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Transboundary pollution control under evolving social norms: a mean-field approach

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Abstract

We analyze a dynamic game of transboundary pollution control under endogenously evolving social norms over a finite time horizon. Each player chooses their emission level in order to minimize the social cost of mitigation, which partly depends on the lack of conformity to the social norm establishing the pollution standards at the local level. We show that social norms per se are unable to favor pollution reductions, but if combined with some public reclamation effort, they become very effective in improving environmental outcomes. Indeed, provided that some minimal public reclamation takes place, social norms promote a reduction in the average of the expected value of the local pollution stocks across locations, both in the case in which players rely on an open loop and a closed loop strategy. Moreover, by explicitly characterizing the equilibrium outcome, we formally confirm the reliability of the mean-field approximation of the finite-population dynamics, despite such an approximation introduces some distortion regarding the difference between open and closed loop strategies. We also show that our results are robust to the introduction of individual abatement efforts and heterogeneity across players.

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Notes

  1. Actually this is the definition of Markovian Nash equilibrium; however, we rely on the closed loop equilibrium terminology, which is most commonly employed in economics.

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Acknowledgements

This work was partially supported by the Gruppo Nazionale per l’Analisi Matematica e le loro Applicazioni (GNAMPA-INdAM).

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Correspondence to Davide La Torre.

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We are grateful to Marco Tolotti and the participants in the International Conference on Sustainability, Environment, and Social Transition in Economics and Finance 2022 (Paris, France) for insightful discussions. All errors and omissions are our own sole responsibility.

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Appendix

Appendix

1.1 Proof of proposition 1

The proof follows the scheme proposed in (Carmona et al. 2015; see section 3.1). Since \(\sigma >0\) is constant, we can rely on the reduced Hamiltonian that for the agent i is given by:

$$\begin{aligned} H^i(x_1, \dots , x_N, p^{i,1}, \dots , p^{i, N}, y_1, \dots , y_N)&=\sum _{j=1}^{N}[(m_N-x_j)+y_j-\tau ]p^{i,j}\nonumber \\&\quad +e^{-\lambda t}[(x_i-m_N)^2+2y_i(x_i-m_N)+y_i^2]. \end{aligned}$$

By the necessary condition of the Pontryagin stochastic maximum principle (PSMP) the value of \(y_i\) minimizing the reduced Hamiltonian when all the other variables (including \(y_j\) for \(j \ne i\)) are fixed, is given by:

$$\begin{aligned} \frac{\partial H^i}{\partial y_i}=0 \quad \longrightarrow \quad \widehat{y}_i=-\frac{p^{i,i}}{2 e^{-\lambda t}}+(m_N-x_i). \end{aligned}$$
(22)

Given an admissible strategy \(y=(y_1, \dots , y_N)\) and the corresponding controlled state, the adjoint processes associated with y are the processes \(P^i(t)=(P^{i,j}(t): j=1,\dots , N)\) and \(Q^i(t)=(Q^{i,j,k}(t): j=1,\dots , N, \, k=0,\dots , N)\) solving the system of backward stochastic differential equations (BSDEs):

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \text {d}P^{i,j}(t)=-\frac{\partial H^i}{\partial x_j} \text {d}t +\sum _{k=0}^N Q^{i,j,k}(t)\, \text {d}B_k(t) \\ \qquad \qquad =\displaystyle -\left[ \sum _{k=1}^N \left( \frac{1}{N}-\delta _{k,j}\right) P^{i,k}(t)+e^{-\lambda t}\left( 2\left( m_N(t)-X_i(t)\right) \left( \frac{1}{N}-\delta _{i,j}\right) -2y_i(t)\left( \frac{1}{N}-\delta _{i,j}\right) \right) \right] \text {d}t\\ \displaystyle \qquad \qquad \quad +\sum _{k=0}^N Q^{i,j,k}(t)\, \text {d}B_k(t),\\ \displaystyle P^{i,j}(T)=2e^{-\lambda T}\phi \left( m_N(T)-X_i(T)\right) \left( \frac{1}{N}-\delta _{i,j}\right) \end{array}\right. } \end{aligned}$$
(23)

