}^\cdot _{T_0}}_t, \zeta ^{0:x,T_0,{\hat{X}}^\cdot _{T_0}}_t; t\in [T_0,T])$, then we repeat and show that after a finite number of patching rounds we are left with the unique solution to (25).

Step 4: Unique solvability of (27) for short time horizons

Let $\tau \in [0, T_0]$ and $E_{[\tau ,T_0]} := C([\tau ,T_0])$. Let $V_{[\tau ,T_0]}$ and $V_{\boxtimes ,[\tau ,T_0]}$ denote $L^2(I; E_{[\tau ,T_0]})$ and $L^2_\boxtimes (\varOmega \times I; E_{[\tau ,T_0]})$, respectively. A fixed-point argument can be made to prove the existence of a $c_1>0$ such that if $T_0-\tau \le c_1$, then there is a unique solution to (27) in $V_{\boxtimes ,T_0}\times V_{T_0}$. Denote the solution by $({\hat{X}}^{1:x,\tau ,\xi ^\cdot _{\tau }}_t, \zeta ^{1:x,\tau ,\xi ^\cdot _{\tau }}_t; t\in [\tau ,T_0])$.

Let $T_1 := T_0 - c_1$. Most importantly (since we aim to use the induction approach) $c_1$ can take any value smaller than a constant ${{\bar{C}}}$ depending only on the time horizon T and the coefficient function bound (from Assumption 1). The fixed-point calculations are tedious and omitted here. The main difficulty comes from that the terminal condition of the backward part of the equation depends on the solution of (26) initiated at the solution of (27). This can however be overcome by using the fixed-point argument for (26) from Step 2.

Step 5: Patching the solutions over $[T_1, T_0]$ and $[T_0, T]$

We now patch the two solutions. Let $\tau \in [T_1,T_0]$ and $\xi _\tau$ be the initial condition vector in (27). For any $s\in [\tau , T]$,

$$\begin{aligned} ({\hat{X}}^x_s, \zeta ^x_s) = {\left\{ \begin{array}{ll} ({\hat{X}}^{1:x,\tau ,\xi ^\cdot _{\tau }}_s, \zeta ^{1:x,\tau ,\xi ^\cdot _{\tau }}_s), &{} s\in [\tau , T_0], \\ ({\hat{X}}^{0:x,T_0,{\hat{X}}^{1:\cdot ,\tau ,\xi _\tau }_{T_0}}_s, \zeta ^{0:x,T_0,{\hat{X}}^{1:\cdot ,\tau ,\xi _\tau }_{T_0}}_s), &{} s\in (T_0, T]. \end{array}\right. } \end{aligned}$$

(28)

Then, ${\mathbb {P}}\boxtimes \lambda$-a.s.,

$$\begin{aligned}&\lim _{s\downarrow T_0} {\hat{X}}^x_s(\omega ) = \lim _{s\downarrow T_0} {\hat{X}}^{0:x,T_0,{\hat{X}}^{1:\cdot ,\tau ,\xi _\tau }_{T_0}}_s(\omega ) = {\hat{X}}^{1:x,\tau ,\xi ^\cdot _\tau }_{T_0}(\omega ) = {\hat{X}}^x_{T_0}(\omega ), \\&\lim _{s\downarrow T_0} \zeta ^x_s = \lim _{s\downarrow T_0} \zeta ^{0:x,T_0,{\hat{X}}^{1:\cdot ,\tau ,\xi _\tau }_{T_0}}_s = \zeta ^{1:x,\tau ,\xi ^\cdot _\tau }_{T_0} = \zeta ^x_{T_0}, \end{aligned}$$

so $({\hat{X}}^x_s, \zeta ^x_s; \tau \le s\le T)$ is ${\mathbb {P}}\boxtimes \lambda$-a.s. continuous and the unique solution to the forward-backward system

$$\begin{aligned} \begin{aligned}&d\begin{bmatrix} {\hat{X}}_t^x \\ \zeta ^x_t \end{bmatrix} = \begin{bmatrix} \varGamma _{11}(x) + \eta ^x_t\varGamma _{12}(x) &{} \varGamma _{12}(x) \\ 0 &{} \varGamma _{22}(x) - \eta ^x_t\varGamma _{12}(x) \end{bmatrix} \begin{bmatrix} {\hat{X}}_t^x \\ \zeta ^x_t \end{bmatrix} dt \\&\quad + \begin{bmatrix} \varGamma _{Z,1}(x) \\ \varGamma _{Z,2}(x) - \eta ^x_t\varGamma _{Z,1}(x) \end{bmatrix}{\hat{Z}}^x_t dt + \begin{bmatrix} 1 \\ 0 \end{bmatrix} dB^x_t,&x\in I,\ t\in [\tau , T], \\&{\hat{X}}^x_{\tau } = \xi ^x_{\tau },\quad \zeta ^x_{T_0} = \zeta ^{0:x,T_0,{\hat{X}}^\cdot _{T_0}}_{T_0}, \quad {\hat{Z}}^x_t = \int _I w(x,y){\mathbb {E}}[{\hat{X}}^y_t]\lambda (dy),&x\in I,\ t\in [\tau ,T]. \end{aligned} \end{aligned}$$

