# A Stochastic Differential Game of Transboundary Pollution under Knightian Uncertainty of Stock Dynamics

Authors: Chunyan Fu; Yongxi Yi; Susu Cheng
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

With the robust control framework of Hansen and Sargent (2001), this paper investigates a stochastic differential game of transboundary pollution between two regions under Knightian uncertainty of stock dynamics. Both regions are assumed to play a noncooperative and a cooperative game, and the worst-case pollution accumulation processes for discrete robustness parameters are characterized. Our objective is to identify both regions’ optimal output and emission levels and analyze the effects of the Knightian uncertainty of pollution stock dynamics on both regions’ optimization behavior. We illustrate the results with some numerical examples.


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## Body

## 1. Introduction

Since the last century, the pollution problem is becoming ever more serious with the rapidly developing industrialization. In particular, many kinds of pollutants can spread incredible distances meaning that it is not contained within the boundaries of any single region or nation, so, the “transboundary pollution” has become a problem across regions and even borders. In each region those who suffer from the pollution wishes that the polluter in neighboring regions would either reduce polluting or compensate for the damages and there is generally a game between the sufferers and the polluter around pollution abatement and benefit compensation [1].In recent years, transboundary pollution problems have raised increasing interest among academic circles and policy makers. Among them, in a dynamic framework, Breton et al. (2010) built a model to analyze how countries join international environmental agreements (IEAs). With a dynamic control model, Li [2] studies the outcome of a pollution game between two neighboring countries. Yeung [3] and Jørgensen et al. (2010) analyze the transboundary pollution problems considering the regional government and industrial firms, simultaneously. Huang et al. [4] develop a model in which there is a Stackelberg game between the industrial firms and their local government while governments can cooperate in transboundary pollution control. Yeung and Petrosyan [1] and Yi et al. [5] develop a cooperative stochastic differential game model to analyze transboundary industrial pollution control, in which the uncertain dynamics of pollution are taken into account.In this paper, we extend Yeung and Petrosyan [1] and Yi et al.’s [5] model to an even more general model. In the models of Yeung and Petrosyan [1] and Yi et al. [5], the uncertain stock dynamics are actually a risk, where the probabilistic structure of pollution stock evolving can be fully captured by a single Bayesian before. However, in our model, the ‘‘uncertainty’’ of evolving stock is seen as a broad term. This ‘‘uncertainty’’ means an inability to posit precise probabilistic structure of stock dynamics. This stems from the concept of uncertainty as introduced by Knight [6] to represent a situation where a decision-maker lacks adequate information to assign probabilities to events. Knight [6] contends that this deeper kind of uncertainty is quite common in economic decision-making and thus deserving of systematic study. Along the way of Knight [6], an axiomatic foundation of maximum expected that utility is established by Gilboa and Schmeidler [7]. They believe that it is sensible, when the underlying uncertainty of an economic system is not well understood and axiomatically compelling, to optimize the worst-case outcome. Klibanoff et al. (2009) built a ‘‘smooth ambiguity’’ model, in this model, different degrees of aversion for uncertainty are explicitly parameterized in agents’ preferences. Hansen and Sargent [8] and Hansen et al. [9] extend Gilboa and Schmeidler’s insight to continuous-time dynamic optimization problems, introducing the concept of robust control to economic environments. They show how standard dynamic programming techniques can be changed to yield robust solutions to problems in which the underlying stochastic nature of the model is not perfectly known. Using the framework of Hansen and Sargent [8], Weitzman (2009) examines the effects of global warming and Athanassoglou and Xepapadeas [10] investigate the problem of pollution controlling with uncertain stock dynamics. In this paper, the framework of Hansen and Sargent [8] is applied to analyze the noncooperative and cooperative optimal emission levels of two neighboring regions and compare and characterize the worst-case pollution accumulation processes for discrete robustness parameter. Further, we consider the problems of the cooperation revenues allocation.The paper is organized as follows. Section2 provides the basic model. In Section 3, we characterize the noncooperative outcomes. Cooperative arrangements and individual rationality are analyzed in Section 4. We illustrate the results of a numerical example in Section 5. Section 6 is the results summarizing the paper.