Note that standard existence and uniqueness results for BSDEs apply to (23) and the existence of the adjoint processes is guaranteed (see Chapter 2 in (Carmona 2016), and references therein). According to the strategy outlined earlier, we replace all the occurrences of the controls \(y_i(t)\) in the forward Eq. (2) and the backward Eq. (23), by:

$$\begin{aligned} \widehat{y}_i(t)=-\frac{P^{i,i}(t)}{2e^{-\lambda t}}+(m_N(t)-X_i(t)). \end{aligned}$$
(24)

If the resulting forward-backward system is solvable then the strategy (24) will characterize the optimal open loop emission level. However, that system is in general extremely difficult to solve, hence we make the following ansatz (i.e., we search for a solution of the form):

$$\begin{aligned} P^{i,j}(t)=2e^{-\lambda t}\eta (t)\left( m_N(t)-X_i(t)\right) \left( \frac{1}{N}-\delta _{i,j}\right) \end{aligned}$$
(25)

for some smooth deterministic function \(\eta :[0,T]\rightarrow \mathbb {R}\) to be determined. Using (24) and (25) the forward Eq. (2) becomes:

$$\begin{aligned} \text {d}{X}_i(t)=\left[ \left( 2+ \left( 1-\frac{1}{N}\right) \eta (t) \right) \left( m_N(t)-{X}_i(t)\right) -\tau \right] \text {d}t+ \sigma \text {d}B_i(t), \end{aligned}$$
(26)

which by summation gives:

$$\begin{aligned} \text {d}m_N(t)=-\tau \, \text {d}t+\sigma \frac{1}{N} \sum _{i=1}^N \text {d}B_i(t), \end{aligned}$$

and hence:

$$\begin{aligned} \text {d} \left( m_N(t)-{X}_i(t)\right)= & {} - \left( 2+ \left( 1-\frac{1}{N}\right) \eta (t) \right) \left( m_N(t)-{X}_i(t)\right) \text {d}t\nonumber \\ {}{} & {} +\sigma \left( \frac{1}{N} \sum _{k=1}^N \text {d}B_k(t)- \text {d}B_i(t)\right) . \end{aligned}$$
(27)

Using instead (24) and (25) in (23) we get:

$$\begin{aligned} \text {d}P^{i,j}(t)=&2e^{-\lambda t}\left( \frac{1}{N}-\delta _{i,j}\right) \left( m_N(t)-{X}_i(t)\right) \left( 2-\frac{1}{N}\right) \eta (t)\text {d}t\nonumber \\&\quad + \sum _{k=0}^N Q^{i,j,k}(t)\, \text {d}B_k(t). \end{aligned}$$
(28)

Differentiating the ansatz (25) and using (27), it follows that:

$$\begin{aligned} \begin{aligned} \text {d}P^{i,j}(t)&= 2e^{-\lambda t}\left( \frac{1}{N}-\delta _{i,j}\right) \left( m_N(t)-{X}_i(t)\right) \\&\left[ -\lambda \eta (t)+\dot{\eta }(t)-\eta (t)\left( 2+\left( 1-\frac{1}{N}\right) \eta (t)\right) \right] {d}t\\&+ 2e^{-\lambda t}\left( \frac{1}{N}-\delta _{i,j}\right) \eta (t)\,\sigma \left( \frac{1}{N} \sum _{k=1}^N \text {d}B_k(t)- \text {d}B_i(t)\right) , \end{aligned} \end{aligned}$$
(29)

where \(\dot{\eta }(t)\) is the time-derivative of \(\eta (t)\). Comparing (28) and (29) we get the process \(Q^{i,j,k}(t)\):

$$\begin{aligned} Q^{i,j,0}(t)=0, \qquad Q^{i,j,k}(t)= 2e^{-\lambda t}\left( \frac{1}{N}-\delta _{i,j}\right) \eta (t)\,\sigma \left( \frac{1}{N}-\delta _{i,k}\right) , \qquad k=1, \dots , N \end{aligned}$$

which turns out to be deterministic and hence adapted. Identifying the drift terms, it follows that \(\eta (t)\) must satisfy the following Bernoulli equation:

$$\begin{aligned} \dot{\eta }(t)=\left( 4+\lambda -\frac{1}{N}\right) \eta (t)+\left( 1-\frac{1}{N}\right) \eta ^2(t) \end{aligned}$$
(30)