Step 6: The induction approach

Consider the following forward-backward system: for some $\tau \in [0, T_1]$,

$$\begin{aligned} \begin{aligned}&d\begin{bmatrix} {\hat{X}}_t^x \\ \zeta ^x_t \end{bmatrix} = \begin{bmatrix} \varGamma _{11}(x) + \eta ^x_t\varGamma _{12}(x) &{} \varGamma _{12}(x) \\ 0 &{} \varGamma _{22}(x) - \eta ^x_t\varGamma _{12}(x) \end{bmatrix} \begin{bmatrix} {\hat{X}}_t^x \\ \zeta ^x_t \end{bmatrix} dt \\&\quad + \begin{bmatrix} \varGamma _{Z,1}(x) \\ \varGamma _{Z,2}(x) - \eta ^x_t\varGamma _{Z,1}(x) \end{bmatrix}{\hat{Z}}^x_t dt + \begin{bmatrix} 1 \\ 0 \end{bmatrix} dB^x_t,&x\in I,\ t\in [\tau , T], \\&{\hat{X}}^x_{\tau } = \xi ^x_{\tau },\quad \zeta ^x_{T_1} =\zeta ^{1:x,T_1,{\hat{X}}^\cdot _{T_1}}_{T_1}, \quad {\hat{Z}}^x_t = \int _I w(x,y){\mathbb {E}}[{\hat{X}}^y_t]\lambda (dy),&x\in I,\ t\in [\tau ,T]. \end{aligned} \end{aligned}$$

(29)

Repeating the analysis that was done for the interval $[T_1,T_0]$ proves unique solvability for (29) whenever $T_1 - \tau < c_1$, with $c_1$ being the same constant that was found above in Step 4. Hence, we have a unique solution to (29) on the time interval $[T-(2c_1+c_0), T-(c_1+c_0)] =: [T_2,T_1]$. This solution can be patched with the solution on $[T_1,T]$ as was done in Step 5. After a finite number N of iterations (an explicit lower bound on N can be found, depending only on T and ${{\bar{C}}}$, the constant from Step 4), the whole interval [0, T] has been covered and the patching has yielded a unique solution to (25) over the interval [0, T] for any finite $T>0$. $\square$

1.3 A.3 Proof of Proposition 3

The proof relies heavily on a bound for the following random variable: for a fixed $x^\infty \in I^\infty$ and $N\in {\mathbb {N}}$, we define

$$\begin{aligned} \zeta ^{x^\infty }_N : [0,T]\times I \ni (t,x) \mapsto \frac{1}{N}\sum _{h=1}^N w(x,x_h)X^{x_h}_t - \int _I w(x,y){\mathbb {E}}[X^y_t]\lambda (dy). \end{aligned}$$

We know $\zeta ^{x^\infty }_N$ is well-defined for all $x^\infty \in I^\infty$ since the graphon game state $X^x$ is defined for all $x\in I$. The proofs of the lemmas below are found in the end of this section.

Lemma 4

For all $x^\infty \in I^\infty$ and $N\in {\mathbb {N}}$ there exists a constant C, independent of $x^\infty$ and N, such that

$$\begin{aligned}&max_{1\le k\le N} \Big ({\mathbb {E}}\Big [\sup _{t\in [0,T]}\big (|X^{k,N}_t - X^{x_k}_t|^2 + |p^{k k,N}_t - p^{x_k}_t|^2\big ) \Big ] + \sup _{t\in [0,T]}{\mathbb {E}}\Big [|Z^{x_k,N}_t - Z^{x_k}_t|^2\Big ]\Big )&\\&\quad \le \sup _{(t,x)\in [0,T]\times I}C\Big ({\mathbb {E}}\Big [|\zeta ^{x^\infty }_N(t,x)|^2\Big ] + \frac{1}{N}\Big ). \end{aligned}$$

Before attending to the first claim of the proposition, we establish a useful estimate. It follows from the Burkholder-Davis-Gundy inequality, Gronwall’s lemma, and the uniform integrability of the initial conditions that

$$\begin{aligned} \sup _{x\in I} {\mathbb {E}}[\Vert X^x\Vert _E^2] \le C(T,w), \end{aligned}$$

(30)

for some finite $C(T,w) > 0$ that depends only on T and w. Adding and subtracting $\frac{1}{N}\sum _{k=1}^N w(x,x_k){\mathbb {E}}[X^{x_k}_t]$ to $\zeta ^{x^\infty }_N(t,x)$ results in

