# A Mean Field Game approach for pollution regulation of competitive firms

Gianmarco Del
Sarto111gianmarco.delsarto$<EMAIL_ADDRESS>Scuola
Normale Superiore, Pisa, Italy  University School for Advanced Studies IUSS,
Pavia, Italy  Marta<EMAIL_ADDRESS>(primary author, alphabetic
order) LUISS Guido Carli, Rome, Italy  Giulia Livieri<EMAIL_ADDRESS>(primary author, alphabetic order)

We are grateful to Prof. René Aïd, Prof. Fausto Gozzi, Prof. Jameson Graber,
Prof. Fujii, Masaaki, and seminar participants at the University of Padova,
LUISS Guido Carli, University of Verona, Durham University, the Mathematics of
subjective probability, Topics in Economics and Probability (Milan, September
11-13, 2023), the DCP24 Dynamics$\&$Complexity (Pisa, June 5-8, 2024), the
$4^{th}$Italian Meeting on Probability and Mathematical Statistics (Rome, June
10-14, 2024) for discussions. Giulia Livieri acknowledges partial support by
the grant Qnt4Green- Quantitative Approaches for Green Bond Market: Risk
Assessment, Agency Problems, and Policy Incentives of the Italian Ministry of
Education and Research. Gianmarco Del Sarto acknowledges the support of the
Italian national inter-university Ph.D. course in sustainable development and
climate change. All errors are our own. The London School of Economics and
Political Science, London, United Kingdom

###### Abstract

We develop a model based on mean-field games of competitive firms producing
similar goods according to a standard AK model with a depreciation rate of
capital generating pollution as a byproduct. Our analysis focuses on the
widely-used cap-and-trade pollution regulation. Under this regulation, firms
have the flexibility to respond by implementing pollution abatement, reducing
output, and participating in emission trading, while a regulator dynamically
allocates emission allowances to each firm. The resulting mean-field game is
of linear quadratic type and equivalent to a mean-field type control problem,
i.e., it is a potential game. We find explicit solutions to this problem
through the solutions to differential equations of Riccati type. Further, we
investigate the carbon emission equilibrium price that satisfies the market
clearing condition and find a specific form of FBSDE of McKean-Vlasov type
with common noise. The solution to this equation provides an approximate
equilibrium price. Additionally, we demonstrate that the degree of competition
is vital in determining the economic consequences of pollution regulation.

Key words: Cap-and-trade; Linear Quadratic Problem; Mean Field Games; Market
Equilibrium; Social Cost Optimization

##  1 Introduction

The problem of excessive firm pollution has long been a part of economic
theory, mainly because it imposes a negative externality on society. In
particular, it is considered as the consequence of the absence of price on
emission, which implies higher volumes than socially optimal levels.
Therefore, from an economic point of view, one possibility is to put a price
on pollution; in this way, polluters will be more conscious about the social
value of their private decisions. One of the most popular measures that help
tackle this problem is the emission trading system, also known as the cap-and-
trade system, which gives the environmental authority direct control on the
overall quantity of emissions and, at the same time, increases the
acceptability of environmental policy for covered companies because they can
make profit from it. The EU-ETS (European Union Emission Trading Scheme) is,
together with the US Sulfur Dioxide Trading System, the most prominent example
of an existing cap-and-trade system deployed in practice (e.g., [16]). Having
made this premise, understanding how the market price of carbon in an emission
trading system is formed through the interaction among a large number of
(indistinguishable rational competitive) firms is significant. This paper
proposes an integrated production-pollution-abatement model in continuous time
and studies cap-and-trade under competition via the Mean Field Game (MFG,
henceforth) approach. The theorethical model is described in detail in Section
3. In particular, we are interested in equilibrium carbon price formation in a
cap-and-trade system, i.e., the pricing of carbon endogenously using a model
of (indistinguishable rational competitive) firms under the market clearing
condition.
It is important to make the following point. In the present work, we consider
two types of competitions. On the one hand, the competition in polluting firms
is because we do not focus on perfect competition or monopoly. However,
instead, we account for firms’ strategic interactions in the output markets by
assuming that firms compete á la Cournot. In other words, competing firms are
trapped in an equilibrium where each firm’s decisions impose not just a
pollution externality on society but also a competitive externality on the
other firms. On the other hand, there is the type of competition in the
continuum limit of an infinity of small players allowed by the MFG framework.
Precisely, each player only sees and reacts to the statistical distribution of
the other players’ states; in turn, their actions determine the evolution of
the state distribution. To avoid confusion, we always refer to the former when
speaking about competition.
MFG models appeared simultaneously and independently in the original works of
[34] and [12], and are, loosely speaking, limits of symmetric stochastic
differential games with a large number of players where each of them interacts
with the average behavior of his/her competitors. In particular, an MFG is an
equilibrium, called $\epsilon$-Nash equilibrium, that occurs when the strategy
employed by a representative agent of a given population is optimal, given the
costs imposed by that population. An increasing stream of research has been
flourishing since 2007, producing theoretical results and a wide range of
applications in many fields, such as economics, finance, crowd dynamics,
social sciences in general, and, only recently, in equilibrium price formation
(e.g., [23] and references therein). We refer to the lecture notes of [14] and
the two-volume monograph by Carmona and Delarue ([20] and [21]) for an
excellent presentation of the MFG theory from analytic and probabilistic
perspective, respectively; in the present paper, we embrace a probabilistic
perspective. A related but distinct concept is that of Mean Field Control
(MFC), where the goal is to assign a strategy to all the agents at once so
that the resulting population behavior is optimal concerning the cost imposed
by a central planner. We refer to the excellent book by [7] for a comparison
between MFGs and MFC.
In general, an optimal control for an MFC is not an equilibrium strategy for
an MFG. Nevertheless, in many cases, the converse holds, and quantifying the
differences between the two approaches is reminiscent of what is known as the
price of anarchy, i.e., the added aggregate cost of allowing all players to
choose their optimal strategy independently. The MFGs for which this happens
are called potential MFGs (e.g., [19]). The model that we propose, while
conceptually constructed as a MFG equilibrium, can be solved via a
reformulation of MFC by using the results in [26]. In particular, our model
belongs to the class of linear-quadratic MFGs (e.g., [8]) with common noise
(e.g., [37] and [38]). The common noise represents an inherent uncertainty in
nature affecting simultaneously all the firms participating in the game (or
being controlled by a central planner). We characterize the solution both in
terms of a stochastic maximum principle (forward-backward system of stochastic
differential equations (FBSDEs)) and Riccati equations. In particular,
similarly to [23], though their work is inspired by financial applications,
when imposing the market clearing condition, we obtain an interesting form of
FBSDEs of McKean-Vlasov type with common noise as a limit problem, involving
the dependence on a conditional expectation. Therefore, the existence of a
unique strong solution is proved by using the well-known Peng-Wu’s
continuation method [36]. In addition, we quantify the relation between the
finite player game and its large population limit, as well as we show that the
solution of the mean-field limit problem actually provides asymptotic market
clearing in the large limit. Instead, if the carbon price process is given
exogenously, the MFG solutions serve as $\epsilon$-Nash equilibria for the
large player game because the game is solved by an optimal control problem
[26].
The last part of the paper presents a numerical study of the proposed model,
which is divided into two parts. In the first part, we analyse the role played
by the environmental authority on the average level of production of a
(representative firm). In the second part, instead, we analyse the economics
of competition. In particular, the representative firm faces a strategic
trade-off between output reduction and pollution abatement under competition.
The latter facilitates synchronization between the representative firm and the
rest of the population in the sense that they agree to reduce output by using
the pollution constraint; naturally, this synchronization mechanism is
expected to work under a suitable range of constraints imposed by the
pollution regulator, the one for which the impact of output reduction on the
representative firm’s profits dominates the cost of pollution abatement, of
trading, and production. Under monopoly, instead, the representative firm can
no longer leverage on the competition with the population of firms to
implement the previously described synchronization mechanism. Whence, the
degree of competition plays a critical role in determining the economic
consequences of pollution regulation. In particular, our model captures a rich
range of competitive markets – with monopoly and Cournot oligopoly as special
cases – and several fundamental elements of pollution generation, abatement
levels and costs, and regulation, which can serve as a basis for future
research.
We proceed as follows. Notation and basic objects are introduced in Section 2.
In Section 3, we provide a precise description of the $N$-player game, where
$N$ denotes the number of firms, together with the definition of
$\epsilon$-Nash equilibria. In Section 4, the limit dynamics for the
$N$-player game is introduced. The corresponding notion of solution of the MFG
is defined and discussed. In Section 5, the MFC problem associated with the
MFG in Section 4 is introduced and discussed; we prove the solvability of the
FBSDE of McKean-Vlasov type and the asymptotic market clearing condition.
Section 7 provides numerical results. Finally, in Section 8, we give
concluding remarks, discuss further extensions of the model and future
directions of research. Additional results on linear-quadratic MFG and MFC are
confined in Appendix A and Appendix B, respectively.

##  2 Notations

Because we are going to derive some broad-gauged results in Appendix A and
Appendix B, the notation in this section will be quite general. Let
$d,d_{0},d_{1},d_{2}\in\mathbb{N}$, where $\mathbb{N}$ is the set of positive
integers, which will be the dimensions of the space of private states, common
noise values, idiosyncratic noise values and control actions, respectively.
The $n$-dimensional Euclidean space $\mathbb{R}^{n}$, with $n\in\mathbb{N}$ a
generic index, is equipped with the standard Euclidean norm, always indicated
by $|\cdot|$. Moreover, we denote by $\mathcal{S}^{n}$ the set of all $n\times
n$ symmetric matrices with real entries. In general, we identify the space of
all $n\times m$ dimensional matrices with real entries with
$\mathbb{R}^{n\times m}$.
Let $N\in\mathbb{N}$. Let
$(\overline{\Omega}^{0},\overline{\mathcal{F}}^{0},\overline{\mathbb{P}}^{0})$
and
$(\overline{\Omega}^{i},\overline{\mathcal{F}}^{i},\overline{\mathbb{P}}^{i})_{i=1}^{N}$
be $(N+1)$ complete probability spaces equipped with filtrations
$(\overline{\mathcal{F}}^{i}_{t})$, $i\in\\{0,\ldots,N\\}$. In particular,
$(\overline{\mathcal{F}}^{0}_{t})$ is the completion of the filtration
generated by the $d_{0}$-dimensional Brownian motion $(W^{0}(t))$, and, for
each $i\in\\{0,\ldots,N\\}$, $(\overline{\mathcal{F}}^{i}_{t})$ is the
complete and right-continuous augmentation of the filtration generated by
$d_{1}$-dimensional Brownian motions $(W^{i}(t))$, as well as a
$(W^{i}(t))$-independent $d$-dimensional square-integrable random variables
$(\xi^{i})_{i=1}^{N}$, which have by assumption the same law. Finally, we
introduce the product probability spaces
$\Omega^{i}=\overline{\Omega}^{0}\times\overline{\Omega}^{i}$,
$\mathcal{F}^{i}$, $(\mathcal{F}_{t}^{i})$, $\mathbb{P}^{i}$,
$i\in\\{1,\ldots,N\\}$, where $(\mathcal{F}^{i},\mathbb{P}^{i})$ is the
completion of
$(\overline{\mathcal{F}}^{0}\otimes\overline{\mathcal{F}}^{i},\overline{\mathbb{P}}^{0}\otimes\overline{\mathbb{P}}^{i})$
and $(\mathcal{F}_{t}^{i})$ is the complete and right-continuous augmentation
of $(\overline{\mathcal{F}}_{t}^{0}\otimes\overline{\mathcal{F}}_{t}^{i})$. In
the same way, we define the complete probability space
$(\Omega,\mathcal{F},\mathbb{P})$ equipped with $(\mathcal{F}_{t})$ satisfying
the usual conditions as a product of
$(\overline{\Omega}^{i},\overline{\mathcal{F}}^{i},\overline{\mathbb{P}}^{i},(\overline{\mathcal{F}}_{t}^{i}))_{i=0}^{N}$.
Let $\Gamma$ be a closed and convex subset of $\mathbb{R}^{d_{2}}$, the set of
control actions, or action space. Moreover, given a probability space
$(\Omega,\mathcal{G},\mathbb{P})$ and a filtration $(\mathcal{G}_{t})$ in
$\mathcal{G}$, let:

  1. (S1)

$\mathbb{L}^{2}(\mathcal{G};\mathbb{R}^{n})$ be the set of
$\mathbb{R}^{n}$-valued $\mathcal{G}$-measurable square-integrable random
variables $U$.

  2. (S2)

$\mathbb{S}^{2}((\mathcal{G}_{t});\mathbb{R}^{n})$ be the set of
$\mathbb{R}^{n}$-valued $(\mathcal{G}_{t})$-adapted continuous processes
$(U(t))$ such that
$\|U\|_{\mathbb{S}^{2}}:=\mathbb{E}\left[\sup_{t\in[0,T]}|U(t)|^{2}\right]^{\frac{1}{2}}<\infty$.

  3. (S3)

$\mathbb{H}^{2}((\mathcal{G}_{t});\mathbb{R}^{n})$ be the set of
$\mathbb{R}^{n}$-valued $(\mathcal{G}_{t})$-progressively measurable processes
$(U(t))$ such that
$\|U\|_{\mathbb{H}^{2}}:=\mathbb{E}\left[\int_{0}^{T}|U(t)|^{2}\,dt\right]<\infty$.

We denote by $\mathcal{L}(U)$ the law of a random variable $U$, and by
$\overline{U}(s)=\mathbb{E}[U(s)|\overline{\mathcal{F}}_{s}^{0}]$ the
conditional expectation of $U(s)$ given $W^{0}(s)$.
For $\mathcal{S}$ a Polish space, let $\mathcal{P}(\mathcal{S})$ denote the
space of probability measures on $\mathcal{B}(\mathcal{S})$, the Borel sets of
$\mathcal{S}$. For $s\in\mathcal{S}$, let $\delta_{s}$ indicate the Dirac
measure concentrated in $s$. Equip $\mathcal{P}(\mathcal{S})$ with the
topology of weak convergence of probability measures. Then
$\mathcal{P}(\mathcal{S})$ is again a Polish space. Let $d_{\mathcal{S}}$ be a
metric compatible with the topology of $\mathcal{S}$ such that
$(\mathcal{S},d_{\mathcal{S}})$ is a complete and separable metric space.
Given a complete compatible metric $d_{\mathcal{S}}$ on $\mathcal{S}$, we also
consider the space of probability measures on $\mathcal{B}(\mathcal{S})$ with
finite $p$-moments, with $p\geq 1$:

$\mathcal{P}_{p}(\mathcal{S})\doteq\left(\nu\in\mathcal{P}(\mathcal{S})\,:\,\exists
s_{0}\in\mathcal{S}\,:\,\int_{\mathcal{S}}d_{\mathcal{S}}(s,s_{0})^{p}\nu(\,ds)<\infty\right).$

In particular, $\mathcal{P}_{p}(\mathcal{S})$ is a Polish space. A compatible
complete metric is given by:

$d_{\mathcal{P}_{p}(\mathcal{S})}(\nu,\tilde{\nu})\doteq\left(\inf_{\begin{subarray}{c}\alpha\in\mathcal{P}(\mathcal{S}\times\mathcal{S})\,:\,[\alpha]_{1}=\nu\,\text{and}\,[\alpha]_{2}=\tilde{\nu}\end{subarray}}\int_{\mathcal{S}\times\mathcal{S}}d_{\mathcal{S}}(s,\tilde{s})^{p}\alpha(\,ds,\,d\tilde{s})\right)^{1/p},$

where $[\alpha]_{1}$ ($[\alpha]_{2}$) denotes the first (second) marginal of
$\alpha$; $d_{\mathcal{P}_{p}(\mathcal{S})}$ is often referred to as the
$p$-Wasserstein (or Vasershtein) metric.