with terminal condition \(\eta (T)=\phi \), obtained applying the ansatz (25) to the the final condition in (23). Equation (30) has a unique solution given by:

$$\begin{aligned} \eta (t)=\frac{\phi \left( 4+\lambda -\frac{1}{N}\right) }{\left( 4+\lambda -\frac{1}{N}\right) e^{\left( 4+\lambda -\frac{1}{N}\right) (T-t)}+\phi \left( 1-\frac{1}{N}\right) \left( e^{\left( 4+\lambda -\frac{1}{N}\right) (T-t)}-1\right) }. \end{aligned}$$
(31)

With (31) in hand, the sufficient part of PSMP implies that an optimal strategy profile is given by:

$$\begin{aligned} \widehat{y}_i(t)=\left[ 1+\left( 1-\frac{1}{N}\right) \eta (t)\right] (m_N(t)-\widehat{X}_i(t)), \end{aligned}$$
(32)

obtained by plugging (25) in (24) and where we denote by \(\widehat{X}_i\) the state of the player i only to stress the fact we are using an open loop equilibrium. We remark that even though the control (32) is in feedback form (since it only depends upon the current value of the state \(\widehat{X}_i(t)\)) we can only claim that it is in an open loop form. Note also that in equilibrium, the state \(\widehat{X}_i(t)\) is Markovian for every \(i=1, \dots , N\) and satisfies the following:

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {d} \widehat{X}_i(t)=\displaystyle \left[ \left( 2+\left( 1-\frac{1}{N}\right) \eta (t)\right) (m_N(t) -\widehat{X}_i(t)) -\tau \right] \text {d}t+\sigma \, \text {d}B_i(t), \quad t \in (0, T]\\ \widehat{X}_i(0)=X_i^0. \end{array}\right. } \end{aligned}$$
(33)

1.2 Proof of proposition 2

Following the proof given by (Carmona et al. 2015; see section 3.2), we characterize in closed-form a closed-loop solution in which players at time t have complete information of the states of all the other players at the same time. Hence when all the other players \(k\ne i\) have chosen strategies in feedback form given by deterministic functions \(y_k(t,x)\) of time and state \(x=(x_1, \dots , x_N)\), player i needs to solve a control problem to find their best response to these choices. The reduced Hamiltonian of their control problem is given by:

$$\begin{aligned} \begin{aligned}&H^i(x, p^{i, N}, \dots , p^{i, N}, y_1(t,x), \dots , y_i(t) , \dots y_N(t,x))\\&\quad = \sum _{k=1, k\ne i}^{N}[(m_N-x_k)+y_k(t,x)-\tau ]p^{i,k}+ [(m_N-x_i)+y_i-\tau ]p^{i,i}\\&\qquad +e^{-\lambda t}\left[ (x_i-m_N)^2+2y_i(x_i-m_N)+y_i^2\right] . \end{aligned} \end{aligned}$$
(34)

As in the proof of Proposition 1, by the necessary condition of the PSMP we get that the value of \(\widetilde{y}_i\) minimizing the reduced Hamiltonian is as in (22). The adjoint processes \(P^i(t)=(P^{i,j}(t): j=1,\dots , N)\) and \(Q^i(t)=(Q^{i,j,k}(t): j=1,\dots , N, \, k=0, \dots , N)\), are the solutions of the same Eq. (23) with \(H^i\) as in (34), while the state dynamics are again as in (2). As before, replacing all the occurrences of the control \(\widetilde{y}_i(t)\) both in the state dynamics and the adjoint equation by (24), gives a forward-backward system which, if solved, provides the optimal closed loop emission level because of the sufficient part of the PSMP. To solve that system we use the ansatz:

$$\begin{aligned} P^{i,j}(t)=2e^{-\lambda t}{\varphi }(t)\left( m_N(t)-X_i(t)\right) \left( \frac{1}{N}-\delta _{i,j}\right) \end{aligned}$$
(35)

for some smooth deterministic function \(\varphi :[0,T]\rightarrow \mathbb {R}\) to be determined. This choice guarantees that (24) is a feedback control. In this way, the state dynamics is as in (26) (apart from replacing \(\eta \) with \(\varphi \)), while the backward equation becomes:

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \text {d}P^{i,j}(t)=2e^{-\lambda t}\left( m_N(t)-X_i(t)\right) \left( \frac{1}{N}-\delta _{i,j}\right) \left[ 2\varphi (t)+\frac{1}{N}\left( 1-\frac{1}{N}\right) \varphi ^2(t)\right] \text {d}t +\sum _{k=0}^N Q^{i,j,k}(t)\, \text {d}B_k(t),\\ \displaystyle P^{i,j}(T)=2e^{-\lambda T}\phi \left( m_N(T)-X_i(T)\right) \left( \frac{1}{N}-\delta _{i,j}\right) \end{array}\right. } \end{aligned}$$
(36)

Differentiating the ansatz (35) we get:

$$\begin{aligned} \begin{aligned} \text {d}P^{i,j}(t)&= 2e^{-\lambda t}\left( \frac{1}{N}-\delta _{i,j}\right) \left( m_N(t)\right. \\&\quad \left. -X_i(t)\right) \left[ -\lambda \varphi (t)+\dot{\varphi }(t)-\varphi (t)\left( 2+\left( 1-\frac{1}{N}\right) \varphi (t)\right) \right] {d}t\\&\quad + 2e^{-\lambda t}\left( \frac{1}{N}-\delta _{i,j}\right) \varphi (t)\,\sigma \left( \frac{1}{N} \sum _{k=1}^N \text {d}B_k(t)- \text {d}B_i(t)\right) . \end{aligned} \end{aligned}$$
(37)

Next, by identifying term by term the first equation of (36) and (37), we obtain:

$$\begin{aligned} Q^{i,j,0}(t)=0, \qquad Q^{i,j,k}(t)= 2e^{-\lambda t}\left( \frac{1}{N}-\delta _{i,j}\right) \varphi (t)\,\sigma \left( \frac{1}{N}-\delta _{i,k}\right) , \qquad k=1, \dots , N \end{aligned}$$

which are deterministic and adapted, while from the drift terms it follows that \(\varphi (t)\) must satisfy the Bernoulli equation:

$$\begin{aligned} \dot{\varphi }(t)=(4+\lambda )\varphi (t)+\left( 1-\frac{1}{N^2}\right) \varphi ^2(t) \end{aligned}$$
(38)

with terminal condition \(\varphi (T)=\phi \), obtained by applying the ansatz (35) to the the final condition in (36). Equation (38) has a unique solution given by:

$$\begin{aligned} \varphi (t)=\frac{(4+\lambda )\phi }{(4+\lambda )e^{(4+\lambda )(T-t)}+\phi \left( 1-\frac{1}{N^2}\right) \left( e^{(4+\lambda )(T-t)}-1\right) } \end{aligned}$$
(39)

Then, an optimal closed loop emission strategy is given by

$$\begin{aligned} \widetilde{y}_i(t)=\left[ 1+\left( 1-\frac{1}{N}\right) \varphi (t)\right] (m_N(t)-\widetilde{X}_i(t)), \end{aligned}$$
(40)

where we denote by \(\widetilde{X}_i\) the state of the player i to stress the fact we are using a closed loop equilibrium. Then, in equilibrium, the state \(\widetilde{X}_i(t)\) satisfies, for every \(i=1, \dots , N\):

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {d} \widetilde{X}_i(t)=\displaystyle \left[ \left( 2+\left( 1-\frac{1}{N}\right) \varphi (t)\right) (m_N(t) -\widetilde{X}_i(t)) -\tau \right] \text {d}t+\sigma \, \text {d}B_i(t), \quad t \in (0, T]\\ \widetilde{X}_i(0)=X_i^0. \end{array}\right. } \end{aligned}$$
(41)

1.3 Proof of proposition 3

Notice that the two functions \(\eta (t)\) and \(\varphi (t)\), given by (31) and (39) respectively, converge to the same limit as \(N\rightarrow +\infty \), i.e.,

$$\begin{aligned} \psi (t)=\frac{(4+\lambda )\phi }{(4+\lambda )e^{(4+\lambda )(T-t)}+\phi \left( e^{(4+\lambda )(T-t)}-1\right) } \end{aligned}$$

which solves the following equation:

$$\begin{aligned} \dot{\psi }(t)=(4+\lambda )\psi (t)+\psi ^2(t), \qquad \psi (T)=\phi . \end{aligned}$$

Consequently, when \(N\rightarrow +\infty \) the open and closed loop emission strategies, given by (32) and (40) respectively, coincide. And it follows that also the open and closed loop optimal states, given by (33) and (41), coincide.