## 2. The Basic Model

Consider a multinational economy, which is comprised of two regions. Following Saltari et al. [11], the output Yi(t) of region i (i=1,2) at time t (t∈[0,+∞)) is a function of emissions Ei(t). That is,(1)Yit=MiEit,where Mi denotes the technology of region i, which is given, and a concave function.Utility is given byUi(Yi(Ei(t))-Di(P(t))), where Di(P(t)) is a damage function of pollution stock P(t), and(2)UiYiEit=ziEit-12Ei2t(3)UiDiPt=gi2P2t,where zi>0 is a utility parameter of Ei and gi>0 (i=1,2) denotes the damage parameter of the pollution stock P(t) which evolves in accordance with the following linear differential equation:(4)P˙t=E1t+E2t-mPtP0=P0where 0<m<1 denotes the environment’s self-cleaning capacity.Risk is introduced to the standard model so that the stock of the pollutant accumulates according to the diffusion process:(5)dPt=E1t+E2t-mPtdt+σdBtP0=P0where {B(t):t≥0} is a Brownian motion on an underlying probability space (Ω,Γ,G) and σ is the instantaneous standard deviation. If in a world without uncertainty, Region i’s objective is to maximize welfare:(6)ViEit=maxEi≥0E∫0+∞e-rtziEit-12Ei2t-gi2P2tdts.t.dPt=E1t+E2t-mPtdt+σdBtViEit=maxEi≥0Pt≥0,ViEit=maxEi≥0P0=P0where r>0 denotes discount rate and we assume the two regions have the same discount rate. Optimization problem (6) is referred to as the benchmark model.If one does not worry about the effects of model misspecification, solving the benchmark problem (6) would be sufficient. However, because there is a Knight Uncertainty, the probabilistic structure implied by stochastic differential equation (5) is distorted and the probability measure G is replaced by another Q. The perturbed model is obtained by performing a change of measure and replacing B(t) in (5) by(7)B^t+∫0tvsdswhere {B^(t):t≥0} is a Brownian motion and {v(t):t≥0} is a measurable drift distortion. Thus, changes to the distribution of B(t) are parameterized as drift distortions to a fixed Brownian motion {B(t):t≥0}. The measurable process v(t) could correspond to any number of misspecified or omitted dynamic effects such as (i) a miscalculation of exogenous sources of emissions, (ii) a miscalculation of the natural pollution decay rate, and (iii) an ignorance of more complex dynamic structure involving irreversibility, feedback, or hysteresis effects. The distortions will be zero when v≡0 and the two measures G and Q coincide. Pollution dynamics under model misspecification are given by(8)dpt=E1t+E2t-mpt+σv11+v21dt+σdB^tWe give the following equation which is the relative entropy with the discrepancy between the two measuresG and Q:(9)RiQ=∫0∞e-rt12ΕQvi2tdt<ηi<+∞where ηi is a entropy constraint. The decision-maker can control the degree of model misspecification by modifying the entropy constraint ηi; then, region i’s robust control problem over continuous time t∈0,∞) can be defined as(10)ViEit,θ=maxEit≥0minvitE∫0+∞e-rtziEit-12Ei2t-gi2P2t+θvi2t2dts.t.dPt=E1t+E2t-mPt+σv11+v21dt+σdB^tViEit,θ=maxEit≥0minvitPt≥0,ViEit,θ=maxEit≥0minvitP0=P0where the parameter θ is a Lagrangian multiplier which associated with entropy constraint Ri(Q)<ηi. Our choice of θ lies in an interval (θ_,+∞), where the lower bound θ_ is a breakdown point beyond which it is fruitless to seek more robustness. On the other hand, when θ→∞ or, equivalently, ηi=0, there are no concerns about model misspecification.The Bellman-Isaacs condition of the robust control problem (10) can be expressed as(11)rVi=maxEi≥0minvitziEit-12Ei2t-gi2P2t+E1t+E2t-mPt+σvtViP+σ22ViPP+θvi2t2Having got the Bellman-Isaacs condition of the robust control problem, next, we will apply the robust control methods to investigate the cooperative and noncooperative strategy between both regions.