##  3 Theoretical model

This section proposes a stochastic equilibrium model for environmental markets
accounting for the design of today’s emission system. Our model is an
integrated production-pollution-abatement model (e.g., [5]) in continuous
time, which combines a model of competing producers with a pollution model
that includes pollution generation and abatement. Precisely, we consider
$N\geq 1$, $N\in\mathbb{N}$, indistinguishable competing, profit-maximizing
firms, whose carbon emissions are regulated in a cap-and-trade fashion.
Although the regulation of carbon allowances occurs over several periods and
allowances can be banked from one period to the other, we follow [6], [16],
[22], [1], [31], [32], and we focus on a single period of $T$ years at the end
of which compliance is assessed. We assume that capital is created according
to a standard AK model with a positive depreciation rate of capital and with a
positive technological level $A_{k}^{i}$; see the term “revenues” in the cost
functional in Equation (12). Let $K^{i}(t)$ be the level of capital at time
$t$ of firm $i$, for $i=1,\ldots,N$. We assume that the dynamics of
$(K^{i}(t))$ is described by the following stochastic differential equation
(SDE)

$dK^{i}(t)=(\kappa_{f}^{i}K^{f,i}(t)+\kappa_{g}^{i}K^{g,i}(t)-\delta^{i}K^{i}(t))\,dt+\sigma
K^{i}(t)\,dW^{1,i}(t),\quad K^{i}(0)=\kappa_{0},$ (1)

where $\kappa_{f}^{i},\kappa_{g}^{i},\sigma,\delta^{i}$ are positive
constants. $K^{f,i}(t)$ and $K^{g,i}(t)$ represent the amount of fossil-fuel
and green energy based level of capital used by firm $i$ for capital creation.
$\delta^{i}$ is the depreciation rate of capital. The quantity $\sigma
K^{i}(t)$ represents the standard deviation of the level of capital of firm
$i$ and depends on $K^{i}(t)$ itself; $(W^{1,i}(t))$ is a standard Brownian
motion. Firm $i$, for $i=1,\ldots,N$, controls the level of capital trend via
$K^{f,i}(t)$ and $K^{g,i}(t)$, which increases at a rate
$\kappa_{f}^{i}K^{f,i}(t)+\kappa_{g}^{i}K^{g,i}(t)-\delta^{i}K^{i}(t)$, while
the volatility is uncontrolled. We assume that
$K^{f,i}(t),K^{g,i}(t)\in\mathbb{H}^{2}((\mathcal{F}_{t}^{i}),\mathbb{R})$.
Moreover, we assume that firm $i$, for $i=1,\ldots,N$, faces quadratic costs,
say $C^{i}(K^{f})$ and $C^{i}(K^{g})$, in the capital levels333Admittedly, we
have not found a precise reference for this assumption, but it seems
reasonable and necessary to obtain a linear quadratic form for our problem..
More precisely, we have:

$C^{i,f}(K^{f})=c_{1,1}^{i}K^{f,i}+c_{1,2}^{i}(K^{f,i})^{2},\,\,\text{and}\,\,C^{i,g}(K^{g})=c_{2,1}^{i}K^{g,i}+c_{2,2}^{i}(K^{g,i})^{2}.$
(2)

Firms generate pollution as a byproduct of the production process. Let $E_{i}$
denote the pollution, in terms of carbon emissions, generated by firm i prior
to any investment in abatement; we name $E_{i}$ as business-as-usual (BAU)
emissions444Henceforth, to keep the reading, we use the term emission instead
of carbon emission.. Clearly, $E^{i}$ must be increasing in the capital level,
and it seems reasonable to assume that it is a function of $K^{f,i}(t)$. We
follow [5], and we impose a linear relation between the BAU emissions and the
latter by assuming:

$dE^{i}(t)={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\kappa_{e}^{i}}K^{f,i}(t)\,dt+\sigma_{1}^{i}d\widetilde{W}^{2,i}(t),\quad
E^{i}(0)=E_{0},$ (3)

where $k_{e}^{i}>0$ characterizes the linear relationship, and
$\widetilde{W}^{2,i}(t)=\sqrt{1-\rho_{i}^{2}}W^{2,i}(t)+\rho_{i}W^{0,1}(t)$,
$\rho_{i}\in[0,1]$, so that the correlation between $\widetilde{W}^{2,i}(t)$
and $\widetilde{W}^{2,j}(t)$ is $r_{ij}:=\rho_{i}\rho_{j}$. The increments of
$W^{2,i}$ and $W^{0,1}$ are independent, and independent from the increments
of $W^{1,i}$. The noise decomposition captures the fact that the emission of
firm $i$ is affected by its own idiosyncratic noise $dW^{2,i}$ and by the
common economic business cycle $dW^{0,1}$. A similar model for the BAU
emissions is employed in [1]. Here, we also assume the presence of a short-
term emission shock $de^{i}(t)=\sigma_{2}^{i}dW^{3,i}(t)$, $e^{i}(0)=e_{0}$
which may represent, e.g., the outage of a carbon-friendly production unit
that is instantaneously replaced by a more polluting one (e.g., [29]).
Increments of $W^{3,i}$ are independents from the increments of $W^{1,i}$,
$W^{2,i}$, $W^{0,1}$. Therefore, the total emission dynamics is given by

$d\widetilde{E}^{i}(t)=dE^{i}(t)+de^{i}(t)=({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\kappa_{e}^{i}}K^{f,i}(t)\,dt+\sigma_{1}^{i}d\widetilde{W}^{2,i}(t))+(\sigma_{2}^{i}dW^{3,i}(t)),$
(4)

where $\widetilde{E}^{i}(0)=\widetilde{E}_{0}$. We describe now our model of
pollution abatement. Firm $i$, for $i=1,\ldots,N$, breaks down emissions via
two complementary notions: (i) abatement level, and (ii) abatement cost. Under
the abatement effort rate
$\alpha^{i}(t)\in\mathbb{H}^{2}((\mathcal{F}_{t}^{i}),\mathbb{R})$, emissions
of firm $i$ becomes:

$d\widetilde{E}^{i,\alpha}(t)=({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\kappa_{e}^{i}}K^{f,i}(t)-\alpha^{i}(t))\,dt+\sigma_{1}^{i}d\widetilde{W}^{2,i}(t))+(\sigma_{2}^{i}dW^{3,i}(t)).$
(5)

In this way, the firm controls its emission trend, which increases at a rate
$({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\kappa_{e}^{i}}K^{f,i}(t)-\alpha^{i}(t))$;
on the other hand, the volatility remains uncontrolled. Notice that, contrary
to [5], our model does not assume that pollution is observable and a
deterministic function of the level of capital. As regards the abatement
costs, the extant literature assumes that pollution abatement costs are
increasing and quadratic (see, e.g., [41]), or at least convex increasing in
the quantity of emission abated, which is $\alpha^{i}(t)$ in our model (see,
e.g., [35]). This is because usually the initial units of emissions are easy
to abate, but once the low-hanging fruit have been exploited, pollution
abatement becomes increasingly difficult (see, e.g., [5]). Thus, we assume the
following quadratic form for the abatement cost function:

$C_{i}(\alpha)=h_{i}\alpha^{i}+\frac{1}{2\eta_{i}}(\alpha^{i})^{2},\quad
h_{i},\eta_{i}>0,$ (6)

where the constant $\eta_{i}$ is positively correlated with the flexibility of
the abatement process and, therefore, with the reversibility of the decision.
Before describing the dynamics for the bank account $\widetilde{X}$, we detail
the competition mechanism in our model. We assume that firm $i$, for
$i=1,\ldots,N$, faces linear inverse demand curve
$p^{i}(t):=p(K^{i}(t),K^{-i}(t))$, which can be derived by a suitable
quadratic utility function555It is not difficult to see that such a linear
demand function can be derived from a quadratic utility function of the
following form – for the sake of simplicity we denote by $q_{i}$ the quantity
produced by firm $i$ –:
$U(q)=a\sum_{i=1}^{N}q_{i}-C_{1}\left(\sum_{i=1}^{N}q_{i}^{2}+C_{2}\sum_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{N-1}q_{i}q_{j}\right)-q\cdot p$, where
$C_{1}=\frac{b}{2}\left(1-\gamma\left(1-\frac{1}{N}\right)\right)$ and
$C_{2}=\frac{\gamma}{N(1-\gamma)+\gamma}$ and solving the utility maximization
problem for a representative consumer; the first-order condition gives
Equation (7)., given by

$p^{i}(t)=a-b{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(1-\gamma)A_{k}^{i}K^{i}(t)}-b\gamma\frac{1}{N}\sum_{\begin{subarray}{c}j=1\end{subarray}}^{N}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A_{k}^{j}K^{j}(t)},$
(7)

where $a$ and $b$ are positive constants, and $\gamma\in[0,1]$ captures the
degree of production substitution, and hence competition, and $A_{k}^{i}$
represents the technological level and hence $A_{k}^{i}K^{i}(t)$ is the
production function of the firm $i$. We continue with the following
observation: the demand function in Equation (7) comprises a range of
competitive markets, in which monopoly and Cournot oligopoly are polar cases.
Indeed, it can be written in the following way:

$\begin{split}p^{i}(t)=a-b\left(1-\gamma\left(1-\frac{1}{N}\right)\right)A_{k}^{i}K^{i}(t)-b\gamma\frac{1}{N}\sum_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{N-1}A_{k}^{j}K^{j}(t)\end{split}$

When $\gamma=0$ (i.e., monopoly), then $p^{i}(t)=a-bA_{k}^{i}K^{i}(t)$. When
$\gamma=1$ (i.e., Cornout oligopoly), then
$p^{i}(t)=a-b\gamma\frac{1}{N}\sum_{\begin{subarray}{c}j=1\end{subarray}}^{N}A_{k}^{j}K^{j}(t)$
and so there is perfect competition. On the other hand, $\gamma\in(0,1)$
captures the degree of substitution. In particular, the price $p^{i}(t)$ is
influenced more by the level of production of the corresponding good $i$ with
respect to the total quantity produced by all the other firms (firm $i$
excluded); indeed
$b\left(1-\gamma\left(1-\frac{1}{N}\right)\right)>b\gamma\frac{1}{N}$ for
every $\gamma\in(0,1)$, which is a natural (although myopic in the sense of
emotions) postulation. Notice that this type of asymmetry is not in contrast
with the symmetry required by the MFG framework. We now turn to the
description of the dynamics for the bank account.
The bank account’s dynamics is specified as in [1]. We assume that the
regulator opens for each firm $i$, $i=1,\ldots,N$, at $t=0$ a bank account
$\widetilde{X}^{i}$ and allocates permits, which are represented by the
cumulative process $\widetilde{A}^{i}$. The dynamics of the bank account is
given by:

$d\widetilde{X}^{i}_{t}=\beta^{i}(t)\,dt+d\widetilde{A}^{i}(t)-d\widetilde{E}^{i,\alpha}(t),$
(8)

where $\beta^{i}(t)\in\mathbb{H}^{2}((\mathcal{F}^{i}_{t}),\mathbb{R})$ is the
trading rate in the liquid allowance market; emissions, trade, and bank
account are measured in tons or in multiples of tons). We assume that
$\widetilde{A}^{i}$ has the following dynamics:

$d\widetilde{A}^{i}(t)=\tilde{a}^{i}(t)\,dt+\tilde{\sigma}_{2}\,dW^{0,2}(t),\quad\widetilde{A}^{i}(0)=\widetilde{A}_{0}.$
(9)

where $(W^{0,2}(t))$ is a standard Brownian motion common to all firms
independent from all the other noises involved in the model. The fact that
$(W^{0,2}(t))$ is independent from $(W^{0,1}(t))$ is admittedly a very heavy
assumption; the case of correlated common noises is an important subject for
future research; see the discussion in Section 8. The choice of a dynamic
allocation mechanism, instead of a static one, is because of the presence of
the common shock in the BAU emissions dynamic, otherwise there would be no
benefit from the implementation of a dynamic allocation scheme; see [1]. The
quantity $\tilde{a}$ represents the rate. Notice that the sign of
$\widetilde{A}^{i}$ can be either negative, meaning that the regulator is
placing a penalty on the firm bank account, or positive, meaning that the
regulator is giving true permits to the firm. Admittedly, the dynamics for
$\widetilde{A}^{i}(t)$ could be more general, by adding, for instance, a pure
jump part (see [1], Section 2). Firm $i$, $i=1,\ldots,N$, controls the trading
rate $\beta^{i}(t)$.
Now, let $(\overline{\omega}_{t})$ be the price of allowances. It can be
either exogenous, for example described by the Black-Scholes model, or
endogenous, in the sense that it is determined by the fundamental condition of
the market. More precisely, the total number of tons of emissions being
purchased by a firm via the emission exchange at a given time must be equal to
the number of those being sold by others via the emission exchange at the same
time. In particular, the balance between the sales and the purchases, called
market clearing conditions, must hold at any point in time. The market
clearing condition reads as:

$\sum_{i=1}^{N}\hat{\beta}^{i}(t)=0,\quad dt\otimes d\mathbb{P}-\text{a.e.},$
(10)

where $\hat{\beta}_{t}^{i}$ is the trading rate of the $i$th firm. We will be
interested in finding an appropriate price process $(\overline{\omega}_{t})$
so that it achieves the market clearing condition among the rational firms.
More precisely, as stated in the introduction, we are interested in finding a
market equilibrium, which is defined as trading strategies and market price
such that each firm has minimized its criteria and the market clears for the
market price. We assume that firms are price-takers, an assumption that is in
line with the large number of companies regulated under today’s emission
trading systems666Indeed, more than 5,500 firms are regulated under the EU ETS
(e.g., [13]), and minimize their cost functional by finding an optimal trade-
off between implementing abatement measures, trading permits in the market,
and taking the risk of penalty payments. Another possibility is to model the
price as an underlying martingale plus a drift representing a form of
permanent price impact (e.g., [17]):

$d\overline{\omega}_{t}=\frac{\tilde{\nu}}{N}\sum_{i=1}^{N}c^{{}^{\prime}}(\beta^{i}(t))\,dt+\sigma_{0}dW_{t},$

which in the particular case $c(\beta)=\beta^{2}$ corresponds to the
influential Almgren-Chriss model ([4]); $\tilde{\nu}>0$. Notice that in this
case, we would obtain a mean-field game of control with common noise. We
follow [23], and we state that the process $(\overline{\omega}_{t})$ is likely
to be given by a $\overline{\mathcal{F}}^{0}$-progressively measurable process
since the effects from the idiosyncratic parts from many firms are expected to
be canceled out. Moreover, we assume that if firm $i$, $i=1,\ldots,N$, places
a (market) order of $\beta^{i}(t)$ when the market price is
$(\overline{\omega}_{t})$, then the cost incurred by the firms is given by

$\beta^{i}(t)\overline{\omega}_{t}+\frac{1}{2\nu}(\beta^{i}(t))^{2},$ (11)

where $\nu>0$ is the (constant) market depth parameter which takes into
account a price impact effect as in the original work of [30]. We make now the
following remark.

######  Remark 3.1.

In our model, we do not enforce constraints on the controls because we give
priority to finding explicit solutions to our problem. Indeed, our goal is to
analyze the qualitative behaviour of the system, a goal achieved in a
satisfactory way in the numerical section. A similar “relaxation” of the
problem is done also in, e.g., [2] and [3].

Let
$\mathcal{H}_{1}^{N}:=\mathbb{H}^{2}((\mathcal{F}_{t}^{i}),\mathbb{R}^{4})$,
and let $\mathcal{H}_{N}^{N}$ the set of all $N$-dimensional vectors
$\boldsymbol{v}^{N}:=(v^{N,1},\ldots,v^{N,N})$ such that
$v^{N,i}\in\mathcal{H}_{1}^{N}$, with the vector $v^{N,i}(t)$ defined as
$v^{N,i}(t):=(K^{f,i}(t),K^{g,i}(t),\alpha^{i}(t),\beta^{i}(t))$, for
$i=1,\ldots,N$. Each element of $\mathcal{H}_{N}^{N}$ is called strategy
vector. Under the assumption of risk neutrality, firm $i$, for $i=1,\ldots,N$,
evaluates a strategy vector $\boldsymbol{v}^{N}\in\mathcal{H}_{N}^{N}$
according to its expected cost (notice that we highlight the dependence on the
vector $\boldsymbol{v}^{N}$ in the term in Equation (7))

$\begin{split}\mathcal{J}^{i}(\boldsymbol{v}^{N})&=\mathbb{E}\Bigg{[}\int_{0}^{T}\underbrace{-p^{i}(t,\boldsymbol{v}^{N}){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A_{k}^{i}K^{i}(t)}}_{-\text{revenues}}+\underbrace{\beta^{i}(t)\overline{\omega}_{t}+\frac{1}{2\nu}(\beta^{i}(t))^{2}}_{\text{cost
of trading
\eqref{eq::costoftrading}}}+\underbrace{C_{i}(\alpha(t))}_{\text{abatement
cost \eqref{eq::costabatement}}}\\\
&+\underbrace{C^{i,f}(K^{f}(t))+C^{i,g}(K^{g}(t))}_{\text{costs of production
\eqref{eq::productioncosts}}}\,dt+\underbrace{\lambda(\widetilde{X}^{i}(T))^{2}}_{\text{final
penalization}}\Bigg{]}\end{split}$ (12)

where the term $\lambda(\widetilde{X}_{T}^{i})^{2}$, with $\lambda>0$, is the
terminal monetary penalty on the bank accounts set by the regulator, which is
a regularized version of the terminal cap penalty function applied in
practice, which is zero if the firm is compliant and linear otherwise.
However, as cited in [15], optimal strategies cannot be found in closed form
in this case. Through the previous penalty, the firm is going to pay both if
its bank account is above or below the compliance zero level. In particular,
notice that the Brownian motions in the dynamics for $\widetilde{A}^{i}$ and
$\widetilde{E}^{i,\alpha}$ are chosen for tractability reason, because of the
additive quadratic penalty $\lambda(\widetilde{X}_{T}^{i})^{2}$ in the cost
functional (12).
Finally, the dynamics for the capital level and the bank account are given by:

$\begin{cases}\begin{split}dK^{i}(t)&=(\kappa_{f}^{i}K^{f,i}(t)+\kappa_{g}^{i}K^{g,i}(t)-\delta^{i}K^{i}(t))\,dt+\sigma
K^{i}(t)\,dW^{1,i}(t),\quad K^{i}(0)=\kappa_{0}.\\\
d\widetilde{X}^{i}(t)&=(\beta^{i}(t)+\tilde{a}^{i}(t)+\alpha^{i}(t)-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\kappa_{e}^{i}}K^{f,i}(t))\,dt+\tilde{\sigma}_{2}\,dW^{0,2}(t)-\sigma_{1}^{i}\rho_{i}\,dW^{0,1}(t)\\\
&-\sigma_{1}^{i}\sqrt{1-\rho_{i}^{2}}\,dW^{2,i}(t)-\sigma_{2}^{i}dW^{3,i}(t),\quad\widetilde{X}^{i}(0)=\widetilde{X}_{0}.\end{split}\end{cases}$
(13)

In addition, see, again, Equation (7), we have:

$p^{i}(t,\boldsymbol{v}^{N})=a-b{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(1-\gamma)A_{k}^{i}K^{i}(t)}-b\gamma\frac{1}{N}\sum_{\begin{subarray}{c}j=1\end{subarray}}^{N}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A_{k}^{j}K^{j}(t)}K^{j}(t).$

In the present paper, we take a non-cooperative game point of view. The aim of
each firm $i$, for $i=1,\ldots,N$, is to minimize the cost in Equation (12) by
controlling the level of capital linked to fossil-fuel and green technologies,
the quantity of emissions abated, and the trading rate in the allowance
market. In a non-cooperative game setting, we are led to the analysis of a
non-zero sum stochastic game with $N$ players and to the search of
$\epsilon$-Nash equilibrium. In the next definition, we use the standard
notation $[v^{N,-i},v]$ to indicate a strategy vector equal to
$\boldsymbol{v}^{N}$ for all firms but the $i$-th, which deviates by playing
$v\in\mathcal{H}_{1}^{N}$ instead.