1.4 Proof of proposition 4

Consider the optimal state (41) (which is the same as (33) apart from substituting \(\varphi \) with \(\eta \)). By applying the sum over i and the ratio over N to both sides of (41) we get

$$\begin{aligned} \text {d}\left( \frac{1}{N}\sum _{i=1}^N \widetilde{X}_i(t)\right) =-\tau \text {d}t+\frac{\sigma }{N} \sum _{i=1}^N \text {d}B_i(t). \end{aligned}$$
(42)

Now, by taking the expectation of both sides of (42) we have that

$$\begin{aligned} \text {d}\left( \frac{1}{N}\sum _{i=1}^N \mathbb {E}[\widetilde{X}_i(t)]\right) =-\tau \text {d}t, \end{aligned}$$

form which follows

$$\begin{aligned} {\overline{X}}(t)=\frac{1}{N}\sum _{i=1}^N \mathbb {E}[\,\widetilde{X}_i(t)]=\frac{1}{N}\sum _{i=1}^N\left( -\tau t +\mathbb {E}[\,\widetilde{X}_i(0)]\right) = X^0-\tau t. \end{aligned}$$
(43)

These considerations hold, as just observed, also for the optimal state (33) in which there is the function \(\eta \) instead of \(\varphi \).

1.5 Proof of proposition 5

The proof follows the same scheme presented in (Carmona et al. 2015; see section 5.1).

In order to characterize the optimal emission strategy, we minimize the reduced Hamiltonian given by:

$$\begin{aligned} H(t,x,p,y)&=\left[ (m(t)-x)+y-\tau \right] p\\ \nonumber&\quad \quad +e^{-\lambda t}\Big ((x-m(t))^2+y^2+2y(x-m(t))\Big ) \end{aligned}$$
(44)

which admits minimum at:

$$\begin{aligned} \widehat{y}(t)=-\frac{p}{2e^{-\lambda t}}+(m(t)-x). \end{aligned}$$

The corresponding adjoint forward-backward equations are given by:

$$\begin{aligned} \text {d} X(t)&= \left[ 2(m(t) -X(t))-\frac{P(t)}{2e^{-\lambda t}}-\tau \right] \text {d}t+\sigma \,\text {d}B(t), \end{aligned}$$
(45)
$$\begin{aligned} \text {d} P(t)&= 2P(t)\,\text {d}t+Q(t)\,\text {d}B(t), \nonumber \\ P(T)&= 2e^{-\lambda T}\phi \left( X(T)-m(T)\right) , \end{aligned}$$
(46)

where Q(t) is an adapted square integrable process. To show the existence of a solution of the above affine forward-backward system we follow the result of (Carmona et al. 2013). In particular, we denote by \(m^X(t)=\mathbb {E}[X(t)]\) and \(m^P(t)=\mathbb {E}[P(t)]\). Using the fact that in equilibrium (i.e., after solving for the fixed point) \(m^X(t)=m(t)\) for every \(t\le T\) which in turn implies \(m^P(T)=2e^{-\lambda T}\phi (m^X(T)-m(T))=0\) and considering what has been done with the corresponding finite player game, we take the ansatz:

$$\begin{aligned} P(t)=-2e^{-\lambda t}\psi (t)\left( m(t)-X(t)\right) \end{aligned}$$
(47)

for some smooth deterministic function \(\psi :[0,T]\rightarrow \mathbb {R}\) to be determined. Using (47), Eq. (45) becomes:

$$\begin{aligned} \text {d} X(t) = \left[ \left( 2+\psi (t)\right) (m(t)-X(t))-\tau \right] \,\text {d}t+\sigma \,\text {d}B(t). \end{aligned}$$
(48)