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\left[ |\zeta ^{x^\infty }_N(t,x)|^2\right]&\le C {\mathbb {E}}\Big [\Big |\frac{1}{N}\sum _{k=1}^N w(x,x_k)\left( X^{x_k}_t - {\mathbb {E}}[X^{x_k}_t]\right) \Big |^2\Big ] \\&\qquad + C \Big |\frac{1}{N}\sum _{k=1}^N w(x,x_k){\mathbb {E}}[X^{x_k}_t] - \int _I w(x,y){\mathbb {E}}[X^y_t]\lambda (dy)\Big |^2. \end{aligned} \end{aligned}$$

(31)

By Lemma 1, $(w(x, x_k)(X^{x_k}_t - {\mathbb {E}}[X^{x_k}_t]))_{k=1}^\infty$ are mutually independent ${\bar{\lambda }}^\infty$-a.s. Thus ${\bar{\lambda }}^\infty \text {-a.s.}$

$$\begin{aligned} {\mathbb {E}}\Big [\Big |\frac{1}{N}\sum _{k=1}^N w(x,x_k)(X^{x_k}_t - {\mathbb {E}}[X^{x_k}_t])\Big |^2\Big ] = \frac{1}{N^2}\sum _{k=1}^N {\mathbb {E}}\left[ \left| w(x,x_k)(X^{x_k}_t - {\mathbb {E}}[X^{x_k}_t])\right| ^2\right] . \end{aligned}$$

(32)

The summands on the right hand side of (32) can be bounded by a constant independent of (t, x), using the estimate derived above. We get

$$\begin{aligned} \sup _{(t,x)\in [0,T]\times I} {\mathbb {E}}\Big [\Big |\frac{1}{N}\sum _{k=1}^N w(x,x_k)(X^{x_k}_t - {\mathbb {E}}[X^{x_k}_t])\Big |^2\Big ] \le \frac{C(T,w)}{N},\quad {\bar{\lambda }}^\infty \text {-a.s.} \end{aligned}$$

for some constant $C(T,w)>0$ depending only on T and w. We move on to the second term of the right hand side of (31). By the strong law of large numbers, it tends to zero almost surely as $N\rightarrow \infty$, proving the first claim of the proposition. To prove the second claim, we prove tightness of the term and then apply the Law of Iterated Logarithms.

Let $m(t,x) := \int _I w(x,y){\mathbb {E}}[X^y_t]\lambda (dy)$ be the mean of $w(x,x_k){\mathbb {E}}[X^{x_k}_t]$ when $x_k$ is $\lambda$-distributed. Let furthermore $\theta _k(x^\infty ) : (t,x) \mapsto w(x,x_k){\mathbb {E}}[X^{x_k}_t]-m(t,x)$. $\theta _k$ is a random variable on $(I^\infty ,{\mathcal {I}}^\infty , \lambda ^\infty )$ into $C([0,T]\times I)$.

Lemma 5

The collection $(\varTheta _N)_{N\in {\mathbb {N}}}$, where $\varTheta _N := \frac{1}{\sqrt{N}}\sum _{k=1}^N\theta _k$, is tight.

By Prokhorov’s theorem (see e.g. [26, Theorem 16.3]), this yields relative compactness in distribution of $(\varTheta _N)_N$. Moreover, the finite-dimensional distributions converge. Indeed, for any $n \in {\mathbb {N}}$ and any $r_1,\dots ,r_n \in [0,T] \times I$, we have that the sequence $(\varTheta _N(r_1), \dots , \varTheta _N(r_n))_{N=1,2,\dots }$ converges in distribution by the standard central limit theorem. Hence, by [26,  Lemma 16.2], $(\varTheta _N)_N$ converges in distribution. By definition, this means that $\theta _1(x^\infty ):(t,x) \mapsto w(x,x_1){\mathbb {E}}[X^{x_1}_t]-m(t,x)$ satisfies CLT (see [31, Section 10.1]). Note that $\theta _k,k=2,3,\dots ,$ are independent copies of $\theta _1$. Thus, $(\varTheta _N)_N$ satisfies a Law of the Iterated Logarithm (see e.g. [31, Theorem 10.12]). More precisely, we obtain the following.

Lemma 6

There exists a constant C such that

$$\begin{aligned} \lambda ^\infty \left( \underset{N\rightarrow \infty }{\lim \sup }\ \frac{\Vert \theta _1 + \dots + \theta _N\Vert _\infty }{\sqrt{2N\log \log N}} = C\right) = 1. \end{aligned}$$