######  Definition 3.2 ($\epsilon$-Nash equilibrium for the $N$-players game).

Let $\epsilon\geq 0$. A strategy vector
$\boldsymbol{v}^{N}\in\mathcal{H}_{N}^{N}$ is called $\epsilon$-Nash
equilibrium for the $N$-player game if for every $i\in\\{1,\ldots,N\\}$ and
for any deviation $v\in\mathcal{H}_{1}^{N}$ we have:

$\mathcal{J}^{i}(\boldsymbol{v}^{N})\leq\mathcal{J}^{i}([v^{N,-i},v])+\epsilon.$

Before proceeding, we summarize our notation in Table 1.

Table 1: The table summarizes and explains the notations used in the theoretical model. The index $i$ generally refers to company $i$ in the economy, and by omitting the $i$ we refer to an economy-wide version of the respective variable. Time dependence is indicated in parentheses. BAU stands for Business As Usual. GBM stands for Geometric Brownian Motion. Notation. | Synthetic description
---|---
$K^{i}(t)$ | Level of capital of company $i$ at time $t$.
$K^{f,i}(t)$ | Fossil-fuel level of capital of company $i$ implemented at time $t$.
$K^{g,i}(t)$ | Green level of capital of company $i$ implemented at time $t$.
$\kappa_{f}^{i}$ | Amount of fossil-fuel level of capital of company $i$ implemented $\forall\,t$.
$\kappa_{g}^{i}$ | Amount of green level of capital of company $i$ implemented $\forall\,t$.
$\sigma$ | Volatility of the level of capital of company $i$.
$C^{i,f}(K^{f})$ | Cost associated to the fossil-fuel level of capital; quadratic; $c_{1,1}^{i},c_{1,2}^{i}$.
$C^{i,g}(K^{g})$ | Cost associated to the green level of capital; quadratic; $c_{2,1}^{i},c_{2,2}^{i}$.
$E^{i}(t)$ | BAU emissions of company $i$ at time $t$.
$\sigma_{1}^{i}$ | Volatility of BAU emissions of company $i$.
$\rho_{i}$ | Correlation between the idiosyncratic and the common noise of $E^{i}(t)$.
$e^{i}(t)$ | Short-term emission shocks of company $i$ at time $t$.
$\sigma_{2}^{i}$ | Volatility of short-term emission shocks of company $i$.
$\widetilde{E}^{i}(t)$ | BAU + short-term emission of company $i$ at time $t$.
$\alpha^{i}(t)$ | Emission abatement effort rate of company $i$ implemented at time $t$.
$C_{i}(\alpha)$ | Abatement cost function function; linear-quadratic; $h_{i}$, $1/(2\eta_{i})$.
$p^{i}(t)$ | Price of the good produced by company $i$ at time $t$; inverse demand; $a$, $b$.
$\gamma$ | Degree of production substitution; $\gamma\in[0,1]$.
$\widetilde{X}^{i}(t)$ | Carbon emission bank account of company $i$ at time $t$.
$\beta^{i}(t)$ | Trading rate in $\widetilde{X}^{i}(t)$ of company $i$ implemented at time $t$.
$\widetilde{A}^{i}(t)$ | Allocated permits of company $i$ at time $t$; GBM, $\tilde{a}^{i}(t),\,\tilde{\sigma}_{2}$.
$\nu$ | Market depth.
$\lambda$ | Terminal monetary penalty on the bank account set by the regulator.
$\delta^{i}$ | Depreciation rate of the capital stock.
$A_{k}^{i}$ | Productivity level of the capital.

##  4 A mean field game approximation with common noise for the $N$ player
game.

In this section, we consider the filtered probability space
$(\Omega,\mathcal{F},\mathbb{P},(\mathcal{F}_{t}))$, $d_{1}=3$ Brownian
motions $(W^{j}(t))$, $1\leq j\leq 3$, which are mutually independent and
independent from the completion of the filtration
$(\overline{\mathcal{F}}_{t}^{0})$, defined in Section 2. In order to find the
expression for the MFG approximation, we follow [33] and we introduce the type
vectors
$\zeta_{i}=(\kappa_{f}^{i},\kappa_{g}^{i},\sigma_{1}^{i},\rho_{i},\sigma_{2}^{i},\delta^{i},A_{k}^{i})$,
for $i=1,\ldots,N$. As said in the introduction, the finite set of firms
becomes a continuum and competes with the rest of the (infinite) population.
In particular, the MFG is defined in terms of a representative firm who is
assigned a random-type vector
$\zeta=(\kappa_{f},\kappa_{g},\sigma_{1},\rho,\sigma_{2},\delta,A_{k})$ at
time zero, which encodes the distribution of the (continuum of) firms’ types.
Formally, cfr. [33], the type vector $\zeta_{i}$ induces an empirical measure
called the type distribution, which is the probability measure on the type
space
$\mathcal{Z}^{e}:=\mathbb{R}\times\mathbb{R}\times\mathbb{R}_{+}\times[0,1]\times\mathbb{R}_{+}\times[0,1]\times\mathbb{R}_{+}$,
given by $m_{N}(A)=\frac{1}{N}\sum_{i=1}^{N}\mathrm{1}_{A}(\zeta_{i}),$ for
Borel sets $A\subset\mathcal{Z}^{e}$. We assume now that as the number of
firms becomes large, $N\rightarrow\infty$, the just introduced empirical
measure $m_{N}$ has a weak limit $m$, in the sense that
$\int_{\mathcal{Z}^{e}}\varphi dm_{N}\rightarrow\int_{\mathcal{Z}^{e}}\varphi
dm$ for every bounded continuous function $\varphi$ on $\mathcal{Z}^{e}$; this
holds almost surely if the $\zeta_{i}^{{}^{\prime}}$ are i.i.d. samples from
$m$. In particular, the probability measure $m$ represents the distribution of
type parameters $\zeta$ among the continuum of firms. At this point, let
$x_{0}=(k_{0},\tilde{x}_{0})$ be a random vector which is independent from
$(\overline{\mathcal{F}}_{t}^{0})$. The representative firm’s level of capital
$K$ and bank account $\widetilde{X}$ solve

$\begin{cases}\begin{split}dK(t)&=(\kappa_{f}K^{f}(t)+k_{g}K^{g}(t)-\delta
K^{i}(t))\,dt+\sigma K(t)\,dt,\quad K(0)=\kappa_{0}.\\\
d\widetilde{X}(t)&=(\beta(t)+\tilde{a}(t)+\alpha(t)-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\kappa_{e}}K^{f}(t))\,dt+\tilde{\sigma}_{2}dW^{0,2}(t)-\sigma_{1}\rho
dW^{0,1}(t)\\\
&-\sigma_{1}\sqrt{1-\rho^{2}}dW^{2}(t)-\sigma_{2}dW^{3}(t),\quad\tilde{X}(0)=\tilde{x}_{0},\end{split}\end{cases}$
(14)

where $v(t):=(K^{f,i}(t),K^{g,i}(t),\alpha^{i}(t),\beta^{i}(t))$ belongs to
the space $\mathbb{H}^{2}((\mathcal{F}_{t}),\mathbb{R}^{4})$. Moreover, by
setting $\overline{K}(t)=\mathbb{E}[K(t)|\overline{\mathcal{F}}_{s}^{0}]$,
with $s\leq t$, we denote

$p^{\overline{K}}(t)=a-b{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(1-\gamma)A_{k}K(t)}-b\gamma{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A_{k}\overline{K}(t)},$

where we assumed that for a large number of firms $N$, we approximate the
dynamics in Equation (7) by the expression in the previous equation, where we
used directly the quantity $\overline{K}(t)$ instead of a generic
$(\overline{\mathcal{F}}_{t}^{0})$ adapted real-valued process since the
dynamics of $p^{i}$ is uncontrolled (see, also, the discussion in [3], Section
3, Page 653).
We now consider the following cost functional:

$\begin{split}\mathcal{J}^{NE}(v;\overline{K})&=\mathbb{E}\Bigg{[}\int_{0}^{T}-p^{\overline{K}}(t)K(t)+\beta(t)\overline{\omega}_{t}+\frac{1}{2\nu}(\beta(t))^{2}+C(\alpha(t))\\\
&+C^{f}(K^{f}(t))+C^{g}(K^{g}(t))\,dt+\lambda(\widetilde{X}(T))^{2}\Bigg{]},\end{split}$
(15)

where the superscript NE stands for Nash equilibrium. Equations (14) and (15)
represent a MFG of linear quadratic type which fits into the framework studied
in [26], Section 3, apart from the presence of terms of order zero in both the
private state dynamics $X(t):=(K(t),\widetilde{X}(t))^{T}$ and the running
cost functional; see the discussion in Appendix A and Appendix B. For the
reader’s convenience and to maintain the present work as self-contained as
possible, Appendix A presents the class of linear quadratic MFGs considered
here. In particular, by using the notation in Appendix A, the non-zero
matrices characterizing the dynamics of $X(t)$ are the following:

$A_{0}(s)=\begin{bmatrix}0\\\ \tilde{a}(s)\\\
\end{bmatrix},\,\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A=\begin{bmatrix}-\delta&0\\\
0&0\\\ \end{bmatrix}},\,\,B=\begin{bmatrix}\kappa_{f}&\kappa_{g}&0&0\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-\kappa_{e}}&0&1&1\\\
\end{bmatrix},\,\,$ (16) $C_{0,2}=\begin{bmatrix}0\\\
-\sigma_{1}\sqrt{1-\rho^{2}}\\\ \end{bmatrix},\,\,C_{0,3}=\begin{bmatrix}0\\\
-\sigma_{2}\\\ \end{bmatrix},\,\,C_{1}=\begin{bmatrix}\sigma&0\\\ 0&0\\\
\end{bmatrix},\,\,$ (17) $F_{0,1}=\begin{bmatrix}0\\\ -\sigma_{1}\rho\\\
\end{bmatrix},\,\,F_{0,2}=\begin{bmatrix}0\\\ \tilde{\sigma}_{2}\\\
\end{bmatrix}.$ (18)

Whereas, the ones characterizing the cost functional are given by:

$Q=\begin{bmatrix}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}b(1-\gamma)A_{k}^{2}}&0\\\
0&0\\\
\end{bmatrix}\quad\overline{Q}=\begin{bmatrix}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{b\gamma
A_{k}^{2}}{2}}&0\\\ 0&0\\\ \end{bmatrix}\quad
R=\begin{bmatrix}c_{1,2}&0&0&0\\\ 0&c_{2,2}&0&0\\\ 0&0&\frac{1}{2\eta}&0\\\
0&0&0&\frac{1}{2\nu}\end{bmatrix}.$ (19)

$q=\begin{bmatrix}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-\frac{aA_{k}}{2}}\\\
0\\\ \end{bmatrix}\quad r(s)=\begin{bmatrix}\frac{c_{1,1}}{2}\\\
\frac{c_{2,1}}{2}\\\ \frac{h}{2}\\\ \frac{\overline{\omega}(s)}{2}\\\
\end{bmatrix}\quad H=\begin{bmatrix}0&0\\\ 0&\lambda\\\ \end{bmatrix}$ (20)

Before proceeding, we make the following remark.

######  Remark 4.1.

In order to satisfy assumptions (N5), the matrix $R$ has to be a positive-
definite matrix, implying that $\nu$ must be in $(0,\infty)$. In particular,
we do not consider directly the case without frictions as done in [1], which,
admittedly, could be a common assumption in the carbon market (see, e.g.,
[31]).

Let us now assume that
$(\overline{\omega}_{t})\in\mathbb{H}^{2}((\overline{\mathcal{F}}^{0}_{t});\mathbb{R})$,
with
$\overline{\omega}_{T}\in\mathbb{L}^{2}(\overline{\mathcal{F}}^{0}_{T};\mathbb{R})$,
is given. Then, we have the following characterization (see also [26],
Proposition 3.2), which is due to the uniform convexity of the functional
$\mathcal{J}^{NE}(v;\overline{K})$ that guarantees the existence of a unique
minimizer (see either [26], Lemma 2.2, or [42], Proposition 2.7). In order to
not burden the reading, we omit the proof of the subsequent proposition
because we will provide in Section 5, Proposition 5.1, the proof of the
characterization of the associated MFC problem, which follows the same line of
argument. Before continuing, let us make the following observation about the
notation. The subsequent results involve the processes
$Z\in\mathbb{H}^{2}((\mathcal{F}_{t});\mathbb{R}^{2\times 3})$ and
$Z_{0}\in\mathbb{H}^{2}((\mathcal{F}_{t});\mathbb{R}^{2\times 2})$. To denote
the entry $(i,j)$ of $Z$ (resp. of $Z_{0}$), we will use the notation
$Z^{(i)}_{j}$ (resp. $Z_{0,j}^{(i)}$).

######  Proposition 4.2.

Let $\overline{K}(t)=\mathbb{E}[K(t)|\overline{\mathcal{F}}_{t}^{0}]$, and
$x_{0}=(k_{0},\tilde{x}_{0})$ be a random vector which is independent from
$(\overline{\mathcal{F}}^{0}_{t})$. Then, there exists a unique control
$v=v(\overline{K},x_{0})$ minimizing the functional in Equation (15).
Furthermore, let $X(s)=(K(s),\widetilde{X}(s))$ be the corresponding
trajectory, i.e., the solution of Equation (14) with control $v$. Then there
exists a unique solution
$(Y,Z,Z_{0})\in\mathbb{S}^{2}((\mathcal{F}_{t});\mathbb{R}^{2})\times\mathbb{H}^{2}((\mathcal{F}_{t});\mathbb{R}^{2\times
3})\times\mathbb{H}^{2}((\mathcal{F}_{t});\mathbb{R}^{2\times 2})$ of the
following BSDE, with $s\in[0,T]$:

$\begin{cases}\begin{split}dY^{(1)}(s)=&-\left({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-\delta
Y^{(1)}(s)\,ds}+\sigma
Z_{1}^{(1)}(s)+b{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{K}}(s)+\frac{b\gamma}{2}\overline{K}(s)-\frac{a}{2}\right)\,ds\\\
&+\sum_{j=1}^{3}Z_{j}^{(1)}dW^{j}(s)+\sum_{j=1}^{2}Z_{0,j}^{(1)}dW^{0,j}(s),\quad
Y^{(1)}(T)=0;\\\
dY^{(2)}(s)=&\sum_{j=1}^{3}Z_{j}^{(2)}dW^{j}(s)+\sum_{j=1}^{2}Z_{0,j}^{(2)}dW^{0,j}(s),\quad
Y^{(2)}(T)=\lambda(\widetilde{X}(T))^{2}\end{split}\end{cases}$ (21)

satisfying the coupling condition, with $s\in[0,T]$, a.s.,

$\begin{cases}K^{f}(s)&=-\frac{\kappa_{f}}{c_{1,2}}Y^{(1)}(s)+\frac{\kappa_{f}}{c_{1,2}}Y^{(2)}(s)-\frac{c_{1,1}}{2c_{1,2}},\\\
K^{g}(s)&=-\frac{\kappa_{g}}{c_{2,2}}Y^{(1)}(s)-\frac{c_{2,1}}{2c_{2,2}},\\\
\alpha(s)&=-2\eta Y^{(2)}(s)-\eta h,\\\ \beta(s)&=-2\nu
Y^{(2)}(s)-\nu\overline{\omega}(s).\end{cases}$ (22)

Conversely, suppose
$(X,v,Y,Z,Z_{0})\in\mathbb{S}^{2}((\mathcal{F}_{t});\mathbb{R}^{2})\times\mathbb{H}^{2}((\mathcal{F}_{t});\mathbb{R}^{4})\times\mathbb{S}^{2}((\mathcal{F}_{t});\mathbb{R}^{2})\times\mathbb{H}^{2}((\mathcal{F}_{t});\mathbb{R}^{2\times
3})\times\mathbb{H}^{2}((\mathcal{F}_{t});\mathbb{R}^{2\times 2})$ is a
solution to the forward-backward system (14), (21) and coupling condition
(22). Then $v$ is the optimal control minimizing
$\mathcal{J}^{NE}(v;\overline{K})$, and $X(s)$ is the optimal trajectory. In
particular, $v$ is a mean field Nash equilibrium.

Before proceeding, let us comment on the coupling condition in Equation (22),
which applies also to the coupling condition in Equation (25) as well as when
$\overline{\omega}_{t}$ is replaced by the corresponding expression for the
endogenous price (see Section 6). First, from the third equation in Equation
(25) we obtain that

$\displaystyle-2Y^{(2)}(s)=\frac{\alpha(s)}{\eta}+h,$

i.e., $-2Y^{(2)}(s)$ is equal to the marginal abatement cost
$C^{{}^{\prime}}(\alpha(t))$; see Equation (6). Plugging it into the last
equation in Equation (25), leads us to

$\beta(s)=\nu\left(\frac{\alpha(s)}{\eta}+h-\overline{\omega}_{t}\right).$

Whence, the firm buys (resp. sells) if its marginal abatement cost is higher
(resp. lower) than the market price, in agreement with the economic intuition.
Similarly, from the second equation in Equation (6) we obtain that

$-Y^{(1)}(s)=\frac{1}{2\kappa_{g}}\left(c_{2,1}+2c_{2,2}K^{g}(s)\right),$

i.e., $-Y^{(1)}(s)$ is proportional to the marginal level of green capital
cost $(C^{g})^{{}^{\prime}}(K^{g})$; see Equation (2). Now plugging the
previous term into the expression for $K^{f}(s)$, leads us to:

$\displaystyle
K^{f}(s)=\frac{1}{2c_{1,2}}\left(\frac{\kappa_{f}}{k_{g}}(C^{g})^{{}^{\prime}}(K^{g})-\kappa_{f}C^{{}^{\prime}}(\alpha(t))-c_{1,1}\right)$

Whence, the firms decide the fossil fuel level of capital $K^{f}(s)$ by
roughly (see the subsequent discussion) comparing the marginal level of green
capital cost $(C^{g})^{{}^{\prime}}(K^{g})$ with the sum of the marginal
abatement cost $C^{{}^{\prime}}(\alpha(t))$ and the baseline cost $c_{1,1}$,
again in line with economic intuition.