## 3. Noncooperative Strategy

### 3.1. Problem Solution

In order to get the noncooperative strategy of both regions gaming for transboundary industrial pollution, first we minimize with respect tovi of the Bellman-Isaacs condition (11) and by which we obtain(12)vi∗=-σViPθSubstituting the results of (12) into (11), differential equations (13a) and (13b) are given as(13a)rV1=maxE1≥0z1E1-12E12-g12P2+V1PE1+E2-mP+σ22V1PP-σ2V1P22θ(13b)rV2=maxE2≥0z2E2-12E22-g22P2+V2PE1+E2-mP+σ22V2PP-σ2V2P22θInvestigating differential equations (13a) and (13b), we find that they are actually the HJB equation adding a negative term, and when θ→∞, the negative term in (13a) and (13b) is also close to zero; then the robust control problems become the benchmark control problems.Next, let us use Proposition1 to investigate the noncooperative strategy and identify the game equilibrium results with subscript “N”.Proposition 1.
If both regions play a noncooperative game, their optimal instantaneous emissions can be given as follows:(14a)E1N∗=z1+2α11θP+α12θ(14b)E2N∗=z2+2α21θP+α22θwhere α11(θ), α12(θ), α21(θ), and α22(θ) are undetermined parameters related to θ.Proof.
Solving the first-order partial derivative of (13a) and (13b) with respect to E1 and E2 and setting them equal to zero, we obtain (15a) and (15b). Then solving the second-order partial derivative of (13a) and (13b) with respect to E1 and E2, (15c) and (15d) are given:(15a)∂rV1∂E1=z1-E1+V1P=0⇒E1N∗=z1+V1P(15b)∂rV2∂E2=z2-E2+V2P=0⇒E2N∗=z2+V2P(15c)∂2rV1∂E12=-1<0(15d)∂2rV2∂E22=-1<0
From (15a) and (15c), seeing that ∀δ>0, V1(E1N∗)>V1(E1N∗±δ), this means to choose the emissions levels E1N∗ are optimal for Region 1. In a similar way, from (15b) and (15d), one can find that Region 2 should choose emissions levels E2N∗ to maximum her value. Furthermore, we find from (13a) and (13b) that if there is a game equilibrium, this equilibrium must be in the case E1N∗=f1N(E2N∗), E2N∗=f2N(E1N∗), where f1N and f2N are response functions of Region 1 to Region 2 and Region 2 to Region 1 when both regions play a noncooperative strategy. In other words, the equilibrium must be a Nash equilibrium.
Substituting (15a) and (15b) into (13a) and (13b), respectively, (16a) and (16b) are obtained:(16a)rV1=z1z1+V1P-12z1+V1P2-g12P2+V1Pz1+V1P+z2+V2P-mP+σ22V1PP-σ2V1P22θ(16b)rV2=z2z2+V2P-12z2+V2P2-g22P2+V2Pz1+V1P+z2+V2P-mP+σ22V2PP-σ2V2P22θ
In order to solve the value functions (16a) and (16b), we assume(17a)V1N=α11θP2+α12θP+α13θ(17b)V2N=α21θP2+α22θP+α23θwhere α11(θ), α12(θ), α13(θ), α21(θ), α22(θ), and α23(θ) are undetermined parameters related to θ.
Maximizing with respect toP in (17a) and (17b), respectively, we obtain(18a)V1NP=∂V1N∂P=2α11θP+α12θ(18b)V1NPP=∂V1NP∂P=2α11θ(18c)V2NP=∂V2N∂P=2α21θP+α22θ(18d)V2NPP=∂V2NP∂P=2α21θ
Substituting (18a)-(18d) into (16a) and (16b), the values of undetermined parameters are given:(19a)α11=2m+r-4α21-2m+r-4α212+4g11-σ2/θ41-σ2/θ≤0(19b)α12=2α11α22+2α11z1+z22α111-σ2/θ+2α21-m-r≤0(19c)α13=1rz122+12-σ22θα122+α12z1+z2+α11α22+α11σ2
α 21 ≤ 0 can be determined by the implicit function (19d):(19d)-21+σ2θα212+2m+rσ2θα21-α214α21-2m-r2+4g11-σ2θ-g221-σ2θ=0(19e)α22=4α11α21z1+z2-2α21z1+z22α111-σ2/θ+2α21-m-r2α111-σ2/θ+2α21-m-r2α211-σ2/θ+2α11-m-r-4α11α21≤0(19f)α23=1r12z22+12-σ22θα222+z1+z2α22+α11α21+α21σ2
Investigating (19a)-(19f), it can be found that the value function is well-defined for θ>σ2 and diverges for θ=σ2. Hence the breakpoint is equal to θ_=σ2 and we from now on only consider θ>θ_=σ2.
Connecting (15a), (15c), (18a), and (18b), we get Proposition 1.
This ends the proof.Investigating the relationship between the value functions and optimal emissions of both regions, we obtain Lemma2.Lemma 2.
The value functions of both regions are inversely related to pollutant stocks; that is,∂ViN/∂P<0, i=1,2.Proof.
From(18a) and (18c), one easily obtain(20)∂V1N∂P=2α11θP+α12θ<0,∂V2N∂P=2α21θP+α22θ<0This ends the proof.