In other words, Lemma 6 says that, for $\lambda ^\infty$-a.e. $x^\infty \in I^\infty$:

$$\begin{aligned} \forall \varepsilon > 0,\ \exists N_\varepsilon (x^\infty ),\ \forall N\ge N_\varepsilon (x^\infty ),\ \ \Vert \theta _1 + \dots + \theta _N\Vert _\infty \le (C+\varepsilon )\sqrt{2N\log \log N}, \end{aligned}$$

and we have for $\lambda ^\infty$-a.e. $x^\infty \in I^\infty$, for all $N \ge N_\varepsilon (x^\infty )$

$$\begin{aligned} \sup _{(t,x)\in [0,T]\times I} \Big |\frac{1}{N}\sum _{k=1}^N \left( w(x,x_k){\mathbb {E}}[X^{x_k}_t] - m(t,x)\right) \Big |^2 \le \frac{(C+\varepsilon )^2\log \log N}{N}. \end{aligned}$$

The last statement also holds a.s. in $(I^\infty , \bar{{\mathcal {I}}}^\infty ,{\bar{\lambda }}^\infty )$, see [21, Section 6]. $\square$

1.3.1 A.3.1 Proof of Lemma 4

From standard estimates for linear SDEs and BSDEs we get

$$\begin{aligned}&{\mathbb {E}}\Big [\sup _{t\in [0,T]}|X^{k,N}_t-X^{i_k}_t|^2 + \sup _{t\in [0,T]}|p^{k k,N}_t-p^{i_k}_t|^2 \Big ] + \sup _{t\in [0,T]} {\mathbb {E}}\left[ |Z^{k,N}_t - Z^{i_k}_t|^2\right] \\&\le C\Big (\sup _{(t,x)\in [0,T]\times I}{\mathbb {E}}\left[ |\zeta ^{x^\infty }_N(t,x)|^2\right] + \max _{1\le \ell ,h \le N: \ell \ne h} {\mathbb {E}}\Big [\sup _{t\in [0,T]}|p^{\ell h,N}_t|^2 \Big ] + \frac{1}{N}\Big ). \end{aligned}$$

Next, we will the estimate for the right hand side term containing off-diagonal adjoint state variables. Consider the following auxiliary BSDE system: for $k=1,\dots , N$, $p^{kk,N}_t = 0$ and $h = 1,\dots , N$, $h\ne k$,

$$\begin{aligned} {\widetilde{p}}^{k h,N}_t = \int _t^T \Big (a(x_k){\widetilde{p}}^{k h, N}_s + \frac{1}{N}\sum _{\ell =1}^N c(x_\ell )w(x_h,x_\ell ){\widetilde{p}}^{k \ell , N}_s\Big )ds - \sum _{\ell =1}^N\int _t^T{\widetilde{q}}^{k h \ell , N}_s dB^{x_\ell }_s. \end{aligned}$$

(33)

The difference $p^{kh,N}_t - {\widetilde{p}}^{kh,N}_t$, $1\le h\ne k \le N$, satisfies the BSDE

$$\begin{aligned}&p^{k h, N}_t - {\widetilde{p}}^{k h, N}_t = \frac{w(x_k,x_h)}{N} \bigg ( {\bar{\varGamma }}^*_T(x_k)\begin{bmatrix} X^{k,N}_T \\ Z^{k, N}_T \end{bmatrix} + \int _t^T {\bar{\varGamma }}(x_k)\begin{bmatrix} X^{k,N}_s \\ p^{kk,N}_s \end{bmatrix} + {\bar{\varGamma }}_Z(x_k)Z^{k,N}_s ds \bigg ) \\&\quad + \int _t^T\Big ( \frac{1}{N}\sum _{\ell =1}^N c(x_\ell )w(x_h,x_\ell )(p^{k\ell ,N}_s-{\widetilde{p}} ^{k\ell ,N}_s) + a(x_h) (p^{kh,N}_s-{\widetilde{p}}^{kh,N}_s) \Big )ds \\&\quad - \sum _{\ell =1}^N\int _t^T (q^{kh\ell ,N}_s - \widetilde{q}^{kh\ell ,N}_s)dB^{i_\ell }_s,\qquad t\in [0,T]. \end{aligned}$$

Standard BSDE estimates (relying on the integrability of $X^{k,N}, Z^{k,N}$, and $p^{kk,N}$) yield

$$\begin{aligned} {\mathbb {E}}\Big [\sup _{t\in [0,T]}|p^{k h, N}_t - {\widetilde{p}}^{k h, N}_t|^2 + \int _0^T|q^{k h \ell , N}_s - {\widetilde{q}}^{k h \ell , N}_s|^2ds\Big ] \le \frac{C}{N}, \end{aligned}$$

with C independent of k, h,  and N. The unique solution to (33) is ${\widetilde{p}}^{k h, N}_t = {\widetilde{q}}^{k h \ell }_t = 0$ for $t\in [0,T]$, $\ell = 1,\dots , N$, $1\le k\ne h\le N$. $\square$