##  5 A mean field control approximation with common noise for the $N$ player
game.

Interestingly, the MFG in Equations (14)-(15) is equivalent to a777See Remark
5.2. mean field type control problem (see [26], Proposition 3.3 and Corollary
3.4), whose general formulation and resolution is given in Appendix B. In our
case, the state dynamics is as in Equation (14) with the associated matrices
as in Equations (16)-(18), and the objective functional
$\mathcal{J}^{LQ}_{x,t}$ is given by:

$\begin{split}\mathcal{J}^{LQ}(v;\overline{K})&=\mathbb{E}\Bigg{[}\int_{0}^{T}-p^{\overline{K}}(t)\overline{K}(t)+\beta(t)\overline{\omega}_{t}+\frac{1}{2\nu}(\beta(t))^{2}+C(\alpha(t))\\\
&+C^{f}(K^{f}(t))+C^{g}(K^{g}(t))\,dt+\lambda(\widetilde{X}(T))^{2}\Bigg{]}.\end{split}$
(23)

Whence, the matrices characterizing the cost functional are in Equations
(19)–(20). In particular, the following proposition, analogous to Proposition
4.2, holds true (see Proposition 2.4 in [26]).

######  Proposition 5.1.

Suppose $v$ is an optimal control minimizing the objective functional
$\mathcal{J}^{LQ}(v;\overline{K})$ in Equation (23) with corresponding
trajectory $X(s)=(K(s),\widetilde{X}(s))$ solution of Equation (14) with
control $v$. Then there exists a unique solution
$(Y,Z,Z_{0})\in\mathbb{S}^{2}((\mathcal{F}_{t});\mathbb{R}^{2})\times\mathbb{H}^{2}((\mathcal{F}_{t});\mathbb{R}^{2\times
3})\times\mathbb{H}^{2}((\mathcal{F}_{t});\mathbb{R}^{2\times 2})$ of the
following BSDE, with $s\in[0,T]$

$\begin{cases}\begin{split}dY^{(1)}(s)=&-\left({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-\delta
Y^{(1)}(s)\,ds}+\sigma
Z_{1}^{(1)}(s)+bK(s)+\frac{b\gamma}{2}\overline{K}(s)-\frac{a}{2}\right)\,ds\\\
&+\sum_{j=1}^{3}Z_{j}^{(1)}dW^{j}(s)+\sum_{j=1}^{2}Z_{0,j}^{(1)}dW^{0,j}(s),\quad
Y^{(1)}(T)=0;\\\
dY^{(2)}(s)=&\sum_{j=1}^{3}Z_{j}^{(2)}dW^{j}(s)+\sum_{j=1}^{2}Z_{0,j}^{(2)}dW^{0,j}(s),\quad
Y^{(2)}(T)=\lambda(\widetilde{X}(T))^{2}\end{split}\end{cases}$ (24)

satisfying the coupling condition, with $s\in[0,T]$, a.s.,

$\begin{cases}K^{f}(s)&=-\frac{\kappa_{f}}{c_{1,2}}Y^{(1)}(s)+\frac{\kappa_{f}}{c_{1,2}}Y^{(2)}(s)-\frac{c_{11}}{2c_{1,2}},\\\
K^{g}(s)&=-\frac{\kappa_{g}}{c_{2,2}}Y^{(1)}(s)-\frac{c_{2,1}}{2c_{2,2}},\\\
\alpha(s)&=-2\eta Y^{(2)}(s)-\eta h,\\\ \beta(s)&=-2\nu
Y^{(2)}(s)-\nu\overline{\omega}(s).\end{cases}$ (25)

Conversely, suppose
$(X,v,Y,Z,Z_{0})\in\mathbb{S}^{2}((\mathcal{F}_{t});\mathbb{R}^{2})\times\mathbb{H}^{2}((\mathcal{F}_{t});\mathbb{R}^{4})\times\mathbb{S}^{2}((\mathcal{F}_{t});\mathbb{R}^{2})\times\mathbb{H}^{2}((\mathcal{F}_{t});\mathbb{R}^{2\times
3})\times\mathbb{H}^{2}((\mathcal{F}_{t});\mathbb{R}^{2\times 2})$ is a
solution to the forward-backward system (14), (24) and coupling condition
(25). Then $v$ is the optimal control minimizing
$\mathcal{J}^{LQ}(v;\overline{K})$, and $X(s)$ is the optimal trajectory.

###### Proof.

The proof follows the steps of Theorem 1.59 in [21]. First, let us write the
MFC problem in Equations (14) and (23) in matrix notation as in Appendix B;
matrices are given in (14)-(15) and (16)-(18).

$\begin{split}\mathcal{J}^{LQ}_{t,x}(v;\overline{X})&=\mathbb{E}\Bigg{[}\int_{t}^{T}\langle
QX(s),X(s)\rangle+\langle\overline{Q}\,\overline{X}(s),\overline{X}(s)\rangle+\langle
R\,v(s),v(s)\rangle\\\ &+2\langle q,X(s)\rangle+\langle
r(s),v(s)\rangle\,ds+\langle H,X(T),X(T)\rangle\Bigg{]}.\end{split}$ (26)
$\begin{split}dX(s)&=(A_{0}(s)+Bv(s))\,ds{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+AX(s)\,ds}\\\
&+C_{0,2}\,dW^{2}(s)+C_{0,3}\,dW^{3}(s)+C_{1}X(s)\,dW^{1}(s)\\\
&+F_{0,1}\,dW^{0,1}(s)+F_{0,2}\,dW^{0,2}(s).\end{split}$ (27)

In the previous equations, as usual, $\bar{X}(s)$ denotes the conditional
expectation of $X(s)$ given $\overline{\mathcal{F}}_{s}^{0}$. Second, let
$(v,X)$ be the optimal pair for the MFC problem, $v^{h}$ the control
$v^{h}=v+h\tilde{v}$, and $X^{h}$ the trajectory associated to $v^{h}$. Then,
let us define the so-called variation process $(V(s))$ as the solution of the
following SDE:

$dV(s)={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}AV(s)\,ds}+Bv(s)\,ds+C_{1}V(s)dW^{1}(s),\quad
s\in[0,T]$

By repeating the computations in [20], Lemma 6.10, we have that the following
limit holds true

$\lim_{\epsilon\rightarrow
0}\mathbb{E}\left[\sup_{s\in[0,T]}\Big{|}\frac{X^{h}(s)-X(s)}{\epsilon}-V(s)\Big{|}^{2}\right]=0.$

We now observe that

$\begin{split}\mathcal{J}^{LQ}_{t,x}(v^{h});\overline{X})&=\mathbb{E}\Bigg{[}\int_{t}^{T}\Big{[}\langle
Q\,X^{h}(s),X^{h}(s)\rangle+\langle\overline{Q}\,\overline{X}^{h}(s),\overline{X}^{h}(s)\rangle+\langle
Rv^{h}(s),v^{h}(s)\rangle\\\ &\hskip 56.9055pt+2\langle
q,X^{h}(s)\rangle+2\langle r(s),v^{h}(s)\rangle\Big{]}\,ds+\langle
HX^{h}(T),X^{h}(T)\rangle\Bigg{]},\end{split}$

from which the Gateaux derivative of
$\mathcal{J}^{LQ}_{t,x}(v^{h};\overline{X})$ in the direction $h$ reads as

$\begin{split}&\frac{d}{dh}\mathcal{J}^{LQ}_{t,x}(v^{h};\overline{X})\\\
&=2\mathbb{E}\Bigg{[}\int_{t}^{T}\langle
QX^{h},V\rangle+\langle\bar{Q}\bar{X}^{h},\bar{V}\rangle+\langle
Rv^{h},\tilde{v}\rangle+\langle q,V\rangle+\langle
r,\tilde{v}\rangle\,ds+\langle HX^{h}_{T},V_{T}\rangle\Bigg{]}.\end{split}$

Whence, the optimal condition for $(v,X)$

$\frac{d}{dh}\mathcal{J}^{LQ}_{t,x}(v^{h};\overline{X})\Big{|}_{h=0}=0$

is

$\mathbb{E}\Bigg{[}\int_{t}^{T}\langle
QX^{h},V\rangle+\langle\bar{Q}\bar{X}^{h},\bar{V}\rangle+\langle
Rv^{h},\tilde{v}\rangle+\langle q,V\rangle+\langle
r,\tilde{v}\rangle\,ds+\langle HX^{h}_{T},V_{T}\rangle\Bigg{]}=0.$ (28)

By [10, 11], we know that the BSDE in Equation (24) admits a unique solution.
Then, by applying Ito’s formula to the process $(\langle Y(t),V(t)\rangle)$,

$\begin{split}\langle
Y(t),V(t)\rangle&=\int_{t}^{T}-\left({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\langle
A^{T}Y(s),V(s)\rangle}+\langle C_{1}^{T}Z_{1}(s),V(s)\rangle+\langle
QX(s),V(s)\rangle\right)\,ds\\\
&-\int_{t}^{T}\langle\left(\overline{Q}\,\overline{X}(s),V(s)\rangle+\langle
q,V(s)\rangle\right)\,ds+\int_{t}^{T}\langle
B^{T}Y(s),\tilde{v}(s)\rangle+\langle C_{1}^{T}Z_{1}(s),V(s)\rangle\,ds\\\
&+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\int_{t}^{T}\langle
AV(s),Y(s)\rangle\,ds}+\text{Martingale}.\\\ \end{split}$

By taking the expectation on both sides, by recalling that
$\mathbb{E}\left[\langle
Q\overline{X}(s),V(s)\rangle\right]=\mathbb{E}\left[\langle
Q\overline{X}(s),\overline{V}(s)\rangle\right]$ and $Y(T)=HX(T)$, we have

$\begin{split}\mathbb{E}\Bigg{[}&\int_{t}^{T}\left(\langle
QX(s),V(s)\rangle+\langle\overline{Q}\overline{X}(s),V(s)\rangle+\langle
q,V(s)\rangle\right)-\langle B^{T}Y(s),\tilde{v}(s)\rangle\,ds\\\ &+\langle
HX(T),V(T)\rangle\Bigg{]}=0.\end{split}$

By using the optimally condition in Equation (28) and the arbitrariness of
$\tilde{v}$ we obtain the coupling condition (25).
To prove the converse, it is sufficient to observe that the functional
$\mathcal{J}^{LQ}_{t,x}(v;\overline{X})$ is strictly convex (see, again, [26],
Lemma 2.2, or [42], Proposition 2.7). Then, given a solution $(X,v,Y,Z,Z_{0})$
to the forward-backward system (14), (24), we know that the Gateaux derivative
of $\mathcal{J}^{LQ}_{t,x}(v;\overline{X})$ at $v$ is zero, which means that
$v$ is a minimizer.

∎

Before proceeding, we make the following remark.

######  Remark 5.2.

The MFC problem in Equations (14) and (23) is not the MFC problem version of
(14) and (15). Indeed, the latter would have the following objective
functional

$\begin{split}\mathcal{J}^{LQ}_{t,x}(v;\overline{X})&=\mathbb{E}\Bigg{[}\int_{t}^{T}\langle
QX(s),X(s)\rangle+\boldsymbol{2}\langle\overline{Q}\,\overline{X}(s),\overline{X}(s)\rangle+\langle
R\,v(s),v(s)\rangle\\\ &+2\langle q,X(s)\rangle+\langle
r(s),v(s)\rangle\,ds+\langle H,X(T),X(T)\rangle\Bigg{]}.\end{split}$ (29)

Nicely, by using the derivations in Appendix B, we can explicitly write down
the solution of problem (14)–(23) in terms of the following system of Riccati
equations, where matrices $C_{1},Q,B,R,H$ and vectors $r(t),q,A_{0}(t)$ are
defined in (16), (17), (18), (19), (20).

$\begin{cases}\overset{\cdot}{P}(t)+C_{1}^{T}P(t)C_{1}+Q-P(t)BR^{-1}B^{T}P(t){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+P(t)A+A^{T}P(t)}=0;\\\
P(T)=H.\\\ \end{cases}$ (30)
$\begin{cases}\overset{\cdot}{\Pi}(t)+C_{1}^{T}P(t)C_{1}+(Q+\bar{Q})-\Pi(t)BR^{-1}B^{T}\Pi(t){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\Pi(t)A+A^{T}\Pi(t)}=0;\\\
\Pi(T)=H.\\\ \end{cases}$ (31)
$\begin{cases}\overset{\cdot}{\phi}(t)-\Pi(t)BR^{-1}r(t)+q+\Pi(t)A_{0}(t)-\Pi(t)BR^{-1}B^{T}\phi(t){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+A^{T}\phi(t)}=0;\\\
\phi(T)=0.\end{cases}$ (32)

In particular, Theorem 2.6 in [26] ensures that the previous Equations
(31)–(32) admit a unique solution, where $P$, $\Pi$ are two deterministic
processes in $\mathcal{S}^{2}$, whereas $\phi$ is a deterministic process in
$\mathbb{R}^{2}$. In addition, both the optimal control and the associated
optimal trajectory for the problem (14) and (26) are expressed in terms of the
solutions $P$, $\Pi$ and $\phi$, $t\in[0,T]$:

$v(t)=-R^{-1}B^{T}P(t)(X(t)-\overline{X}(t))-R^{-1}(B^{T}\Pi(t)\overline{X}(t)+r(t)+B^{T}\phi(t)).$
(33)
$\begin{split}dX(t)&=(A_{0}(s)-BR^{-1}B^{T}P(t)(X(t)-\overline{X}(t)))\,ds\\\
&-BR^{-1}(B^{T}\Pi(t)\overline{X}(t)+B^{T}\phi(t)+r(t)))\,ds\\\
&+C_{1}X(s)\,dW^{1}(s)+C_{0,2}dW^{2}(s)+C_{0,3}dW^{3}(s)\\\
&+F_{0,1}dW^{0,1}(s)+F_{0,2}dW^{0,2}(s)\end{split}$ (34)

Proposition 5.1 ensures that there exists a unique adapted solution of the
forward-backward system (14), (24), and therefore such a solution coincides
with the solution constructed with the Riccati Equations (31)–(32).
We make now the following observation regarding the so-called Price of Anarchy
(PoA, henceforth).

######  Observation 5.3 (PoA).

The objective functional $\mathcal{J}^{NE}(v;\overline{K})$ (Equation (15))
and $\mathcal{J}^{LQ}(v;\overline{K})$ (Equation (23)) are not precisely the
same, even though $v$ solves both a fixed point Nash equilibrium and a mean
field type control problem. The difference between
$\mathcal{J}^{NE}(v;\overline{K})$ and $\mathcal{J}^{LQ}(v;\overline{K})$ is
called PoA since it represents the added aggregate cost of allowing all
players to choose their optimal strategy independently.

$\begin{split}\mathcal{J}^{NE}(v;\overline{K})-\mathcal{J}^{LQ}(v;\overline{K})&=b\frac{\gamma}{2}\mathbb{E}\left[\int_{t}^{T}(\mathbb{E}[K(s)|\mathcal{\overline{F}}^{0}_{s}])^{2}\,ds\right]\end{split}$

It is strictly positive as soon as the conditional expectation of the
equilibrium capital level is different from zero, and strictly increasing in
$\gamma$, reflecting the fact that greater competition yields a greater cost
of non-cooperation.

##  6 Market clearing condition and equilibrium price.

We start this section with the definition of market equilibrium for the finite
player game.

######  Definition 6.1.

For the $N$ player game a market equilibrium is, for every $t\in[0,T]$, a
$N$-dimensional vector
$\mathbf{\beta}^{\star,N}(t)=(\beta^{\star,1}(t),\ldots,\beta^{\star,N}(t))$
such that: (1) each $\beta^{\star,i}(t)\in\mathcal{H}_{1}^{N}$; (2)
$\beta^{\star,i}(t)$ is the $i^{th}$ component of the $\epsilon$-Nash
equilibrum for the $N$-player game (see Definition 3.2), and (3) the
asymptotic market clearing condition
$\lim_{N\rightarrow\infty}\sum_{i=1}^{N}\beta^{\star,i}(t)=0$ holds true.

At this point, we observe that because our MFG is equivalent to an optimal
control problem, the mean field Nash equilibrium is an $\epsilon$-Nash
equilibrium for the $N$-player game, in the sense of Definition 3.2; see
Theorem 3.6 in [26]. By using Proposition 5.1, the optimal trading rate
$\beta^{\star,i}(t)$ of each firm is given by:

$\beta^{\star,i}(t)=-2\nu Y^{(2),i}(t)-\nu\overline{\omega}(t),\quad
t\in[0,T].$

Because we model the trading mechanism as part of the firms’ decision problem,
the equilibrium (market-clearing) price of emission allowances emerges
endogenously. In the present situation, Equation (35) is equivalent to the
following condition:

$\frac{1}{N}\sum_{i=1}^{N}\beta^{\star,i}(t)=\frac{1}{N}\sum_{i=1}^{N}\left(-2\nu
Y^{(2),i}(t)-\nu\overline{\omega}(t)\right)=0,$ (35)

from which we have that

$\overline{\omega}(t)=-\frac{2}{N}\sum_{i=1}^{N}Y^{(2),i}(t).$ (36)

The previous solution is of course inconsistent with our standing assumption
that the price process $(\overline{\omega}_{t})$ is a
$(\overline{\mathcal{F}}^{0})$-adapted process. However, we can argue as in
[23], Page 267, and expect that in the large-$N$ limit, the market price of
allowances may be given by
$\overline{\omega}_{t}=-2\mathbb{E}[Y^{(2)}(t)|\overline{\mathcal{F}}^{0}_{t}]$.
Therefore, we need to consider a different BSDE system with respect to the one
in Proposition 5.1. More precisely, the coupling condition for $\beta(s)$ in
Equation (25) is now given by:

$\beta(s)=-2\nu
Y^{(2)}(s)+2\nu\mathbb{E}[Y^{(2)}(s)|\overline{\mathcal{F}}_{t}^{0}],$

which leads to the following matrix-based representation of the coupling
condition

$v(t)=-R^{-1}(B^{T}\,Y(t)+\tilde{r}+D\,\overline{Y}(s)),\quad
t\in[0,T],\,\,a.s.$ (37)

where

$\tilde{r}=\begin{bmatrix}\frac{c_{1,1}}{2}\\\ \frac{c_{2,1}}{2}\\\
\frac{h}{2}\\\ 0\end{bmatrix},\quad D=\begin{bmatrix}0&0\\\ 0&0\\\ 0&0\\\
0&-\frac{1}{2}\end{bmatrix}.$ (38)

Equations (14) and (24), instead, remain unchanged; notice that now $v(t)$ is
as in Equation (37). Their matrix-based representation is given by the
following equations:

$\begin{split}dX(s)&=(A_{0}(s){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+AX(s)}+Bv(s))\,ds\\\
&+C_{0,2}\,dW^{2}(s)+C_{0,3}\,dW^{3}(s)+C_{1}X(s)\,dW^{1}(s)\\\
&+F_{0,1}\,dW^{0,1}(s)+F_{0,2}\,dW^{0,2}(s),\quad X(0)=x_{0}.\end{split}$ (39)
$\begin{split}dY(s)&=-({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}A^{T}Y(s)}+C_{1}^{T}Z_{1}(s)+QX(s)+\overline{Q}\,\overline{X}(s)+q)\,ds\\\
&+\sum_{j=1}^{3}Z_{j}(s)dW^{j}(s)+\sum_{j=1}^{2}Z_{0,j}(s)dW^{0,j}(s),\quad
Y(T)=HX(T).\end{split}$ (40)

We now state and prove the following short-term existence result.