### 3.2. Characterizing the Worst-Case Pollution Accumulation Process

Substituting (18a) and (18c) into (12), we get(21a)v1N∗=-σθ2α11θP+α12θ(21b)v2N∗=-σθ2α21θP+α22θSubstituting (21a) and (21b) into (9), the pollution dynamics under model misspecification became the following form:(22)dP=E1+E2-σ2θα21θ+α22θ-m+2σ2θα11θ+α21θPdt+σdB^tThere are two negative effects of model misspecification in (22); one is the additional constant drift term equal to -σ2/θ(α12(θ)+α22(θ))>0; this suggests the presence of exogenous sources of pollution beyond those responsible for preindustrial pollution stock P0; another is the term of 2σ2/θ(α11(θ)+α21(θ))<0; this term says that the environment’s self-cleaning capacity has been reduced by an amount.The regioni (i=1,2) reacts to this worst-case scenario by adopting an emissions strategy E1N∗ and E2N∗ given by Proposition 1. Therefore, at optimality the worst-case pollution process, PN∗, can be given by the following stochastic differential equation:(23)dPN∗=E1N∗+E2N∗-mPN∗+σv1N∗+v2N∗dt+σdB^Substituting (14a), (14b), (21a), and (21b) into (23) reduces to(24)dPN∗=ξNθμNθ-PN∗+σdB^where μN(θ) and ξN(θ) are given by(25)μNθ=z1+z1+α12θ+α22θ1-σ2/θ-2α11θ+α21θ1-σ2/θ-mξNθ=-2α11θ+α21θ1-σ2θ-mNext, let us apply Proposition3 to probe the solutions of the stochastic differential equation (23).Proposition 3.
(i)
{ P N ∗ ( θ , t ) : t ≥ 0 } has expectation and variance(26)EPN∗θ,t=P0e-ξNθt+μNθ1-e-ξNθt(27)VarPN∗θ,t=σ22ξNθ1-e-2ξNθt2 (ii)
{ P N ∗ ( θ , t ) : t ≥ 0 } has a stationary distribution:(28)NNμNθ,σ22ξNθFrom Proposition3, we see that the expected value and variance of the worst-case pollution levels are decreasing in θ, and when t→∞, we obtain the following functions:(29)limt→∞EPN∗θ,t=μNθ=z1+z2+α12θ+α22θ1-σ2/θ-2α11θ+α21θ1-σ2/θ-m(30)limt→∞VarPN∗θ,t=σ22ξNθ=σ2-22α11θ+α21θ1-σ2/θ-mUsing Proposition3, the entropy of the worst-case model misspecification can be given:(31a)R1NQ∗θ=∫0+∞12e-rtEQ∗v1N∗2dt=σ22θ2∫0+∞e-rt4α112θEPN∗θ,t2+VarPN∗θ,t+4α11θα12θEPN∗θ,t+α122θdt(31b)R2NQ∗θ=∫0+∞12e-rtEQ∗v2N∗2dt=σ22θ2∫0+∞e-rt4α212θEPN∗θ,t2+VarPN∗θ,t+4α21θα22θEPN∗θ,t+α222θdt