1.3.2 A.3.2 Proof of Lemma 5

We will apply the tightness criterion provided by [26, Corollary 16.9], which stems from the Kolmogorov-Chentsov criterion. We first note that

$$\begin{aligned} \varTheta _N(0,0) = \frac{1}{\sqrt{N}}\sum _{k=1}^N\theta _k = \frac{1}{\sqrt{N}}\sum _{k=1}^N\left[ w(0,x_k){\mathbb {E}}[X^{x_k}_0] - \int _I w(0,y){\mathbb {E}}[X^y_0]\lambda (dy) \right] , \end{aligned}$$

where the random variables are i.i.d. (and with mean 0). So the sequence $(\varTheta _N(0,0))_N$ is tight. Moreover, we prove that there exists a constant $C>0$ and there exists a positive integer $N_0\in {\mathbb {N}}$ such that for every $(t,x), (t',x') \in [0,T]\times I$, $N \ge N_0$,

$$\begin{aligned} \int _{I^\infty }\Big | \frac{1}{\sqrt{N}}\sum _{k=1}^N\theta _k(x^\infty )(t,x) - \theta _k(x^\infty )(t',x')\Big |^4 \lambda ^\infty (dx^\infty ) \le C\left( |t-t'|^4 + |x-x'|^4\right) . \end{aligned}$$

As a consequence of [26, Corollary 16.9], we will obtain that the sequence of $(\varTheta _N)_N$ is tight and the limiting processes are $\lambda ^\infty$-a.s. locally Hölder continuous with exponent 1/2.

We now prove the claim. Let $(t,x), (t',x') \in [0,T]\times I$, $N\in {\mathbb {N}}$. Letting ${\mathbf {T}}_k := \theta _k(i^\infty )(t,x) - \theta _k(x^\infty )(t',x')$, we note that

$$\begin{aligned}&\int _{I^N}\Big | \frac{1}{\sqrt{N}}\sum _{k=1}^N\theta _k(x^\infty )(t,x) - \theta _k(x^\infty )(t',x')\Big |^4 \otimes _{k=1}^N\lambda (dx_k) \\&\quad = \frac{1}{N^2} \int _{I^N} \left[ \sum _{k_1,k_2,k_3,k_4=1}^N {\mathbf {T}}_{k_1} {\mathbf {T}}_{k_2} {\mathbf {T}}_{k_3} {\mathbf {T}}_{k_4} \right] \otimes _{k=1}^N\lambda (dx_k) \\&\quad = \underbrace{\frac{1}{N^2} \int _{I^N} \sum _{k=1}^N {\mathbf {T}}_{k}^4 \otimes _{k=1}^N\lambda (dx_k)}_{\rightarrow 0 \hbox { as} ~N \rightarrow +\infty } + \frac{1}{N^2} \sum _{\begin{array}{c} k,k'=1 \\ k\ne k' \end{array}}^N \int _{I^N} \left[ {\mathbf {T}}_{k}^2 {\mathbf {T}}_{k'}^2 \right] \otimes _{k=1}^N\lambda (dx_k) \\&\qquad + \frac{1}{N^2} \sum _{k_1=1}^N \underbrace{\int _{I^N} {\mathbf {T}}_{k_1} \otimes _{k=1}^N\lambda (dx_k)}_{{ = 0 }} \sum _{\begin{array}{c} k_2,k_3,k_4=1 \\ k_2,k_3,k_4 \ne k_1 \end{array}}^N \int _{I^N} \left[ {\mathbf {T}}_{k_2} {\mathbf {T}}_{k_3} {\mathbf {T}}_{k_4} \right] \otimes _{k=1}^N\lambda (dx_k). \end{aligned}$$

The first term can be made arbitrarily small by taking N large enough. The third term is zero. To bound the second term from above, we observe that:

$$\begin{aligned} {\mathbf {T}}_{k}^2&= \big (\theta _k(x^\infty )(t,x) - \theta _k(x^\infty )(t',x')\big )^2 \le C \Big ( |w(x,x_k) - w(x',x_k)|^2 {\mathbb {E}}[X^{x_k}_t]^2 \\&\quad + w(x',x_k)^2 \left| {\mathbb {E}}[X^{x_k}_t] - {\mathbb {E}}[X^{x_k}_{t'}] \right| ^2 + |m(t,x) - m(t',x')|^2 \Big ) \\&\le C \Big ( |x - x'|^2 + \left| t - t' \right| ^2 + |m(t,x) - m(t',x')|^2 \Big ). \end{aligned}$$

Furthermore, for the last term, we have by definition of m,

$$\begin{aligned}&|m(t,x) - m(t',x')|^2 \\&\le \int _I\left( \left| w(x,y) - w(x',y) \right| ^2 {\mathbb {E}}[X^y_t]^2 + w(x',y)^2 \left| {\mathbb {E}}[X^y_t] - {\mathbb {E}}[X^y_{t'}]\right| ^2 \right) \lambda (dy) \\&\le C \left( \left| x - x' \right| ^2 + \left| t - t'\right| ^2 \right) . \end{aligned}$$