######  Theorem 6.2.

There exists some constant $\tau>0$ which depends only on the matrices
$A_{0}(s),B,C_{0,2},C_{0,3},C_{1},F_{0,1},F_{0,2},Q,\overline{Q},q$ such that
for any $T\leq\tau$, there exists a unique strong solution
$(X,Y,Z,Z_{0})\in\mathbb{S}^{2}((\mathcal{F}_{t});\mathbb{R}^{2})\times\mathbb{S}^{2}((\mathcal{F}_{t});\mathbb{R}^{2})\times\mathbb{H}^{2}((\mathcal{F}_{t});\mathbb{R}^{2\times
3})\times\mathbb{H}^{2}((\mathcal{F}_{t});\mathbb{R}^{2\times 2})$ to the
FBSDE (39)–(40).

###### Proof.

The proof is an adaption of the arguments used in [20], Theorem 4.24. The main
difference is that there exists a term involving
$\mathbb{E}[Y^{(2)}(s)|\overline{\mathcal{F}}_{t}^{0}]$ via the coupling
condition in (37). Let $\Phi$ be the map constructed in the following way. For
any element
$(X,Y)\in\mathbb{S}^{2}((\overline{\mathcal{F}}_{t});\mathbb{R}^{2})$, let
$(Y,Z)$ be the solution of the following BSDE, where $s\in(0,T]$:

$\begin{cases}dY(s)=-\left({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}AY(s)}+C_{1}^{T}Z_{1}(s)+QX(s)+\overline{Q}\,\overline{X}(s)+q\right)\,ds\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\sum_{j=1}^{3}Z_{j}(s)dW^{j}(s)+\sum_{j=1}^{2}Z_{0,j}(s)dW^{0,j}(s)\\\
Y(T)=HX(T)\end{cases}$ (41)

Notice that $X\in\mathbb{S}^{2}((\mathcal{F}_{t});\mathbb{R}^{2})$ and the
pair
$(Y,Z)\in\mathbb{S}^{2}((\mathcal{F}_{t});\mathbb{R}^{2})\times\mathbb{H}^{2}((\overline{\mathcal{F}}_{t});\mathbb{R}^{2\times
3})$ are progressively measurable with respect the completion of the
filtration generated by $(W(s)-W(t))_{s\in[t,T]}$ and
$(W^{0}(s)-W^{0}(t))_{s\in[t,T]}$. Then, we associate to the couple $(Y,Z)$
the solution $(\widetilde{X}(s))$ of the following SDE, where $s\in(0,T]$

$\begin{split}d\widetilde{X}(s)&=\left(A_{0}(s){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+A\widetilde{X}(s)}-BR^{-1}(B^{T}Y(s)+\tilde{r}+D\overline{Y}(s)\right)\,ds\\\
&+C_{0,2}dW^{2}(s)+C_{0,3}dW^{3}(s)+C_{1}\widetilde{X}(s)dW^{1}(s)\\\
&+F_{0,1}dW^{0,1}(s)+F_{0,2}dW^{0,2}(s),\quad\quad\widetilde{X}(0)=x_{0}.\end{split}$
(42)

The map $\Phi$ is given by $\Phi:X\rightarrow(Y,Z)\rightarrow\widetilde{X}$.
The aim is to show that $\Phi$ is a contraction for small $T$. To this end,
let $X^{1}$ and $X^{2}\in\mathbb{S}^{2}((\mathcal{F}_{t});\mathbb{R}^{2})$ and
denote by $(Y^{1},Z^{1})$ and $(Y^{2},Z^{2})$ be the associated solution of
the BSDE in (41). In addition, set $\widetilde{X}^{1}=\Phi(X^{1})$ and
$\widetilde{X}^{2}=\Phi(X^{2})$. The fact that $\Phi$ is a contraction follows
from the following standard estimates for SDEs and BSDEs:

$\begin{split}&\mathbb{E}\left[\sup_{s\in[0,T]}\Big{|}\widetilde{X}^{1}(s)-\widetilde{X}^{2}(s)\Big{|}\right]+\mathbb{E}\left[\sup_{s\in[0,T]}\Big{|}\overline{\widetilde{X}^{1}}(s)-\overline{\widetilde{X}^{2}}(s)\Big{|}\right]\\\
&\leq
CT\mathbb{E}\Bigg{[}\sup_{s\in[0,T]}\Big{|}Y^{1}(s)-Y^{2}(s)\Big{|}^{2}+\sup_{s\in[0,T]}\Big{|}\overline{Y^{1}}(s)-\overline{Y^{2}}(s)\Big{|}^{2}\\\
&+\sum_{j=1}^{3}\int_{0}^{T}\Big{|}Z_{j}^{1}(s)-Z_{j}^{2}(s)\Big{|}\,ds+\sum_{j=1}^{2}\int_{0}^{T}\Big{|}Z_{0,j}^{1}(s)-Z_{0,j}^{2}(s)\Big{|}\,ds\Bigg{]}\\\
&\leq
CT\left(\mathbb{E}\Bigg{[}\sup_{s\in[0,T]}\Big{|}X^{1}(s)-X^{2}(s)\Big{|}^{2}\Bigg{]}+\mathbb{E}\Bigg{[}\sup_{s\in[0,T]}\Big{|}\overline{X^{1}}(s)-\overline{X^{2}}(s)\Big{|}^{2}\Bigg{]}\right)\end{split}$

∎

Before proceeding, we make the following observation. Should the solution of
the FBSDE (39)–(40) be linked to that of some MFC problem, a term of the form
$D\,\bar{v}(s)$ would be present in the state dynamics because of the presence
of a term like $D\mathbb{E}[Y(t)|\overline{\mathcal{F}}_{t}^{0}]$ in the
coupling condition. However, this is not the case for our FBSDE.
The next theorem gives us the unique existence of solutions to the FBSDE
(39)–(40) for general $T$.

######  Theorem 6.3.

Under the assumption that

$\left(\frac{\kappa_{f}^{2}}{c_{1,2}}+\frac{\kappa_{g}^{2}}{c_{2,2}}-\frac{\kappa_{f}\kappa_{e}}{c_{1,2}}\right)>0,\quad\left(2\eta+\nu+\frac{\kappa_{e}^{2}}{c_{1,2}}-\frac{\kappa_{f}\kappa_{e}}{c_{1,2}}\right)>0,$

there exists a unique strong solution
$(X,Y,Z,Z_{0})\in\mathbb{S}^{2}((\mathcal{F}_{t});\mathbb{R}^{2})\times\mathbb{S}^{2}((\mathcal{F}_{t});\mathbb{R}^{2})\times\mathbb{H}^{2}((\mathcal{F}_{t});\mathbb{R}^{2\times
3})\times\mathbb{H}^{2}((\mathcal{F}_{t});\mathbb{R}^{2\times 2})$ to the
FBSDE (39)–(40).

###### Proof.

The proof hinges on the continuation method of [36] and reduces to verify that
assumption (H2.1) in the previous paper holds true for our system, in
expectation. Notice that in our case their $2\times 2$ full-rank matrix $G$ is
the identity matrix. In order to facilitate the comparison, we rewrite the
FBSDE (39)–(40) in terms of the following functional
$b,f,\sigma,\sigma_{0},\Phi$ and of a vector $\theta$ where all the static
parameters are collected

$\begin{split}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}b(s,Y(s),X(s),\tilde{a}(s),\theta):=\left(A_{0}(s)+AX(s)-BR^{-1}(B^{T}Y(s)+\tilde{r}+D\overline{Y}(s)\right)}\\\
&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}f(s,Z_{1}(s),Y(s),X(s),\theta):=\left(A^{T}Y(s)+C_{1}^{T}Z_{1}(s)+QX(s)+\overline{Q}\overline{X}(s)+q\right)}\\\
&\Phi(X(T)):=HX(T),\\\ &\sigma(X^{(1)}(s),\theta):=\begin{bmatrix}\sigma
X^{(1)}(s)&0&0\\\ 0&-\sigma_{1}\sqrt{1-\rho^{2}}&-\sigma_{2}\end{bmatrix}\\\
&\sigma_{0}(\theta):=\begin{bmatrix}0&0\\\
-\sigma_{1}\rho&\tilde{\sigma}_{2}\\\ \end{bmatrix}\\\
&\theta=(\kappa_{f},\kappa_{g},c_{1,2},c_{2,2},\eta,\nu,c_{1,1},c_{2,1},h,\sigma_{1},\rho,\sigma_{2},\sigma,\tilde{\sigma}_{2},b,\gamma,a,\lambda)\end{split}$

as

$\begin{split}dX(s)&={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}b(s,Y(s),X(s),\tilde{a}(s),\theta)}\,ds+\sigma(X^{(1)}(s),\theta)dW(s)+\sigma_{0}(\theta)dW^{0}(s).\\\
dY(s)&={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-f(s,Z_{1}(s),Y(s),X(s),\theta)}\,ds+Z(s)dW(s)+Z_{0}(s)dW^{0}(s),\end{split}$

where $X(0)=x_{0}$ and $Y(T)=\Phi(X(T))$. We use the following notation:

$u=\begin{bmatrix}x\\\ y\\\ z\\\ z_{0}\end{bmatrix},\quad\quad
A(s,u)=\begin{bmatrix}-f\\\ b\\\ \sigma\\\ \sigma_{0}\\\ \end{bmatrix}(s,u).$

Besides, for all pairs
$(x,y,z,z_{0}),(x^{{}^{\prime}},y^{{}^{\prime}},z^{{}^{\prime}},z^{{}^{\prime}}_{0})\in\mathbb{L}^{2}(\mathcal{F};\mathbb{R}^{2}\times\mathbb{R}^{2}\times\mathbb{R}^{2\times
3}\times\mathbb{R}^{2\times 2})$, we denote by $\hat{x}=x-x^{{}^{\prime}}$,
$\hat{y}=y-y^{{}^{\prime}}$, $\hat{z}=z-z^{{}^{\prime}}$, and
$\hat{z}_{0}=z_{0}-z_{0}^{{}^{\prime}}$. We have:

$\begin{split}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\langle-f(s,z_{1},y,x,\theta)-(-f(s,z_{1}^{{}^{\prime}},y^{\prime},x^{{}^{\prime}},\theta)),x-x^{{}^{\prime}}\rangle}\\\
&=-\sigma\hat{z}_{1}^{(1)}\hat{x}^{(1)}-b(1-\gamma)A_{k}^{2}(\hat{x}^{(1)})^{2}-\frac{b\gamma
A_{k}^{2}}{2}\hat{\overline{x}}^{(1)}\hat{x}^{(1)}+\delta\hat{y}^{(1)}\hat{x}^{(1)}\end{split}$
(43) $\begin{split}&\langle
b(s,y,\tilde{a},\theta)-b(s,y^{{}^{\prime}},\tilde{a},\theta),y-y^{{}^{\prime}}\rangle\\\
&=-\left(\frac{\kappa_{f}^{2}}{c_{1,2}}+\frac{\kappa_{g}^{2}}{c_{2,2}}\right)(\hat{y}^{(1)})^{2}+2\frac{\kappa_{f}\kappa_{e}}{c_{1,2}}\hat{y}^{(1)}\hat{y}^{(2)}-\left(2(\eta+\nu)+\frac{\kappa_{e}^{2}}{c_{1,2}}\right)(\hat{y}^{(2)})^{2}+\nu\hat{\overline{y}}^{(2)}\hat{y}^{(2)}-\delta\hat{x}^{(1)}\hat{y}^{(1)}\end{split}$
(44)

and

$\langle\sigma(x,\theta)-\sigma(x^{\prime},\theta),z\rangle=\sigma\hat{x}^{(1)}\hat{z}_{1}^{(1)};$
(45)

notice that $\sigma_{0}$ does not lead to any contribution since it is state-
independent. By combining (43), (44), (45), we get

$\begin{split}&\langle A(s,u)-A(s,u^{\prime}),u-u^{\prime}\rangle=\\\
&-b(1-\gamma)A_{k}^{2}(\hat{x}^{(1)})^{2}-\frac{b\gamma
A_{k}^{2}}{2}\hat{\overline{x}}^{(1)}\hat{x}^{(1)}-\left(\frac{\kappa_{f}^{2}}{c_{1,2}}+\frac{\kappa_{g}^{2}}{c_{2,2}}\right)(\hat{y}^{(1)})^{2}+2\frac{\kappa_{f}\kappa_{e}}{c_{1,2}}\hat{y}^{(1)}\hat{y}^{(2)}\\\
&-\left(2(\eta+\nu)+\frac{\kappa_{e}^{2}}{c_{1,2}}\right)(\hat{y}^{(2)})^{2}+\nu\hat{\overline{y}}^{(2)}\hat{y}^{(2)}.\end{split}$

Now, by taking the expectation, by the law of total expectation and Jensen
inequality:

$\displaystyle\mathbb{E}\left[\langle
A(s,u)-A(s,u^{\prime}),u-u^{\prime}\rangle\right]\leq-b(1-\gamma)A_{k}^{2}\mathbb{E}[(\hat{x}^{(1)})^{2}]-\frac{b\gamma
A_{k}^{2}}{2}\mathbb{E}[(\hat{x}^{(1)})^{2}]$
$\displaystyle-\left(\frac{\kappa_{f}^{2}}{c_{1,2}}+\frac{\kappa_{g}^{2}}{c_{2,2}}-\frac{\kappa_{f}\kappa_{e}}{c_{1,2}}\right)\mathbb{E}[(\hat{y}^{(1)})^{2}]$
$\displaystyle\quad-\left(2\eta+\nu+\frac{\kappa_{e}^{2}}{c_{1,2}}-\frac{\kappa_{f}\kappa_{e}}{c_{1,2}}\right)\mathbb{E}[(\hat{y}^{(2)})^{2}]$
$\displaystyle\leq$
$\displaystyle-\left(\frac{\kappa_{f}^{2}}{c_{1,2}}+\frac{\kappa_{g}^{2}}{c_{2,2}}-\frac{\kappa_{f}\kappa_{e}}{c_{1,2}}\right)\mathbb{E}[(\hat{y}^{(1)})^{2}]$
$\displaystyle\quad-\left(2\eta+\nu+\frac{\kappa_{e}^{2}}{c_{1,2}}-\frac{\kappa_{f}\kappa_{e}}{c_{1,2}}\right)\mathbb{E}[(\hat{y}^{(2)})^{2}]$
$\displaystyle\leq$
$\displaystyle-\min\left(\left(\frac{\kappa_{f}^{2}}{c_{1,2}}+\frac{\kappa_{g}^{2}}{c_{2,2}}-\frac{\kappa_{f}\kappa_{e}}{c_{1,2}}\right),\left(2\eta+\nu+\frac{\kappa_{e}^{2}}{c_{1,2}}-\frac{\kappa_{f}\kappa_{e}}{c_{1,2}}\right)\right)\mathbb{E}[|\hat{y}|^{2}]$

In addition, we have:

$\langle\Phi(x_{T})-\Phi(x_{T}^{{}^{\prime}}),\hat{x}\rangle=\lambda(\hat{x}^{(1)})^{2}\geq
0.$

In particular, the monotone conditions in (H2.3) in [36] hold with
$\beta_{1}=0$ and
$\beta_{2}=\min\left(\left(\frac{\kappa_{f}^{2}}{c_{1,2}}+\frac{\kappa_{g}^{2}}{c_{2,2}}-\frac{\kappa_{f}\kappa_{e}}{c_{1,2}}\right),\left(2\eta+\nu+\frac{\kappa_{e}^{2}}{c_{1,2}}-\frac{\kappa_{f}\kappa_{e}}{c_{1,2}}\right)\right)$.
Now, the continuation method hinges on the following steps. First, one has to
introduce a family of FBSDE indexed by a parameter $\varrho\in[0,1]$,

$\begin{split}&dx_{t}^{\varrho}=[-(1-\varrho)\beta_{2}(y_{t}^{\varrho})+\varrho
b(t,u_{t}^{\varrho},\tilde{a}(t),\theta)+\phi_{t}]\,dt\\\
&+[\varrho\sigma(t,u_{t})+\psi_{t}]dW(t)+\sigma_{0}(\theta)\,dW^{0}(t)\\\
&dy_{t}^{\varrho}=-[\varrho
f(t,u_{t}^{\varrho})+\gamma_{t}]\,dt+z_{t}^{\varrho}dW(t)+z_{0,t}^{\varrho}\,dW^{0}(t),\\\
&x_{0}^{\varrho}=x_{0},\quad
y_{T}^{\varrho}=\varrho\Phi(x_{T}^{\varrho})+(1-\varrho)x_{T}^{\varrho}+\xi,\end{split}$
(46)

where $\phi,\psi$ and $\gamma$ are given processes in
$\mathbb{H}^{2}((\mathcal{F}_{t});\mathbb{R}^{2})$ and
$\mathbb{H}^{2}((\mathcal{F}_{t});\mathbb{R}^{2\times 3})$, respectively, and
$\xi\in\mathbb{L}^{2}(\mathcal{F}_{T};\mathbb{R}^{2})$. Notice that the
previous system is constructed in such a way that for $\rho=0$ its solution is
straightforward, whereas for $\rho=1$ it implies the existence of a unique
strong solution to the FBSDE (39)–(40). In particular, the monotone conditions
above allow to extend the existence from $\varrho=0$ to $\varrho=1$. The proof
follows directly from the one of Theorem 2.2 in [36], and it is therefore
omitted. ∎

It is important to note that the existence of a unique strong solution to the
FBSDE (39)–(40) holds for every level of competition $\gamma\in[0,1]$.