Taking the expectation of both sides of (48) we get \(\text {d}m(t)=-\tau \, \text {d}t\), from which it follows that:

$$\begin{aligned} m(t)=X^0-\tau t=m^X(t), \qquad m^P(t)=0, \qquad t\in [0,T]. \end{aligned}$$
(49)

Moreover, we have:

$$\begin{aligned} \text {d} \left( m(t)-X(t)\right) =- \left( 2+\psi (t)\right) \left( m(t)-X(t)\right) \text {d}t- \sigma \text {d}B(t). \end{aligned}$$
(50)

Using (47) in (46) we get:

$$\begin{aligned} \text {d}P(t)=-4e^{-\lambda t}\psi (t)\left( m(t)-X(t)\right) \text {d}t+ Q(t)\, \text {d}B(t). \end{aligned}$$
(51)

Differentiating the ansatz (47) and using (50), it follows that:

$$\begin{aligned} \text {d}P(t)=2e^{-\lambda t}\left[ \lambda \psi (t)-\dot{\psi }(t)+\psi (t)(2+\psi (t))\right] \left( m(t)-X(t)\right) \text {d}t+2e^{-\lambda t}\sigma \psi (t)\,\text {d}B(t). \end{aligned}$$
(52)

By the comparison of (51) with (52) we get that \(Q(t)= 2e^{-\lambda t}\sigma \psi (t)\) and \(\psi (t)\) must satisfy the Bernoulli equation:

$$\begin{aligned} \dot{\psi }(t)=(4+\lambda )\psi (t)+\psi ^2(t), \qquad \psi (T)=\phi \end{aligned}$$
(53)

whose solution is

$$\begin{aligned} \psi (t)=\frac{(4+\lambda )\phi }{(4+\lambda )e^{(4+\lambda )(T-t)}+ \phi \left( e^{(4+\lambda )(T-t)}-1\right) }\,, \end{aligned}$$

In conclusion, as consequence of Proposition 3, the optimal control is:

$$\begin{aligned} \widehat{y}(t)=(1+\psi (t))(m(t)-X(t)). \end{aligned}$$
(54)

1.6 The optimal mean-field state

The optimal state correspondent to the optimal strategy (54), reads as:

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {d} \widehat{X}(t) = \left[ \left( 2+\psi (t)\right) (m(t)-\widehat{X}(t))-\tau \right] \text {d}t+\sigma \,\text {d}B(t), \quad t \in (0, T]\\ \widehat{X}(0)=X^0. \end{array}\right. } \end{aligned}$$
(55)

By applying the expectation to both members of the first equation of (55) ( and using (49)) we get:

$$\begin{aligned} \overline{X}(t)=\mathbb {E}[\widehat{X}(t)]=X^0-\tau t. \end{aligned}$$
(56)

1.7 Proof of proposition 6

The proof follows the same reasoning as Proposition 5, so we will avoid repetitions where possible. To characterize the optimal emission strategy, we minimize the reduced Hamiltonian given by:

$$\begin{aligned} H(t,x,p,y)&=\left[ (1+\theta (t))(m(t)-x)+y-\tau (t)\right] p\\&+e^{-\lambda t}\Big ((1+\theta (t))^2(x-m(t))^2+y^2+2y(1+\theta (t))(x-m(t))\Big ) \end{aligned}$$

which admits minimum at:

$$\begin{aligned} {y^{*}}(t)=-\frac{p}{2e^{-\lambda t}}+(1+\theta (t))(m(t)-x). \end{aligned}$$
(57)

The corresponding adjoint forward-backward equations are given by:

$$\begin{aligned} \text {d} X(t)&= \left[ 2(1+\theta (t))(m(t) -X(t))-\frac{P(t)}{2e^{-\lambda t}}-\tau (t)\right] \text {d}t+\sigma \,\text {d}B(t), \end{aligned}$$
(58)
$$\begin{aligned} \text {d} P(t)&= 2P(t)(1+\theta (t))\,\text {d}t+Q(t)\,\text {d}B(t), \nonumber \\ P(T)&= 2e^{-\lambda T}\phi \left( X(T)-m(T)\right) , \end{aligned}$$
(59)

where Q(t) is an adapted square integrable process. The existence of a solution of the above affine forward-backwards system follows the result of Carmona et al. (2013). In particular, with the same considerations made above (47), we take the ansatz:

$$\begin{aligned} P(t)=2e^{-\lambda t}\Psi (t)\left( X(t)-m(t)\right) \end{aligned}$$
(60)

for some smooth deterministic function \(\Psi :[0,T]\rightarrow \mathbb {R}\) to be determined. Using (60), Eq. (58) becomes:

$$\begin{aligned} \text {d} X(t) = \left[ \left( 2(1+\theta (t))+\Psi (t)\right) (m(t)-X(t))-\tau (t)\right] \,\text {d}t+\sigma \,\text {d}B(t). \end{aligned}$$
(61)