## 3.1. Problem Solution

In order to get the noncooperative strategy of both regions gaming for transboundary industrial pollution, first we minimize with respect tovi of the Bellman-Isaacs condition (11) and by which we obtain(12)vi∗=-σViPθSubstituting the results of (12) into (11), differential equations (13a) and (13b) are given as(13a)rV1=maxE1≥0z1E1-12E12-g12P2+V1PE1+E2-mP+σ22V1PP-σ2V1P22θ(13b)rV2=maxE2≥0z2E2-12E22-g22P2+V2PE1+E2-mP+σ22V2PP-σ2V2P22θInvestigating differential equations (13a) and (13b), we find that they are actually the HJB equation adding a negative term, and when θ→∞, the negative term in (13a) and (13b) is also close to zero; then the robust control problems become the benchmark control problems.Next, let us use Proposition1 to investigate the noncooperative strategy and identify the game equilibrium results with subscript “N”.Proposition 1.
If both regions play a noncooperative game, their optimal instantaneous emissions can be given as follows:(14a)E1N∗=z1+2α11θP+α12θ(14b)E2N∗=z2+2α21θP+α22θwhere α11(θ), α12(θ), α21(θ), and α22(θ) are undetermined parameters related to θ.Proof.
Solving the first-order partial derivative of (13a) and (13b) with respect to E1 and E2 and setting them equal to zero, we obtain (15a) and (15b). Then solving the second-order partial derivative of (13a) and (13b) with respect to E1 and E2, (15c) and (15d) are given:(15a)∂rV1∂E1=z1-E1+V1P=0⇒E1N∗=z1+V1P(15b)∂rV2∂E2=z2-E2+V2P=0⇒E2N∗=z2+V2P(15c)∂2rV1∂E12=-1<0(15d)∂2rV2∂E22=-1<0
From (15a) and (15c), seeing that ∀δ>0, V1(E1N∗)>V1(E1N∗±δ), this means to choose the emissions levels E1N∗ are optimal for Region 1. In a similar way, from (15b) and (15d), one can find that Region 2 should choose emissions levels E2N∗ to maximum her value. Furthermore, we find from (13a) and (13b) that if there is a game equilibrium, this equilibrium must be in the case E1N∗=f1N(E2N∗), E2N∗=f2N(E1N∗), where f1N and f2N are response functions of Region 1 to Region 2 and Region 2 to Region 1 when both regions play a noncooperative strategy. In other words, the equilibrium must be a Nash equilibrium.
Substituting (15a) and (15b) into (13a) and (13b), respectively, (16a) and (16b) are obtained:(16a)rV1=z1z1+V1P-12z1+V1P2-g12P2+V1Pz1+V1P+z2+V2P-mP+σ22V1PP-σ2V1P22θ(16b)rV2=z2z2+V2P-12z2+V2P2-g22P2+V2Pz1+V1P+z2+V2P-mP+σ22V2PP-σ2V2P22θ
In order to solve the value functions (16a) and (16b), we assume(17a)V1N=α11θP2+α12θP+α13θ(17b)V2N=α21θP2+α22θP+α23θwhere α11(θ), α12(θ), α13(θ), α21(θ), α22(θ), and α23(θ) are undetermined parameters related to θ.
Maximizing with respect toP in (17a) and (17b), respectively, we obtain(18a)V1NP=∂V1N∂P=2α11θP+α12θ(18b)V1NPP=∂V1NP∂P=2α11θ(18c)V2NP=∂V2N∂P=2α21θP+α22θ(18d)V2NPP=∂V2NP∂P=2α21θ
Substituting (18a)-(18d) into (16a) and (16b), the values of undetermined parameters are given:(19a)α11=2m+r-4α21-2m+r-4α212+4g11-σ2/θ41-σ2/θ≤0(19b)α12=2α11α22+2α11z1+z22α111-σ2/θ+2α21-m-r≤0(19c)α13=1rz122+12-σ22θα122+α12z1+z2+α11α22+α11σ2
α 21 ≤ 0 can be determined by the implicit function (19d):(19d)-21+σ2θα212+2m+rσ2θα21-α214α21-2m-r2+4g11-σ2θ-g221-σ2θ=0(19e)α22=4α11α21z1+z2-2α21z1+z22α111-σ2/θ+2α21-m-r2α111-σ2/θ+2α21-m-r2α211-σ2/θ+2α11-m-r-4α11α21≤0(19f)α23=1r12z22+12-σ22θα222+z1+z2α22+α11α21+α21σ2
Investigating (19a)-(19f), it can be found that the value function is well-defined for θ>σ2 and diverges for θ=σ2. Hence the breakpoint is equal to θ_=σ2 and we from now on only consider θ>θ_=σ2.
Connecting (15a), (15c), (18a), and (18b), we get Proposition 1.
This ends the proof.Investigating the relationship between the value functions and optimal emissions of both regions, we obtain Lemma2.Lemma 2.
The value functions of both regions are inversely related to pollutant stocks; that is,∂ViN/∂P<0, i=1,2.Proof.
From(18a) and (18c), one easily obtain(20)∂V1N∂P=2α11θP+α12θ<0,∂V2N∂P=2α21θP+α22θ<0This ends the proof.