######  Corollary 6.4.

The solution $(X,Y,Z^{0},Z)$ to the FBSDE (39)–(40) satisfies the following
estimate

$\mathbb{E}\left[\sup_{t\in[0,T]}|X(t)|^{2}+\sup_{t\in[0,T]}|Y(t)|^{2}+\sum_{j=1}^{3}\int_{0}^{T}|Z_{j}(t)|^{2}\,dt+\sum_{j=1}^{2}\int_{0}^{T}|Z_{0,j}(t)|^{2}\,dt\right]\leq
C,$

where $C$ is a constant depending only on $T$ and on the matrices of the
system $A_{0}(s)$, $B$, $C_{0,2}$,$C_{0,3}$, $C_{1}$, $F_{0,1}$, $F_{0,2}$,
$Q$, $\overline{Q}$, $q$.

###### Proof.

By applying Ito’s formula to $|Y(t)|^{2}$ we obtain:

$\begin{split}&\mathbb{E}[|Y(t)|^{2}]+\mathbb{E}\left[\sum_{j=1}^{3}\int_{t}^{T}|Z_{j}(s)|^{2}\,ds\right]+\mathbb{E}\left[\sum_{j=1}^{2}\int_{t}^{T}|Z_{0,j}(s)|^{2}\,ds\right]\\\
&\leq\mathbb{E}\left[H|X(T)|^{2}\right]+\epsilon\,\mathbb{E}\left[\int_{t}^{T}|Z_{1}(s)|^{2}\,ds\right]\\\
&+C\mathbb{E}\left[\int_{t}^{T}|Y(s)|^{2}\,ds+\int_{t}^{T}|X(s)|^{2}\,ds+\int_{t}^{T}|\overline{X}(s)|^{2}\,ds\right].\end{split}$
(47)

Then, by choosing $\epsilon>0$ small, say $\epsilon<1$, and applying
Gronwall’s inequality, we obtain

$\begin{split}&\mathbb{E}[|Y(t)|^{2}]+\mathbb{E}\left[\sum_{j=1}^{3}\int_{t}^{T}|Z_{j}(s)|^{2}\,ds\right]+\mathbb{E}\left[\sum_{j=1}^{2}\int_{t}^{T}|Z_{0,j}(s)|^{2}\,ds\right]\\\
&\leq\mathbb{E}\left[H|X(T)|^{2}\right]+C\mathbb{E}\left[\int_{t}^{T}|X(s)|^{2}\,ds+\int_{t}^{T}|\overline{X}(s)|^{2}\,ds\right].\end{split}$

Now, by using again Ito’s formula, a simple application of the Burkholder-
Davis-Gundy inequality, Young’s inequality, and the triangular inequality
gives, for new constants $C_{H}$, $C_{Q}$, $C_{\overline{Q}}$, $C>0$:

$\begin{split}&\mathbb{E}\left[\sup_{t\in[0,T]}|Y(t)|^{2}+\sum_{j=1}^{3}\int_{t}^{T}|Z_{j}(s)|^{2}\,ds+\sum_{j=1}^{2}\int_{t}^{T}|Z_{0,j}(s)|^{2}\,ds\right]\\\
&\leq C_{H}\mathbb{E}\left[|X(T)|^{2}+\int_{0}^{T}|Y(s)|^{2}\,ds\right]\\\
&+\mathbb{E}\left[\epsilon\int_{0}^{T}|Z(s)|^{2}\,ds+C_{Q}\int_{0}^{T}|X(s)|^{2}\,ds+C_{\overline{Q}}\int_{0}^{T}|\overline{X}(s)|^{2}\,ds\right]\\\
&+\epsilon_{1}\mathbb{E}\left[\sup_{t\in[0,T]}|Y(t)|^{2}\right]+C\mathbb{E}\left[\sum_{j=1}^{3}\int_{0}^{T}|Z_{j}(s)|^{2}\,ds\right]\\\
&+\epsilon_{2}\mathbb{E}\left[\sup_{t\in[0,T]}|Y(t)|^{2}\right]+C\mathbb{E}\left[\sum_{j=1}^{2}\int_{0}^{T}|Z_{0,j}(s)|^{2}\,ds\right]\\\
\end{split}$

Choosing now $\epsilon_{1},\epsilon_{2}$ such that
$\epsilon_{1}+\epsilon_{1}<1$ and using Equation (47), we get for a new
constant $C>0$:

$\begin{split}&\mathbb{E}\left[\sup_{t\in[0,T]}|Y(t)|^{2}+\sum_{j=1}^{3}\int_{t}^{T}|Z_{j}(s)|^{2}\,ds+\sum_{j=1}^{2}\int_{t}^{T}|Z_{0,j}(s)|^{2}\,ds\right]\\\
&\leq
C\mathbb{E}\left[|X(T)|^{2}+\int_{0}^{T}|Y(s)|^{2}\,ds+C\int_{0}^{T}|X(s)|^{2}\,ds+C\int_{0}^{T}|\overline{X}(s)|^{2}\,ds\right]\end{split}$
(48)

Analogous computations can be performed on $(\bar{Y}(t))$, which lead, for a
new constant $C>0,$ to:

$\begin{split}&\mathbb{E}\left[\sup_{t\in[0,T]}|\overline{Y}(t)|^{2}+\sum_{j=1}^{3}\int_{t}^{T}|\overline{Z}_{j}(s)|^{2}\,ds+\sum_{j=1}^{2}\int_{t}^{T}|\overline{Z}_{0,j}(s)|^{2}\,ds\right]\\\
&\leq
C\mathbb{E}\left[|X(T)|^{2}+\int_{0}^{T}|X(s)|^{2}\,ds+\int_{0}^{T}|\overline{X}(s)|^{2}\,ds\right]\\\
&+C\mathbb{E}\left[\int_{0}^{T}|X(s)|^{2}\,ds+\int_{0}^{T}|\overline{Y}(s)|^{2}\,ds\right]\end{split}$
(49)

By combining Equation (48) and (49), for a new constant $C_{\epsilon}>0$, we
get:

$\begin{split}&\mathbb{E}\left[\sup_{t\in[0,T]}|Y(t)|^{2}+\sup_{t\in[0,T]}|\overline{Y}(t)|^{2}+\overline{C}_{\epsilon}\sum_{j=1}^{3}\int_{0}^{T}|Z_{j}(s)|^{2}\,ds+\overline{C}_{\epsilon}\sum_{j=1}^{2}\int_{0}^{T}|Z_{0,j}(s)|^{2}\,ds\right]\\\
&\leq
C\mathbb{E}\left[|X(T)|^{2}+|\overline{X}(T)|^{2}\right]+C\mathbb{E}\left[\int_{0}^{T}|X(s)|^{2}\,ds+\int_{0}^{T}|\overline{X}(s)|^{2}\,ds\right]\\\
&+C\mathbb{E}\left[\int_{0}^{T}|Y(s)|^{2}\,ds+\int_{0}^{T}|\overline{Y}(s)|^{2}\,ds\right]\end{split}$
(50)

where $\overline{C}_{\epsilon}:=(1-C_{\epsilon})$. On the other hand, the
standard estimates for SDEs give, for a new constant $C>0$,

$\begin{split}&\mathbb{E}\left[\sup_{t\in[0,T]}|X(t)|^{2}\right]+\mathbb{E}\left[\sup_{t\in[0,T]}|\overline{X}(t)|^{2}\right]\\\
&\leq|X(0)|^{2}+\mathbb{E}\left[\int_{0}^{T}|Y(s)|^{2}\,ds+\int_{0}^{T}|\overline{Y}(s)|^{2}\,ds\right]+C.\end{split}$
(51)

Combining the inequalities (50) and (51) and a simple application of the
Burkholder-Davis-Gundy inequality establishes the claim. ∎

We are now ready to investigate if our FBSDE (39)–(40) actually provides an
approximation of the market price and if so, how accurate it is. In
particular, if we use
$(-2\mathbb{E}[Y^{(2)}(t)|\overline{\mathcal{F}}_{t}^{0}])$ as the input
$(\overline{\omega}_{t})$, where $(Y^{(2)}(t))$ is the unique solution to the
FSBDE (39)–(40), then by Theorem 3.6 in [26] the optimal strategy for the
individual firm is given by

$\beta^{\star,i}(t):=-2\nu
Y^{(2),i}(t)+2\nu\mathbb{E}[Y^{(2)}(t)|\overline{\mathcal{F}}_{t}^{0}],$ (52)

where $(Y^{(2),i})$ is the (second component of the) solution to (13) and (24)
with
$(\overline{\omega}_{t}=-2\mathbb{E}[Y^{(2)}(t)|\overline{\mathcal{F}}_{t}^{0}])$
and $W^{1}\equiv W^{1,i}$ and $W^{3}\equiv W^{3,i}$. The next theorem shows
that the market clearing condition in the large-$N$ limit, i.e.,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{i=1}^{N}\beta^{\star,i}(t)=0,\quad
dt\otimes d\mathbb{P}-a.s.$

holds.

######  Theorem 6.5.

Let $T>0$ and $(\beta^{\star,i}(t))$ be defined as in Equation (52). Then

$\lim_{N\rightarrow\infty}\mathbb{E}\left[\Bigg{|}\frac{1}{N}\sum_{i=1}^{N}\beta^{\star,i}(t)\Bigg{|}^{2}\right]=0.$
(53)

Moreover, there exists some constant $C$ independent of $N$ such that:

$\mathbb{E}\left[\Bigg{|}\frac{1}{N}\sum_{i=1}^{N}\beta^{\star,i}(t)\Bigg{|}^{2}\right]\leq\frac{C}{N}$
(54)

###### Proof.

The proof is similar to the one of Lemma 5.1 in [24]. Because
$(Y^{(2),i}(t))_{i=1}^{N}$ are $(\overline{\mathcal{F}}^{0}_{t})$
conditionally i.i.d., the ladder property of the conditional expectation
yields

$\begin{split}\mathbb{E}\left[\Bigg{|}\frac{1}{N}\sum_{i=1}^{N}\beta^{\star,i}(t)\Bigg{|}^{2}\right]&=\mathbb{E}\left[\Bigg{|}\frac{1}{N}\sum_{i=1}^{N}\left(-2\nu
Y^{(2),i}(t)+2\nu\mathbb{E}[Y^{(2)}(t)|\overline{\mathcal{F}}_{t}^{0}]\right)\Bigg{|}^{2}\right]\\\
&\leq\frac{4\nu^{2}}{N^{2}}\sum_{i=1}^{N}\mathbb{E}\left[\Big{|}Y^{(2),i}(t)-\mathbb{E}[Y^{(2)}(t)|\overline{\mathcal{F}}_{t}^{0}]\Big{|}^{2}\right]\end{split}$

∎

Since
$\sup_{t\in[0,T]}\mathbb{E}\left[\Big{|}Y^{(2),i}(t)-\mathbb{E}[Y^{(2)}(t)|\overline{\mathcal{F}}_{t}^{0}]\Big{|}^{2}\right]\leq
2\sup_{t\in[0,T]}\mathbb{E}\left[\Big{|}Y^{(2),1}(t)\Big{|}^{2}\right]$, the
conclusion follows from the estimates in Corollary 6.4. Finally, we conclude
this section by providing explicit solutions to the FSBDE (39)–(40) in terms
of the following system of Riccati equations; the proof of its derivation
follows the same line of arguments as in Appendix B and it is, therefore,
omitted. In particular, Theorem 6.2 guarantees that solutions to (39)–(40) are
uniquely determined in terms of these Riccati equations.

$\begin{cases}\overset{\cdot}{P}(t)+C_{1}^{T}P(t)C_{1}+Q-P(t)BR^{-1}B^{T}P(t){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+P(t)A+A^{T}P(t)}=0;\\\
P(T)=H.\\\ \end{cases}$ (55)
$\begin{cases}\overset{\cdot}{\Pi}(t)+C_{1}^{T}P(t)C_{1}+(Q+\bar{Q})-\Pi(t)BR^{-1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(B^{T}+D)}\Pi(t)+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\Pi(t)A+A^{T}\Pi(t)}=0;\\\
\Pi(T)=H.\\\ \end{cases}$ (56)
$\begin{cases}\overset{\cdot}{\phi}(t)-\Pi(t)BR^{-1}\tilde{r}+q+\Pi(t)A_{0}(t)-\Pi(t)BR^{-1}(B^{T}+D)\phi(t){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+A^{T}\phi(t)}=0\\\
\phi(T)=0.\end{cases}$ (57)

Matrices $C_{1},Q,\overline{Q},B,R,H,q,A_{0}(t)$ are as in Section 5, whereas
$\tilde{r}$ and $D$ are defined in Equation (38). In particular, $(X(t))$ and
solves the following equation:

$\begin{split}dX(t)&=(A_{0}(s)+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}AX(s)}-BR^{-1}B^{T}P(t)(X(t)-\overline{X}(t)))\,ds\\\
&-BR^{-1}((B^{T}+D)\Pi(t)\overline{X}(t)+(B^{T}+D)\phi(t)+\tilde{r})\,ds\\\
&+C_{1}X(s)\,dW^{1}(s)+C_{0,2}dW^{2}(s)+C_{0,3}dW^{3}(s)\\\
&+F_{0,1}dW^{0,1}(s)+F_{0,2}dW^{0,2}(s),\quad\quad X(0)=x_{0}.\end{split}$
(58)

In addition, the equilibrium price is given by

$\overline{\omega}_{t}=-2\left((\Pi(t))_{2,1}\overline{K}(t)+(\Pi(t))_{2,2}\overline{\widetilde{X}}(t)\right)-2(\phi(t))_{2},$
(59)

where $(\Pi(t))_{\ell,m}$ denotes the entry $(\ell,m)$ of the matrix $\Pi(t)$
and $(\phi(t))_{2}$ the second component of the vector $\phi(t)$, $t\in[0,T]$.

##  7 Numerical Illustration

We illustrate here the firm’s behavior in the policy scheme described in the
previous sections. We consider the objective of reducing carbon emission over
$T=5$ years. The solutions of the Riccati equations (55)–(57) are computed by
using the MATLAB numerical integrator ode45 with a temporal resolution of
$\Delta t=10^{-3}$, which is also employed to simulate the SDE (58) via the
Euler-Maruyama method. All the expectations below are computed via the
classical Monte Carlo method using $5\cdot 10^{3}$ trajectories. Table 2
reports the employed numerical value for the parameters along with a synthetic
yet exhaustive description in the caption.

Table 2: Numerical values for the parameters employed in the numerical illustration. The values of the parameter for which the source is not reported are taken for the sake of the exercise. The value for the volatility of the BAU emissions $\sigma_{1}$ is taken from [1]. The average correlation to the common shock of emissions is taken from [1]. The previous two values are estimated by considering the six main sectors covered by the EU ETS (Public Power and Heat, Pulp and Paper, Cement, Lime and Glass, Metals, Oil and Gas, Other) and the corresponding yearly verified emissions from 2008 to 2012. The $\nu$ parameter is the current value of the market depth in the EEX exchange (e.g., [9]). The flexibility parameter $\eta$ is taken from [9], where it is obtained by updating the approximation in [25], Section 4.4. The value of $h$ is taken from [1] and it roughly corresponds to the running maximum of the EU ETS carbon price, reached in February 2023. The initial level of BAU emissions is taken from [9] and it corresponds to the order of magnitude of the energy and industrial pollution in the past four years, as per the European Environmental Agency and the data provider Statista.com. The default value for $\tilde{a}$ has been chosen according to the following rationale. In [1] the initial firm’s allowances endowment is $A_{0}=0.1$. By assuming that the same amount of allowances are distributed over $T=5$ years and a constant value for $\tilde{a}(t)$ we have that $0.1=\int_{0}^{T}\tilde{a}(t)\,dt=\tilde{a}T$, whence $\tilde{a}=0.1/T$. Parameter | Numerical value and short description
---|---
$\kappa_{f}$ |  $5$; parameter linked to the fossil-fuel level of capital;
$\kappa_{g}$ |  $3\gamma+0.2$; parameter linked to the green level of capital;
$\delta$ |  $1\%$; depreciation rate of the level of capital;
$\sigma$ |  $0.5\%$ Gtons/year${}^{\frac{1}{2}}$; volatility of the level of capital.
$\kappa_{e}$ |  $2$ Gtons/(level of capital); proportionality factor production - emissions.
$\tilde{a}$ |  $[0,20]$ Gtons/year; growth rate emissions permits. Default: $0.5/T$ Gtons/year.
$\tilde{\sigma}_{2}$ |  $0.2$ Gtons/year${}^{\frac{1}{2}}$;volatility of the emissions permits.
$\sigma_{1}$ | 0.2 Gtons/year${}^{\frac{1}{2}}$; volatility of the BAU emissions.
$\sigma_{2}$ | 0.5 Gtons/year${}^{\frac{1}{2}}$; volatility of the short-term emissions.
$\rho$ | 0.92; average correlation to the common shock of emissions.
$a$ |  $50$ Eur; parameter of the inverse demand curve.
$b$ |  $0.07$; parameter of the inverse demand curve.
$A_{k}$ |  $2$; productivity level of the capital.
$\nu$ |  $285.713$ Gton2/Eur a year; market depth.
$\eta$ |  $0.211$ Gton2/Eur a year; flexibility parameter.
$h$ |  $80$ Eur/ton; abatement cost coefficient, liner part.
$c_{1,1}$ |  $0.01$ Eur/(level of capital); cost fossil-fuel level of capital, linear part.
$c_{1,2}$ |  $3$ Eur $\cdot$ year/(level of capital)2;cost fossil-fuel level of capital, quadratic part.
$c_{2,1}$ |  $0.02$ Eur/(level of capital); cost green level of capital, linear part.
$c_{2,2}$ |  $4$ Eur $\cdot$ year/(level of capital)2;cost green level of capital, quadratic part.
$\lambda$ |  $7.5\cdot 10^{-*}$ Eur/ton2 with $*\in\\{-7,-5-3\\}$; final penalty.
$\kappa_{0}$ |  $30$; initial level of capital.
$\widetilde{E}_{0}$ |  $4$ Gtons; initial level of BAU emissions.
$A_{0}$ |  $0.1$ Gtons; initial level of emissions permits.
$\widetilde{X}_{0}$ |  $A_{0}-\widetilde{E}_{0}$; initial level of the bank account.