Hence:

$$\begin{aligned} \frac{1}{N^2} \sum _{\begin{array}{c} k,k'=1 \\ k\ne k' \end{array}}^N \int _{I^N} \left[ {\mathbf {T}}_{k}^2 {\mathbf {T}}_{k'}^2 \right] \otimes _{k=1}^N\lambda (di_k) \le C \left( \left| x - x' \right| ^4 + \left| t - t'\right| ^4 \right) . \end{aligned}$$

$\square$

1.3.3 A.3.3 Proof of Lemma 6

Using (30), we have that

$$\begin{aligned} \Vert \theta _N\Vert ^2_\infty&= \sup _{(t,x)\in [0,T]\times I}|w(x,x_N){\mathbb {E}}[X^{x_N}_t]-m(t,x)|^2 \\&\le C(w)\left( {\mathbb {E}}[\Vert X^{x_N}\Vert _\infty ^2]+{\mathbb {E}}^\boxtimes [\Vert X\Vert _\infty ^2]\right) \le C(w,T), \end{aligned}$$

where $C(w,T) > 0$ is a finite constant depending only on the graphon w and T. Let $Lt := \max (1,\log t)$ for $t\ge 0$. Using the uniform bound derived above, we see that

$$\begin{aligned} \int _{I^\infty }\left( \frac{\Vert \theta _N\Vert ^2_\infty }{LL\Vert \theta _N\Vert _\infty }\right) d\lambda ^\infty (x^\infty ) \le \int _{I^\infty }\Vert \theta _N\Vert ^2_\infty d\lambda ^\infty (x^\infty ) < \infty . \end{aligned}$$

The claim now follows from the Law of Iterated Logarithms in Banach spaces, see, e.g., [31, Theorem 10.12]. $\square$

1.4 A.4 Proof of Proposition 4

We first prove the convergence claim. Recall that the two equilibria are linear functions of state, costate, and aggregate, so the propagation of chaos from Proposition 3 immediately yields

$$\begin{aligned} \max _{1\le k\le N}{\mathbb {E}}\Big [\int _0^T|{\hat{\alpha }}^{k,N}_t - {\hat{\alpha }}^{x_k}_t|^2dt\Big ] \le \varepsilon _N^2,\quad N\ge {\underline{N}},\ {\bar{\lambda }}^\infty \text {-a.s.} \end{aligned}$$

Moving on to the approximation claim, let $\hat{\alpha }^{-k,N} := (\hat{\alpha }^{1,N},\dots , \hat{\alpha }^{k-1,N},\hat{\alpha }^{k+1,N},\dots , \hat{\alpha }^{N,N})$. Since $(\hat{\alpha }^{1,N},\dots , {\hat{\alpha }}^{N,N})$ is a Nash equilibrium for the N-player game

$$\begin{aligned}&J^{k,N}(\hat{\alpha }^{x_k}; \bar{\alpha }^{-k,N}) - J^{k,N}(\beta ; \bar{\alpha }^{-k,N}) \\&\le |J^{k,N}({\hat{\alpha }}^{x_k}; {\bar{\alpha }}^{-k,N}) - J^{k,N}({\hat{\alpha }}^{k,N}; {\hat{\alpha }}^{-k,N})| + |J^{k,N}(\beta ; {\bar{\alpha }}^{-k,N}) - J^{k,N}(\beta ; {\hat{\alpha }}^{-k,N})|. \end{aligned}$$

Let $\beta$ be an admissible control for player 1 in the N-player game. Let $({{\bar{X}}}^{k,N,\beta })_{k=1}^N$ be the player states when player 1 is using $\beta$ and the others are using the graphon game equilibrium control

$$\begin{aligned}&d{{\bar{X}}}^{1,N,\beta }_t = \Big (a(x_k){{\bar{X}}}^{1,N,\beta }_t + b(x_k)\beta _t + c(x_k)\frac{1}{N}\sum _{h=1}^Nw(x_k,x_h)\bar{X}^{h,N,\beta }_t\Big )dt + dB^{x_1}_t, \\&d{{\bar{X}}}^{k,N,\beta }_t = \Big (a(x_k){{\bar{X}}}^{k,N,\beta }_t + b(x_k){\hat{\alpha }}^{x_k}_t + c(x_k)\frac{1}{N}\sum _{h=1}^Nw(x_k,x_h){{\bar{X}}}^{h,N,\beta }_t\Big )dt + dB^{x_k}_t, \\&{{\hat{X}}}^{1,N,\beta }_0 = \xi ^{x_1},\quad {{\bar{X}}}^{k,N,\beta }_0 = \xi ^{x_k},\quad k\ge 2. \end{aligned}$$