## 3.2. Characterizing the Worst-Case Pollution Accumulation Process

Substituting (18a) and (18c) into (12), we get(21a)v1N∗=-σθ2α11θP+α12θ(21b)v2N∗=-σθ2α21θP+α22θSubstituting (21a) and (21b) into (9), the pollution dynamics under model misspecification became the following form:(22)dP=E1+E2-σ2θα21θ+α22θ-m+2σ2θα11θ+α21θPdt+σdB^tThere are two negative effects of model misspecification in (22); one is the additional constant drift term equal to -σ2/θ(α12(θ)+α22(θ))>0; this suggests the presence of exogenous sources of pollution beyond those responsible for preindustrial pollution stock P0; another is the term of 2σ2/θ(α11(θ)+α21(θ))<0; this term says that the environment’s self-cleaning capacity has been reduced by an amount.The regioni (i=1,2) reacts to this worst-case scenario by adopting an emissions strategy E1N∗ and E2N∗ given by Proposition 1. Therefore, at optimality the worst-case pollution process, PN∗, can be given by the following stochastic differential equation:(23)dPN∗=E1N∗+E2N∗-mPN∗+σv1N∗+v2N∗dt+σdB^Substituting (14a), (14b), (21a), and (21b) into (23) reduces to(24)dPN∗=ξNθμNθ-PN∗+σdB^where μN(θ) and ξN(θ) are given by(25)μNθ=z1+z1+α12θ+α22θ1-σ2/θ-2α11θ+α21θ1-σ2/θ-mξNθ=-2α11θ+α21θ1-σ2θ-mNext, let us apply Proposition3 to probe the solutions of the stochastic differential equation (23).Proposition 3.
(i)
{ P N ∗ ( θ , t ) : t ≥ 0 } has expectation and variance(26)EPN∗θ,t=P0e-ξNθt+μNθ1-e-ξNθt(27)VarPN∗θ,t=σ22ξNθ1-e-2ξNθt2 (ii)
{ P N ∗ ( θ , t ) : t ≥ 0 } has a stationary distribution:(28)NNμNθ,σ22ξNθFrom Proposition3, we see that the expected value and variance of the worst-case pollution levels are decreasing in θ, and when t→∞, we obtain the following functions:(29)limt→∞EPN∗θ,t=μNθ=z1+z2+α12θ+α22θ1-σ2/θ-2α11θ+α21θ1-σ2/θ-m(30)limt→∞VarPN∗θ,t=σ22ξNθ=σ2-22α11θ+α21θ1-σ2/θ-mUsing Proposition3, the entropy of the worst-case model misspecification can be given:(31a)R1NQ∗θ=∫0+∞12e-rtEQ∗v1N∗2dt=σ22θ2∫0+∞e-rt4α112θEPN∗θ,t2+VarPN∗θ,t+4α11θα12θEPN∗θ,t+α122θdt(31b)R2NQ∗θ=∫0+∞12e-rtEQ∗v2N∗2dt=σ22θ2∫0+∞e-rt4α212θEPN∗θ,t2+VarPN∗θ,t+4α21θα22θEPN∗θ,t+α222θdt


## 4. Cooperative Arrangements

Now consider the case when both regions cooperate in pollution control. To uphold the cooperative scheme, both group rationality and individual rationality are required to be satisfied at any time.