###  7.1 Cap-and-trade system: the role of the regulator.

This subsection emphasizes the role played by the regulator, which may be
separated into two components, namely the dynamical allocation of emission
allowances $\widetilde{A}(t)$ and the severity of the cap, reflected into the
parameter $\lambda$. Naturally, ceteris paribus, the average level of
production increases as $\tilde{a}$ increases, although it does not play a
first-order role to the representative firm’s production; see Figure 1.
According to our model, this dependence may be explained by the fact that the
optimal production features a linear dependence on $\tilde{a}$, with a slope
that depends on the solution of the Riccati equations; see Equation (58).
Also, the average pollution abatement rate $\alpha(t)$, the optimal average
trading rate $\beta(t)$, and the average price of permits $\bar{\omega}_{t}$
naturally decreases as $\tilde{a}$ increases; see the sub-figures in Figure 2,
from left to right and from top to bottom. We also report the simulation of
one trajectory of the just mentioned quantities. In particular, apart from the
level, the dynamics of $\alpha(t)$ and $\beta(t)$ looks very similar. We
interpret this result as being symptomatic of the type of dependence on
$Y^{(2)}(t)$. More precisely, from the (optimal) coupling condition in
Equation (37) we have that

$\beta(t)=2\nu(\overline{Y}^{(2)}(t)-Y^{(2)}(t))\quad\text{and}\quad\alpha(t)=2\eta
Y^{(2)}(t)-\eta h.$

Whence, both $\beta(t)$ and $\alpha(t)$ depends linearly on $Y^{(2)}(t)$ and
are both affected by the idiosyncratic noise. Nonetheless, $\alpha(t)$ depends
on the common shocks too. Finally, by construction,
$\overline{\omega}=-\mathbb{E}[\overline{Y}^{(2)}(t)|\overline{\mathcal{F}}_{t}^{0}]$,
and therefore the dependence on $\overline{Y}^{(2)}(t)$ is, again, linear.
Moreover, as in [1], we observe large oscillations of the price near the
maturity; see the last sub-figure in Figure 2.
Furthermore, since the market power of the representative firm is (almost)
equal to the one of the population ($\gamma=0.5$), the representative firm
cannot charge higher prices to compensating possible lower sales; see also the
discussion in the next Subsection 7.2. Therefore, if the regulator tightens
the cap, i.e., if $\lambda$ increases, all the other things being equal, then
the production of the representative firm unambiguously decreases (Figure 1).
Notice that this fact generalizes the results for monopoly of [39] and the one
for Cournot oligopoly under taxes of [40]. Consistently, the representative
firm increases the use of the green level of capital at the expenses of the
fossil-fuel level of capital; see Figure 3. Finally, Figure 4 plots three
trajectories, one for every considered $\lambda$, of the bank account
$\widetilde{X}(t)$. It seems that the representative firm pays more for a
lower level of $\lambda$; in other words, a relaxation in the final penalty
implicitly induces the firm to emit more. However, in the present work, the
dynamic allocation of the regulator is exogenous and we do not have any
compliance constraint, neither on the expected emissions nor on a point-wise
value on the terminal net emissions of the representative firm, as in the very
recent research paper [9]. Extending our model to such a setting is an
interesting direction for future research; see Section 8.

Figure 1: Effect of $\tilde{a}$ on the average level of production
$\mathbb{E}\left[\int_{0}^{T}A_{k}K(t)\,dt\right]$. The figure plots the
average level of production as a function of the growth rate emission permits
$\tilde{a}$ for different levels of $\lambda$. Parameters value as in Table 2.

Figure 2: Effect of $\tilde{a}$ on the average pollution abatement rate
$\alpha(t)$, average trading rate $\beta(t)$, and average price of permits
$\bar{\omega}_{t}$. From left to right, from top to bottom: the figure plots
1. the optimal average level of pollution abatement $\alpha(t)$; 2. the
optimal average level of optimal trading $\beta(t)$; 3. the optimal average
price of permits, as a function of the growth rate emission permits
$\tilde{a}$, and 4. the variance of the price of permits as a function of
time. We also report a simulation of one trajectory of the corresponding
quantities. Parameters value as in Table 2.

Figure 3: Effect of $\lambda$ on the average level of capital, on the average
fossil-fuel level of capital, and on the average green level of capital. From
left to right, from top to bottom: The figure plots the average difference
between the average level of capital, the average fossil-fuel level of
capital, and the average green level of capital for two different values of
the parameter $\lambda$ over the interval $[0,T]$. Parameters value as in
Table 2. Figure 4: Effect of $\lambda$ on the average value of the bank
account. The figure plots the average value of the bank account for three
different values of the parameter $\lambda$ over the interval $[0,T]$.
Parameters value as in Table 2.

###  7.2 The economics of competition.

Using a static model of imperfect market competition, [5] emphasizes the
importance of the degree of competition in determining the economic
consequences of pollution regulation. It is therefore natural to ask whether
their findings are also recovered in our (dynamic) setting with stochastic
emissions888Stochastic emissions are considered as a direction for future
research in [5], Section 7.2. and production costs. Naturally, the pollution
regulator wants to encourage pollution abatement and discourage output
reduction. Indeed, should the output be lower, consumer surplus and welfare
would be hurt because of an increase in goods prices. Figure 5 shows that the
expected average output increases with $\gamma$, whereas the average price of
goods decreases. This is because when $\gamma=0$, the representative firm has
significant market power and can charge higher prices, compensating it for
lower sales. As $\gamma$ increases, competition with the rest of the
population intensifies, thus reducing the firm’s market power. This is made
clear by the following relation:
$p^{\overline{K}}(t)=a-b(1-\gamma)A_{k}K(t)-b\gamma A_{k}\overline{K}(t)$.
Indeed, if the representative firm lowers the output, then this has a limited
effect on the price because the rest of the population would increase its
output in response. Consequently, the representative firm prefers pollution
abatement over output reduction, with the caveat that even though the first-
order partial effect is positive, it does not have the same compensatory
dynamic as the one of the price. Hence, increasing the competition helps align
the firm incentives with the goal of the regulation of pollution abatement;
this is in line with [5]. Consistently, Figure 6 shows that ceteris paribus,
the average level of capitals, and the average trading activity increase with
an increase in the level of competition. In particular, the latter causes an
increase in the average market price of permits because of increased
liquidity.
Figure 6 plots the value function along with its components. Generally, the
cap-and-trade mechanism has both a direct pollution abatement effect and an
indirect output-reduction effect. Consistently with our theoretical argument
in the previous paragraph, competition, i.e., a value of $\gamma\in(0,1]$,
induces the representative firm to overproduce. From a population perspective,
it would be beneficial if every (representative) firm lowers the corresponding
output to keep prices high. This is an unlikely scenario because no
representative firm could credibly commit to such a lower output, as one would
expect. The cap-and-trade mechanism should coordinate the previous mechanism
in such a way that the population of firms agrees to reduce output by using
the pollution constraint; naturally, this synchronization mechanism is
expected to work under a suitable range of constraints imposed by the
pollution regulator, the one for which the impact of output reduction on the
representative firm’s profits dominates the cost of pollution abatement, of
trading, and production. In particular, in our numerical example costs
dominate revenues, and therefore profits, as $\gamma\in(0,1]$ increases. When
$\gamma=0$ (monopoly), the representative firm has significant market power
and it can optimize its output. However, should the regulator decide to
tighten the cap, the output of the representative firm would further reduce
and the representative firm can no longer leverage on the competition with the
population of firms to implement the previously described synchronization
mechanism. Therefore, a cap-and-trade mechanism hurts more monopoly than
competitive firms, which is in agreement with the findings for competitive
markets in [5].

Figure 5: Effect of the level of competition $\gamma$ on the average level of
production $\mathbb{E}\left[\int_{0}^{T}A_{k}K(t)\,dt\right]$, on the average
price of good $\mathbb{E}\left[\int_{0}^{T}p^{\overline{K}}(t)\,dt\right]$,
and on the average price of permits
$\mathbb{E}\left[\int_{0}^{T}\overline{\omega}(t)\,dt\right]$. From left to
right, from top to bottom: The figure plots the average value of the
production, the average price of good, and the average price of permits as a
function of the level of competition $\gamma$. Parameters value as in Table 2.

Figure 6: From left to right, from top to bottom: Effect of the level of
competition $\gamma$ on the value function $\mathcal{J}^{NE}(v,\overline{K})$,
on the average final penalty $\mathbb{E}[\lambda X^{2}_{T}]$, on the average
abatement cost $\mathbb{E}[\int_{0}^{T}C(\alpha(t))\,dt]$, on the average
level of fossil-fuel $\mathbb{E}[\int_{0}^{T}K^{f}(t)\,dt]$ and green capital
$\mathbb{E}[\int_{0}^{T}K^{f}(t)\,dt]$, and on the average pollution abatement
rate $\mathbb{E}[\int_{0}^{T}\alpha(t)\,dt]$ and average trading rate
$\mathbb{E}[\int_{0}^{T}\beta(t)\,dt]$. Parameters value as in Table 2.

##  8 Conclusion and future research

The model proposed in this paper introduces several fundamental elements
regarding pollution generation, abatement and costs, and regulation, which can
serve as a basis for future research.
The model assumes that firms produce products using a standard AK model with a
positive depreciation rate of capital. Future research could consider relaxing
this assumption and exploring a more realistic, namely non-linear, production
function. Additionally, it assumes that the business cycle affecting the BAU
carbon emissions does not correlate with the one affecting emission
allowances. Said differently, we are assuming that the regulator has access to
limited information and it is affected by a macroeconomic shocks driver that
it is independent from the one influencing the emissions. Investigating
correlated business cycles and the impact of asymmetric information on
production and abatement costs could be potential areas for future research;
the latter, in particular, can lead to interesting agency problems. In the
present work, the dynamic allocation of the regulator is exogenous and we do
not consider any compliance constraint, neither on the expected emissions nor
on a point-wise value on the terminal net emissions of the representative
firm, as done in the very recent research paper [9]. Extending our model to
such a setting is an interesting direction for future research. The model also
assumes that all firms share the same cost and coefficient functions.
Extending the model to incorporate multiple populations, where firms within
each population share the same cost and coefficient functions, but differ
across populations, is an area for future exploration. This will provide an
important tool to study the market equilibrium price in the presence of
different types of firms. In addition, it’s important for future research to
consider relaxing the assumption of the carbon price being
$(\overline{\mathcal{F}}^{0})$-adapted. Also, it might be an interesting
avenue for future research to account for the way in which the regulator
allocates allowances to individual firms, in particular analysing the case
when the initial allocation is through auctions, as initially intended by the
European Union, which switched back to grandfathering in the third phase
(after 2012). Extending the model to account for multiple compliance periods
and the specific design of current cap-and-trade systems could lead to clear-
cut predictions about permit prices and related derivatives. Finally, future
research could consider integrated production-pollution-abatement models in
continuous time and study other types of policies, such as the policy rules
under the Market Stability Reserve (MSR), launched in 2019 by the EU.

##  Appendix A Linear quadratic mean field games with common noise.

In this section, we present the general formulation of a linear quadratic mean
field game class with common noise, which our framework fits into.
Let $(\overline{\xi}(s))$ and $(\overline{\psi}(s))$ be given processes
adapted to the filtration $(\overline{\mathcal{F}}^{0}_{t})$. We consider the
following dynamics:

$\begin{split}dX(s)&=(A_{0}(s)+A(s)X(s)+\overline{A}(s)\overline{\xi}(s)+B(s)v(s)+\overline{B}(s)\overline{\psi}(s))\,ds\\\
&+\sum_{j=1}^{d_{1}}\left(C_{0,j}(s)+C_{j}(s)X(s)+\overline{C}_{j}(s)\overline{\xi}(s)+D_{j}(s)v(s)+\overline{D}_{j}(s)\overline{\psi}(s)\right)\,dW^{j}(s)\\\
&+\sum_{\ell=1}^{d_{0}}\left(F_{0,j}(s)+F_{j}(s)X(s)+\overline{F}_{j}(s)\overline{\xi}(s)+G_{j}(s)v(s)+\overline{G}_{j}(s)\overline{\psi}(s)\right)\,dW^{0,j}(s),\\\
\end{split}$ (60)

with $X(t)=x$, and objective functional:

$\begin{split}\mathcal{J}_{x,t}^{NE}(v)=\mathbb{E}\Bigg{[}\int_{t}^{T}&\Bigg{(}Q_{0}(s)+\langle
Q(s)X(s),X(s)\rangle+2\langle\overline{Q}(s)\overline{\xi}(s),\overline{X}(s)\rangle+\langle
R(s)v(s),v(s)\rangle\\\
&+2\langle\overline{R}(s)\overline{\psi}(s),v(s)\rangle+2\langle
S(s)X(s),v(s)\rangle+2\langle\overline{S}_{1}(s)\overline{\xi}(s),v(s)\rangle\\\
&+2\langle\overline{S}_{2}(s)X(s),\overline{\psi}(s)\rangle\\\ &+2\langle
q(s),X(s)\rangle+2\langle\overline{q}(s),\overline{\xi}(s)\rangle+2\langle
r(s),v(s)\rangle+2\langle\overline{r}(s),\overline{\psi}(s)\rangle\Bigg{)}\,ds\\\
&+\langle
HX(T),X(T)\rangle+2\langle\overline{H}\overline{\xi}(T),X(T)\rangle\Bigg{]},\end{split}$
(61)

where $\langle\cdot,\cdot\rangle$ denotes the inner product on Euclidean
space. The goal is to find a control $\hat{v}(s)$ with corresponding state
process $\hat{X}$ such that
$\mathcal{J}_{x,t}^{NE}(\hat{v};\overline{\xi},\overline{\psi})=\inf_{v}\mathcal{J}_{x,t}^{NE}(v;\overline{\xi},\overline{\psi})$
and $\mathbb{E}[\hat{X}(s)|\overline{\mathcal{F}}_{s}^{0}]=\overline{\xi}$,
$\mathbb{E}[\hat{v}(s)|\overline{\mathcal{F}}_{s}^{0}]=\overline{\psi}$. The
process $\hat{v}$ is called a mean field Nash equilibrium. We state the
following assumption on the coefficient matrices (cfr. [26], Assumption 3.1)

  1. (N1)

$A_{0},C_{0,j},F_{0,j}\in L^{\infty}([0,T];\mathbb{R}^{d})$, $1\leq j\leq
d_{1}$ and $\ell\leq 1\leq d_{0}$, and $Q_{0}(s)\in
L^{\infty}([0,T];\mathbb{R})$.

  2. (N2)

$A,\overline{A},C,\overline{C},F,\overline{F}\in
L^{\infty}([0,T];\mathbb{R}^{d\times d})$.

  3. (N3)

$B,\overline{B},D,\overline{D},G,\overline{G}\in
L^{\infty}([0,T];\mathbb{R}^{d\times d_{2}})$.

  4. (N4)

$Q,\overline{Q}\in L^{\infty}([0,T];\mathcal{S}^{d})$, $R,\overline{R}\in
L^{\infty}([0,T];\mathcal{S}^{d_{2}})$, $H,\overline{H}\in\mathcal{S}^{d}$.

  5. (N5)

$H\geq 0$ and for some $\delta_{1}\geq 0$, $\delta_{2}>0$,
$Q,Q\geq\delta_{1}I_{d}$ and $R\geq\delta_{2}I_{d}$.

  6. (N6)

$S,\overline{S}_{1},\overline{S}_{2}\in
L^{\infty}([0,T];\mathbb{R}^{d_{2}\times d})$; $q,\overline{q}\in
L^{\infty}([0,T];\mathbb{R}^{d})$; $r,\overline{r}\in
L^{\infty}([0,T];\mathbb{R}^{d_{2}})$.

  7. (N7)

$\|S\|_{\infty}^{2}<\delta_{1}\delta_{2}$ if $\delta_{1}>0$,
$S=\overline{S}=0$ otherwise.

##  Appendix B Linear quadratic mean field type control with common noise.

In this section, we provide explicit solutions of a class of linear quadratic
mean field type control problems in terms of a system of Riccati equations.
The class of problems we consider is a generalization of the one analyzed in
[26]. In the latter, both the private states dynamics and the running cost
appearing in the cost functional do not contain (possibly time-dependant)
terms of order zero, and both common and idiosyncratic noise values are uni-
dimensional, i.e., $d_{0}=d_{1}=1$. Instead, we consider the following
dynamics:

$\begin{split}dX(s)&=(A_{0}(s)+A(s)X(s)+\overline{A}(s)\overline{X}(s)+B(s)v(s)+\overline{B}(s)\overline{v}(s))\,ds\\\
&+\sum_{j=1}^{d_{1}}\left(C_{0,j}(s)+C_{j}(s)X(s)+\overline{C}_{j}(s)\overline{X}(s)+D_{j}(s)v(s)+\overline{D}_{j}(s)\overline{v}(s)\right)\,dW^{j}(s)\\\
&+\sum_{\ell=1}^{d_{0}}\left(F_{0,j}(s)+F_{j}(s)X(s)+\overline{F}_{j}(s)\overline{X}(s)+G_{j}(s)v(s)+\overline{G}_{j}(s)\overline{v}(s)\right)\,dW^{0,j}(s),\\\
\end{split}$ (62)

with $X(t)=x$. In addition, the objective cost functional is given by:

$\begin{split}\mathcal{J}_{x,t}^{LQ}(v)=\mathbb{E}\Bigg{[}\int_{t}^{T}&\Bigg{(}Q_{0}(s)+\langle
Q(s)X(s),X(s)\rangle+\langle\overline{Q}(s)\overline{X}(s),\overline{X}(s)\rangle+\langle
R(s)v(s),v(s)\rangle\\\
&+\langle\overline{R}(s)\overline{v}(s),\overline{v}(s)\rangle+2\langle
S(s)X(s),v(s)\rangle+2\langle\overline{S}(s)\overline{X}(s),\overline{v}(s)\rangle+2\langle
q(s),X(s)\rangle\\\ &+2\langle\overline{q}(s),\overline{X}(s)\rangle+2\langle
r(s),v(s)\rangle+2\langle\overline{r}(s),\overline{v}(s)\rangle\Bigg{)}\,ds\\\
&+\langle
HX(T),X(T)\rangle+2\langle\overline{H}\overline{X}(T),X(T)\rangle\Bigg{]},\end{split}$
(63)

where $\langle\cdot,\cdot\rangle$ denotes the inner product on Euclidean
space.