Let $({{\hat{X}}}^{k,N,\beta })_{k=1}^N$ be the player states when player 1 is using $\beta$ and the others are using the N-player game equilibrium control

$$\begin{aligned}&d{{\hat{X}}}^{1,N,\beta }_t = \Big (a(x_k){{\hat{X}}}^{1,N,\beta }_t + b(x_k)\beta _t + c(x_k)\frac{1}{N}\sum _{h=1}^Nw(x_k,x_h){{\hat{X}}}^{h,N,\beta }_t\Big )dt + dB^{x_1}_t, \\&d{{\hat{X}}}^{k,N,\beta }_t = \Big (a(x_k){{\hat{X}}}^{k,N,\beta }_t + b(x_k){\hat{\alpha }}^{k,N}_t + c(x_k)\frac{1}{N}\sum _{h=1}^Nw(x_k,x_h){{\hat{X}}}^{h,N,\beta }_t\Big )dt + dB^{x_k}_t, \\&{{\hat{X}}}^{1,N,\beta }_0 = \xi ^{x_1},\quad {{\hat{X}}}^{k,N,\beta }_0 = \xi ^{x_k},\quad k\ge 2. \end{aligned}$$

We have for any admissible $\beta$ and ${\bar{\lambda }}^\infty$-a.e. $x^\infty \in I^\infty$ that

$$\begin{aligned}&|J^{1,N}(\beta ; {\bar{\alpha }}^{-1,N}) - J^{1,N}(\beta ; {\hat{\alpha }}^{-1,N})| \\&\le {\mathbb {E}}\Big [\int _0^T\big |f^{x_1}(\bar{X}^{1,N,\beta }_t,\beta _t, {{\bar{Z}}}^{1,N,\beta }_t) - f^{x_1}({{\hat{X}}}^{1,N,\beta }_t, \beta _t, {{\hat{Z}}}^{1,N,\beta }_t)\big |dt\Big ] \\&\qquad + {\mathbb {E}}\Big [\big |h^{x_1}({{\bar{X}}}^{1,N,\beta }_T, \bar{Z}^{1,N,\beta }_T) - h^{x_1}({{\hat{X}}}^{1,N,\beta }_T, {{\hat{Z}}}^{1,N,\beta }_T)\big |\Big ] \\&\le C \max _{1\le k\le N}{\mathbb {E}}\Big [\int _0^T\big |{\hat{\alpha }}^{x_k}_t - {\hat{\alpha }}^{k,N}_t\big |^2dt\Big ]^{1/2} \le \varepsilon _N,\quad N\ge {\underline{N}}(x^\infty ). \end{aligned}$$

In the same way we can perturb the other players’ actions and that concludes the proof. $\square$

1.5 A.5 Conditions for Well-Posedness of the Time-Varying Riccati Equation (20)

Let

$$\begin{aligned} {\mathbb {A}}_t := \varGamma _{Z,2} - \varGamma _{Z,1}\eta _t, \ {\mathbb {B}}^k_t := -\varGamma _{12}\eta _t -\frac{1}{2}(\varGamma _{11} - \varGamma _{22} + \varGamma _{Z,1}\lambda _k), \ {\mathbb {C}}^k := -\varGamma _{12}\lambda _k, \ t\in [0,T]. \end{aligned}$$

Set ${\mathbb {D}}^k := ({\mathbb {B}}^k_t)^2 + {\mathbb {C}}^k {\mathbb {A}}_t - \dot{{\mathbb {B}}}^k_t$. Note that ${\mathbb {D}}^k$ does not depend on t. Indeed, using the ODE satisfied by $\eta$ (i.e.,  (25) but with constant coefficients) and the identity $\varGamma _{11} = -\varGamma _{22}$, we have

$$\begin{aligned} {\mathbb {D}}^k&= \frac{1}{4}(2\varGamma _{11} + \varGamma _{Z,1}\lambda _k)^2 -\varGamma _{12}\lambda _k\varGamma _{Z,2} + \varGamma _{12} \varGamma _{21}. \end{aligned}$$

Last, we let:

$$\begin{aligned} {\mathbb {F}}^k := \frac{{\mathbb {B}}^k_T + {\mathbb {C}}^k \varGamma _{T,2} + \sqrt{{\mathbb {D}}^k}}{{\mathbb {B}}^k_T + {\mathbb {C}}^k \varGamma _{T,2} - \sqrt{{\mathbb {D}}^k}}. \end{aligned}$$

Proposition 5

Assume ${\mathbb {D}}^k \ge 0$ and, for all $t \in [0,T]$, ${\mathbb {F}}^k e^{\sqrt{{\mathbb {D}}^k}t} - e^{-\sqrt{{\mathbb {D}}^k}t} \ne 0$. Then the Riccati equation (20) has a unique solution.