Taking the expectation of both sides of (61) we get \(\text {d}m(t)=-\tau (t)\, \text {d}t\), from which it follows that:

$$\begin{aligned} m(t)=X^0-\int \limits _0^t\tau (s)\, \text {d}s=m^X(t), \qquad m^P(t)=0, \qquad t\in [0,T]. \end{aligned}$$
(62)

Moreover,

$$\begin{aligned} \text {d} \left( m(t)-X(t)\right) =- \left( 2(1+\theta (t))+\Psi (t)\right) \left( m(t)-X(t)\right) \text {d}t- \sigma \text {d}B(t). \end{aligned}$$
(63)

Using (60) in (59) we get:

$$\begin{aligned} \text {d}P(t)=-4e^{-\lambda t}\Psi (t)(1+\theta (t))\left( m(t)-X(t)\right) \text {d}t+ Q(t)\, \text {d}B(t). \end{aligned}$$
(64)

Differentiating the ansatz (60) and using (63), it follows that:

$$\begin{aligned} \text {d}P(t)&=2e^{-\lambda t}\left[ \lambda \Psi (t)-\dot{\Psi }(t)+\Psi (t)(2(1+\theta (t))+\Psi (t))\right] \left( m(t)\right. \nonumber \\&\quad \left. -X(t)\right) \text {d}t+2e^{-\lambda t}\sigma \Psi (t)\,\text {d}B(t). \end{aligned}$$
(65)

By the comparison of (64) with (65) we get that \(Q(t)= 2e^{-\lambda t}\sigma \Psi (t)\) and \(\Psi (t)\) must satisfy the Bernoulli equation:

$$\begin{aligned} \dot{\Psi }(t)=(4(1+\theta (t))+\lambda )\Psi (t)+\Psi ^2(t), \qquad \Psi (T)=\phi \end{aligned}$$

whose solution is:

$$\begin{aligned} \Psi (t)=\displaystyle \frac{\phi }{ e^{(4+\lambda )(T-t)+\int _t^T 4\theta (\xi ) \text {d}\,\xi }+\phi \,e^{-(4+\lambda )(t-1)-\int _1^t 4\theta (\xi ) \text {d}\,\xi }\left( \int _t^T e^{(4+\lambda )(s-1)+\int _1^s 4\theta (\xi )\text {d}\,\xi }\,\, \text {d}\,s \right) }. \end{aligned}$$

In conclusion, by applying the ansatz (60) to (57) the mean field optimal emission strategy is given by:

$$\begin{aligned} {y^{*}}(t)=(1+\theta (t)+\Psi (t))(m(t)-X(t)). \end{aligned}$$
(66)

1.8 The optimal mean-field state of the extended model

The optimal state correspondent to (66), is:

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {d} {X^{*}}(t) = \left[ \left( 2(1+\theta (t))+\Psi (t)\right) (m(t)-{X^{*}}(t))-\tau (t)\right] \text {d}t+\sigma \,\text {d}B(t), \quad t \in (0, T]\\ {X^{*}}(0)=X^0. \end{array}\right. } \end{aligned}$$
(67)

By applying the expectation to both members of the first equation of (67) [ and using (62)] we get:

$$\begin{aligned} \overline{X}(t)=\mathbb {E}[{X^{*}}(t)]=X^0-\int _0^t\tau (s)\, \text {d}s, \qquad t \in [0, T]. \end{aligned}$$

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La Torre, D., Maggistro, R. & Marsiglio, S. Transboundary pollution control under evolving social norms: a mean-field approach. Decisions Econ Finan (2024). https://doi.org/10.1007/s10203-024-00459-9

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