We state the following assumption on the coefficient matrices (cfr. [26],
Assumption 2.1)

  1. (M1)

$A_{0},C_{0,j},F_{0,j}\in L^{\infty}([0,T];\mathbb{R}^{d})$, $1\leq j\leq
d_{1}$ and $\ell\leq 1\leq d_{0}$, and $Q_{0}(s)\in
L^{\infty}([0,T];\mathbb{R})$.

  2. (M2)

$A,\overline{A},C,\overline{C},F,\overline{F}\in
L^{\infty}([0,T];\mathbb{R}^{d\times d})$.

  3. (M3)

$B,\overline{B},D,\overline{D},G,\overline{G}\in
L^{\infty}([0,T];\mathbb{R}^{d\times d_{2}})$.

  4. (M4)

$Q,\overline{Q}\in L^{\infty}([0,T];\mathcal{S}^{d})$, $R,\overline{R}\in
L^{\infty}([0,T];\mathcal{S}^{d_{2}})$, $H,\overline{H}\in\mathcal{S}^{d}$.

  5. (M5)

$H,H+\overline{H}\geq 0$ and for some $\delta_{1}\geq 0$, $\delta_{2}>0$,
$Q,Q+\overline{Q}\geq\delta_{1}I_{d}$ and
$R,R+\overline{R}\geq\delta_{2}I_{d}$.

  6. (M6)

$S,\overline{S}\in L^{\infty}([0,T];\mathbb{R}^{d_{2}\times d})$;
$q,\overline{q}\in L^{\infty}([0,T];\mathbb{R}^{d})$; $r,\overline{r}\in
L^{\infty}([0,T];\mathbb{R}^{d_{2}})$.

  7. (M7)

$\|S\|_{\infty}^{2},\|S+\overline{S}\|_{\infty}^{2}<\delta_{1}\delta_{2}$ if
$\delta_{1}>0$, $S=\overline{S}=0$ otherwise.

The procedure used in [26], Section 2.2, uses a technique developed by [42].
In order to facilitate the reader, we will highlight in bold font the
additional terms with respect [26], Theorem 2.6, Equations (2.31)– (2.32) and
the subsequent non-numbered one, linked to the terms of order zero999Notice
that the expression for $\Sigma_{0}$ and $\phi(s)$ derived in [26] presents
some inaccuracies. First, the term $G^{T}PG$ is missed in the expression for
$\Sigma_{0}$ (see Equation (79)). Second, there is an extra term in the
equation for $\phi$; nonetheless, the equation remains linear and, therefore,
essentially trivial to solve (see Equation (84)). We suppose that:

$Y(s)=P(s)(X(s)-\overline{X}(s))+\Pi(s)\overline{X}(s)+\phi(s),$ (64)

where $P$ and $\Pi$ are $\mathcal{S}^{d}$-valued processes such that they
satisfy the following terminal conditions:

$P(T)=H,\quad\Pi(T)=H+\overline{H},$

and $\phi(s)$ is an $\mathbb{R}^{d}$-valued process; $P$, $\Pi$, and $\phi$
are deterministic. Hereafter, in order to ease the notation, we suppress the
time indexes and we work under the assumption that $d_{0}=d_{1}=1$; we will
provide the expressions for the case $d_{0}>1$ and $d_{1}>1$ at the end of the
present section. By taking the conditional expectation in Equation (64), we
obtain:

$\overline{Y}=\Pi\overline{X}+\phi\,\,\text{and}\,\,Y-\overline{Y}=P(X-\overline{X}).$
(65)

Moreover, by taking the conditional expectation in Equation (62), we obtain:

$d\overline{X}=\left(\boldsymbol{A_{0}}+(A+\overline{A})\overline{X}+(B+\overline{B})\overline{v}\right)\,ds+\left(\boldsymbol{F_{0}}+(F+\overline{F})\overline{X}+(G+\overline{G})\overline{v}\right)dW^{0}.$
(66)

By subtracting the previous equation from Equation (62), we have:

$\begin{split}d(X-\overline{X})&=\left(A(X-\overline{X})+B(v-\overline{v})\right),ds\\\
&+\left(\boldsymbol{C_{0}}+C(X-\overline{X})+(C+\overline{C})\overline{X}+D(v-\overline{v})+(D+\overline{D})\overline{v}\right)\,dW\\\
&+\left(F(X-\overline{X})+G(v-\overline{v})\right)\,dW^{0}.\end{split}$ (67)

Proposition 2.4, Equation (2.4), in [26] gives us101010Proposition 2.4,
Equation (2.4), in [26] is not affected by the presence of zero-order terms.:

$\begin{split}dY&=-\Bigl{(}A^{T}(Y-\overline{Y})+(A^{T}+\overline{A}^{T})\overline{Y}+C^{T}(Z-\overline{Z})+(C^{T}+\overline{C}^{T})\overline{Z}\\\
&+F^{T}(Z_{0}-\overline{Z}_{0})+(F^{T}+\overline{F}^{T})\overline{Z}_{0}+Q(X-\overline{X})+(Q+\overline{Q})\overline{X}\\\
&+S^{T}v+\overline{S}^{T}\overline{v}+q+\overline{q}\Bigr{)}\,ds+Z\,dW+Z_{0}\,dW^{0}\\\
\end{split}$ (68)

Now, on one hand we have:

$\begin{split}d(Y-\overline{Y})&=\left(\overset{\cdot}{P}(X-\overline{X})+PA(X-\overline{X})+PB(v-\overline{v})\right)\,ds\\\
&+P\left(\boldsymbol{C_{0}}+C(X-\overline{X})+(C+\overline{C})\overline{X}+D(v-\overline{v})+(D+\overline{D})\overline{v}\right)\,dW\\\
&+P\left(F(X-\overline{X})+G(v-\overline{v})\right)\,dW^{0}.\end{split}$ (69)

On the other hand, it holds that (see Equation (65)):

$\begin{split}d\overline{Y}&=(\overset{\cdot}{\phi}+\overset{\cdot}{\Pi})\,ds+\Pi
d\overline{X}\\\
&=\left(\overset{\cdot}{\phi}+\overset{\cdot}{\Pi}\overline{X}+\boldsymbol{\Pi
A_{0}}+\Pi(A+\overline{A})\overline{X}+\Pi(B+\overline{B})\overline{v}\right)\,ds\\\
&+\Pi\left(\boldsymbol{F_{0}}+(F+\overline{F})\overline{X}+(G+\overline{G})\overline{v}\right)\,dW^{0}\end{split}$
(70)

Noting that $dY=d(Y-\overline{Y})+d\overline{Y}$, we compare the diffusion
terms of the left and right hand side of this equation. We get:

$Z=P\left(\boldsymbol{C_{0}}+C(X-\overline{X})+(C+\overline{C})\overline{X}+D(v-\overline{v})+(D+\overline{D})\overline{v}\right)$
(71)
$Z_{0}=P\left(F(X-\overline{X})+G(v-\overline{v})\right)+\Pi\left(\boldsymbol{F_{0}}+(F+\overline{F})\overline{X}+(G+\overline{G})\overline{v}\right)$
(72)

which implies

$\overline{Z}=P\left(\boldsymbol{C_{0}}+(C+\overline{C})\overline{X}+(D+\overline{D})\overline{v}\right)$
(73) $Z-\overline{Z}=P\left(C(X-\overline{X})+D(v-\overline{v})\right)$ (74)
$\overline{Z}_{0}=\Pi\left(\boldsymbol{F_{0}}+(F+\overline{F})\overline{X}+(G+\overline{G})\overline{v}\right)$
(75)

and

$Z_{0}-\overline{Z}_{0}=P\left(F(X-\overline{X})+G(v-\overline{v})\right)$
(76)

At this point, the coupling condition in [26], Proposition 2.4, Equation (2.5)
reads as111111On the other hand, Equation (2.5) is affected by zero-order
terms since it depends on $\overline{Z}$ and $\overline{Z}_{0}$.:

$\begin{split}&B^{T}(Y-\overline{Y})+(B^{T}+\overline{B}^{T})\overline{Y}+D^{T}(Z-\overline{Z})+(D+\overline{D}^{T})\overline{Z}\\\
&+G^{T}(Z_{0}-\overline{Z}_{0})+(G^{T}+\overline{G}^{T})\overline{Z}_{0}+R(v-\overline{v})+(R+\overline{R})\overline{v}+S(X-\overline{X})+(S+\overline{S})\overline{X}(s)+r+\overline{r}\\\
&=B^{T}P(X-\overline{X})+(B^{T}+\overline{B}^{T})\Pi\overline{X}+(B^{T}+\overline{B}^{T})\phi+D^{T}P\left(C(X-\overline{X})+D(v-\overline{v})\right)\\\
&+(D+\overline{D}^{T})\left(\boldsymbol{C_{0}}+(C+\overline{C})\overline{X}+(D+\overline{D})\overline{v}\right)+G^{T}P\left(F(X-\overline{X})+G(v-\overline{v})\right)\\\
&+(G^{T}+\overline{G}^{T})\Pi\left(\boldsymbol{F_{0}}+(F+\overline{F})\overline{X}+(G+\overline{G})\overline{v}\right)\\\
&+R(v-\overline{v})+(R+\overline{R})\overline{v}+S(X-\overline{X})+(S+\overline{S})\overline{X}+r+\overline{r},\\\
\end{split}$ (77)

which can be rewritten in the following way

$\Lambda_{0}(X-\overline{X})+\Lambda_{1}\overline{X}+\Sigma_{0}(v-\overline{v})+\Sigma_{1}\overline{v}+(B^{T}+\overline{B}^{T})\phi+r+\overline{r}+\boldsymbol{(D+\overline{D}^{T})C_{0}+(G^{T}+\overline{G}^{T})\Pi
F_{0}}=0$ (78)

by setting

$\begin{split}\Lambda_{0}&=B^{T}P+D^{T}PC+G^{T}PF+S;\\\
\Lambda_{1}&=(B^{T}+\overline{B}^{T})\Pi+(D+\overline{D}^{T})P(C+\overline{C})+(G^{T}+\overline{G}^{T})\Pi(F+\overline{F})+(S+\overline{S})\\\
\Sigma_{0}&=D^{T}PD+G^{T}PG+R\\\
\Sigma_{1}&=(D+\overline{D}^{T})P(D+\overline{D})+(G^{T}+\overline{G}^{T})\Pi(G+\overline{G})+R+\overline{R}\\\
\end{split}$ (79)

Taking the conditional expectation in Equation (78), assuming $\Sigma_{1}$
invertible, and making the term $\overline{v}$ explicit in Equation (78), we
deduce

$\overline{v}=-\Sigma_{1}^{-1}\left(\Lambda_{1}\overline{X}+r+\overline{r}+(B^{T}+\overline{B}^{T})\phi+\boldsymbol{(D+\overline{D}^{T})C_{0}+(G^{T}+\overline{G}^{T})\Pi
F_{0}}\right)$ (80)

Assuming $\Sigma_{0}$ is also invertible and observing that
$v=v-\overline{v}+\overline{v}$ , we have:

$\begin{split}v=&-\Sigma_{0}^{-1}\Lambda_{0}(X-\overline{X})\\\
&-\Sigma_{1}^{-1}\left(\Lambda_{1}\overline{X}+r+\overline{r}+(B^{T}+\overline{B}^{T})\phi+\boldsymbol{(D+\overline{D}^{T})C_{0}+(G^{T}+\overline{G}^{T})\Pi
F_{0}}\right)\\\ \end{split}$ (81)

At this point, we compare the drift terms from (68) to those of (69) and (70).
Using the relations (73), (74), (75), (76), (80), and (81) proved above. By
noticing that $v-\overline{v}=-\Sigma_{0}^{-1}\Lambda_{0}(X-\overline{X})$,
after some algebra, we deduce that $P$ and $\Pi$ should satisfy the following
Riccati equations:

$\begin{cases}\begin{split}&\overset{\cdot}{P}+PA+A^{T}P+C^{T}PC+F^{T}PF+Q-(PB+C^{T}PD+F^{T}PG+S^{T})\Sigma_{0}^{-1}\Lambda_{0}=0\\\
&\Lambda_{0}=B^{T}P+D^{T}PC+G^{T}PF+S;\\\ &\Sigma_{0}=D^{T}PD+G^{T}PG+R\\\
&P(T)=H.\end{split}\end{cases}$ (82)

$\begin{cases}\begin{split}&\overset{\cdot}{\Pi}+\Pi(A+\overline{A})+(A^{T}+\overline{A}^{T})\Pi+(C^{T}+\overline{C}^{T})P(C+\overline{C})+(F^{T}+\overline{F}^{T})\Pi(F+\overline{F})+(Q+\overline{Q})\\\
&-\left(\Pi(B+\overline{B})+(C^{T}+\overline{C}^{T})P(D+\overline{D})+(F^{T}+\overline{F}^{T})\Pi(G+\overline{G})+\overline{S}^{T}+S^{T}+\right)\Sigma_{1}^{-1}\Lambda_{1}=0\\\
&\Lambda_{1}=(B^{T}+\overline{B}^{T})\Pi+(D+\overline{D}^{T})P(C+\overline{C})+(G^{T}+\overline{G}^{T})\Pi(F+\overline{F})+(S+\overline{S})\\\
&\Sigma_{1}=(D+\overline{D}^{T})P(D+\overline{D})+(G^{T}+\overline{G}^{T})\Pi(G+\overline{G})+R+\overline{R}\\\
&\Pi(T)=H+\overline{H}\end{split}\end{cases}$ (83)

Once we have $P$ and $\Pi$ solution to Equation (82) and (83), we set:

$\begin{split}&\overset{\cdot}{\phi}-\Bigl{(}\Pi(B+\overline{B})+(C^{T}+\overline{C}^{T})P(D+\overline{D})+(F^{T}+\overline{F}^{T})\Pi(G+\overline{G})+S^{T}+\overline{S}^{T}\Bigr{)}\Sigma_{1}^{-1}\cdot\\\
&\Big{(}r+\overline{r}+\boldsymbol{(D+\overline{D}^{T})C_{0}+(G^{T}+\overline{G}^{T})\Pi
F_{0}}\\\ &+q+\overline{q}+\boldsymbol{\Pi
A_{0}-(C^{T}+\overline{C}^{T})PC_{0}-(F^{T}+\overline{F}^{T})\Pi
F_{0}\Big{)}}\\\
&-\Bigl{[}\Bigl{(}\Pi(B+\overline{B})+(C^{T}+\overline{C}^{T})P(D+\overline{D})+(F^{T}+\overline{F}^{T})\Pi(G+\overline{G})+S^{T}+\overline{S}^{T}\Bigr{)}\Sigma_{1}^{-1}(B^{T}+\overline{B}^{T})\\\
&+(A^{T}+\overline{A}^{T})\Bigr{]}\phi=0\end{split}$ (84)

Finally, we obtain the optimal trajectory (using Equation (81)), a formula for
the process $Z$ (using Equations (74), (75) and (81)), and a formula for the
process $\overline{Z}_{0}$ (using Equations (74) and (75)):

$\begin{split}dX&=\left(A(X-\overline{X})+(A+\overline{A})\overline{X}+B(v-\overline{v})+(B+\overline{B})\overline{v}\right)\,ds\\\
&+\left(C(X-\overline{X})+(C+\overline{C})\overline{X}+D(v-\overline{v})+(D+\overline{D})\overline{v}\right)\,dW\\\
&+\left(F(X-\overline{X})+(F+\overline{F})\overline{X}+G(v-\overline{v})+(G+\overline{G})\overline{v}\right)\,dW^{0}\\\
\end{split}$ (85)
$\begin{split}Z&=P\left(C(X-\overline{X})-D\Sigma_{0}^{-1}\Lambda_{0}(X-\overline{X})\right)\\\
&+P\Bigg{(}\boldsymbol{C_{0}}+(C+\overline{C})\overline{X}-(D+\overline{D})\Sigma_{1}^{-1}\Bigg{(}\Lambda_{1}\overline{X}+r+\overline{r}+(B^{T}+\overline{B}^{T})\phi\\\
&+\boldsymbol{(D+\overline{D}^{T})C_{0}+(G^{T}+\overline{G}^{T})\Pi
F_{0}}\Bigg{)}\Bigg{)}\end{split}$ (86)
$\begin{split}Z_{0}&=P\left(F-G\Sigma_{0}^{-1}\Lambda_{0}\right)(X-\overline{X})\\\
&+\Pi\left(\boldsymbol{F_{0}}+(F+\overline{F})-(G+\overline{G})\Sigma_{1}^{-1}\Lambda_{1}\right)\overline{X}\\\
&-\Pi(G+\overline{G})\Sigma_{1}^{-1}(r+\overline{r}+(B^{T}+\overline{B}^{T})\phi+\boldsymbol{(D+\overline{D}^{T})C_{0}+(G^{T}+\overline{G}^{T})\Pi
F_{0}}).\end{split}$ (87)

Equations for the general case are easily obtained by using the summations
where necessary (e.g., $C^{T}PC$ is replaced by
$\sum_{j=1}^{d_{1}}C_{j}^{T}PC_{j}$).

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