Remark 4

Note that we can rewrite ${\mathbb {D}}^k$ as:

$$\begin{aligned} {\mathbb {D}}^k&= \bigg (\varGamma _{11} -\frac{\varGamma _{12} \varGamma _{Z,2}}{\varGamma _{Z,1}} + \frac{1}{2}\varGamma _{Z,1} \lambda _k \bigg )^2 - \bigg (\varGamma _{11} -\frac{\varGamma _{12} \varGamma _{Z,2}}{\varGamma _{Z,1}}\bigg )^2 + (\varGamma _{11})^2 + \varGamma _{12}\varGamma _{21}. \end{aligned}$$

So, to ensure ${\mathbb {D}}^k \ge 0$ independently of the value of $\lambda _k$, a sufficient condition is:

$$\begin{aligned} (\varGamma _{11})^2 \ge \bigg (\varGamma _{11} -\frac{\varGamma _{12}\varGamma _{Z,2}}{\varGamma _{Z,1}}\bigg )^2 - \varGamma _{12} \varGamma _{21}. \end{aligned}$$

Proof

Existence: We first consider the ODE with the mixed initial condition

$$\begin{aligned} \ddot{\nu }^k_t - {\mathbb {D}}^k \nu ^k_t = 0,\quad t\in [0,T],\qquad \nu ^k_0 ({\mathbb {B}}^k_T + {\mathbb {C}}^k \varGamma _{T,2}) - {\dot{\nu }}^k_0 = 0. \end{aligned}$$

(34)

A solution is given by $\nu ^k_t = {\mathbb {F}}^k e^{\sqrt{{\mathbb {D}}^k}t} - e^{-\sqrt{{\mathbb {D}}^k}t}$. Let $\theta ^k_t = \nu ^k_{T-t}$, for $t \in [0,T]$. It solves the ODE with the mixed terminal condition:

$$\begin{aligned} \ddot{\theta }^k_t = {\mathbb {D}}^k \theta ^k_t,\quad t\in [0,T],\qquad \theta ^k_T ({\mathbb {B}}^k_T + {\mathbb {C}}^k \varGamma _{T,2}) + {\dot{\theta }}^k_T = 0. \end{aligned}$$

To conclude, let

$$\begin{aligned} \pi ^k_t = -\frac{1}{{\mathbb {C}}^k}\bigg ( \frac{{\dot{\theta }}^k_t}{\theta ^k_t} + {\mathbb {B}}^k_t \bigg ), \qquad {\dot{\pi }}^k_t = -\frac{1}{{\mathbb {C}}^k}\bigg ( \frac{\ddot{\theta }^k_t}{\theta ^k_t} - \bigg (\frac{{\dot{\theta }}^k_t}{\theta ^k_t}\bigg )^2 + \dot{{\mathbb {B}}}^k_t \bigg ), \qquad t \in [0,T]. \end{aligned}$$

Then it can be checked that $\pi ^k_t$ solves

$$\begin{aligned} {\dot{\pi }}^k_t = - {\mathbb {A}}_t + 2 {\mathbb {B}}^k_t \pi ^k_t + {\mathbb {C}}^k (\pi ^k_t)^2, \quad \pi ^k_T = [C_h]_{12}, \end{aligned}$$

which is equivalent to (20).

Uniqueness: Let us consider $\pi ^k$ and ${\tilde{\pi }}^k$ solving (20). Reverting the above change of variables yields solutions $\nu ^k$ and ${\tilde{\nu }}^k$ to (34) such that ${\tilde{\nu }}^k_t \ne 0$ and $\nu ^k_t \ne 0$ for all $t \in [0,T]$. For any such solutions, there exist constants $C_1, C_2, {{\tilde{C}}}_1, {{\tilde{C}}}_2$ such that

$$\begin{aligned} \nu ^k_t = C_1 e^{\sqrt{{\mathbb {D}}^k}t} + C_2 e^{-\sqrt{{\mathbb {D}}^k}t}, \qquad {\tilde{\nu }}^k_t = {{\tilde{C}}}_1 e^{\sqrt{{\mathbb {D}}^k}t} + {{\tilde{C}}}_2 e^{-\sqrt{{\mathbb {D}}^k}t}. \end{aligned}$$

Due to (34), we have:

$$\begin{aligned}&C_1 ({\mathbb {B}}^k_T + {\mathbb {C}}^k \varGamma _{T,2} - \sqrt{{\mathbb {D}}^k}) + C_2 ({\mathbb {B}}^k_T + {\mathbb {C}}^k \varGamma _{T,2} + \sqrt{{\mathbb {D}}^k}) = 0, \\&{{\tilde{C}}}_1 ({\mathbb {B}}^k_T + {\mathbb {C}}^k \varGamma _{T,2} - \sqrt{{\mathbb {D}}^k}) + {{\tilde{C}}}_2 ({\mathbb {B}}^k_T + {\mathbb {C}}^k \varGamma _{T,2} + \sqrt{{\mathbb {D}}^k}) = 0. \end{aligned}$$

As a consequence, we necessarily have $C_1/C_2 = {{\tilde{C}}}_1 / \tilde{C}_2 = - {\mathbb {F}}^k$. We deduce that $\pi ^k_t = {\tilde{\pi }}^k_t$ for all $t \in [0,T]$. $\square$