### 4.1. Group Optimality and Cooperative State Trajectory

To secure group optimality, the participating two regions would seek to maximize their joint expected payoff by solving following stochastic control problem:(32)VP,θ=maxE1≥0,E2≥0E∫0+∞e-rtz1E1+z2E2-12E12-12E22-g1+g22P2+θv12+v222dts.t.dP=E1+E2-mP+σv1+v2dt+σdB^P0=p0,P≥0The Bellman-Isaacs condition of the robust control problem (32) is given by(33)rV=maxE1≥0,E2≥0minv1,v2z1E1+z2E2-12E12-12E22-g1+g22P2+θv12+v222+VPE1+E2-mP+σv1+v2+σ22VPPwhere VP=∂V/∂P, VPP=∂2V/∂P2.Minimizing with respect tov1 and v2 of the Bellman-Isaacs condition (33), respectively, we obtain(34)v1∗=v2∗=-σVPθSubstituting the results of (34) into (33), it takes the following form:(35)rV=maxE1≥0,E2≥0z1E1+z2E2-12E12-12E22-g1+g22P2+σ2VP2θ+VPE1+E2-mP-2σ2VPθ+σ22VPPNext, we will apply Proposition4 to investigate the cooperative strategy of both regions and identify the game equilibrium results with subscript “C”.Proposition 4.
If both regions use cooperative strategy, their optimal instantaneous emissions can be given as(36a)E1C∗=z1+2β1θP+β2θ(36b)E2C∗=z2+2β1θP+β2θwhere β1(θ), β2(θ), and β3(θ) are undetermined parameters related to θ.Proof.
Solving the first-order partial derivative of (35) with respect to E1 and E2 and setting it equal to zero, respectively, we get(37a)∂rV∂E1=z1-E1+VP=0⇒E1C∗=z1+VP(37b)∂rV∂E2=z2-E2+VP=0⇒E2C∗=z2+VP
Then identify following second-order partial derivatives:∂2(rV)/∂(E1)2=-1<0; ∂2(rV)/∂(E2)2=-1<0; ∂2(rV)/∂(E1)∂(E2)=0; ∂2(rV)/∂(E1)2∂2(rV)/∂(E2)2-∂2(rV)/∂(E1)∂(E2)>0. According to these second-order conditions, one can conclude that E1C∗ and E2C∗ maximize the cooperative value V; in other words, E1C∗ and E2C∗ are optimal for the partners.
Substituting (37a) and (37b) into (35), we get(38)rV=z1+z2-mPVP+1-σ2θVP2+σ22VPP-g1+g22P2+z12+z222
In order to solve the value functions (38), we make the following assumption:(39)VC=β1θP2+β2θP+β3θwhere undetermined parameters β1(θ), β2(θ), and β3(θ) can be given by(40a)β1θ=m+r-m+r2+81-σ2/θg1+g281-σ2/θ≤0(40b)β2θ=2β1θz1+z241-σ2/θβ1θ-m-r≤0(40c)β3θ=1rz12+z222+z1+z2β2θ+1-σ2θβ22θ+σ2β1θ
Inspecting (40a)-(40c), we find that the value function is well-defined for θ>σ2 and diverges for θ=σ2. Hence the breakpoint is equal to θ_=σ2 and we from now on only consider θ>θ_=σ2.
Differentiating (39) with respect to P yields(41)VCP=2β1θP+β2θ
Substituting the results of (41) into (37a) and (37b), Proposition 4 is obtained.
This ends the proof.Investigating the relationship between the value functions and optimal emissions of both regions under cooperative strategy, we obtain the following Lemma.Lemma 5.
(i) The difference between both regions’ optimal emissions under cooperative strategy is equal to the difference between both regions’ utility parameter; that is,E1C∗-E2C∗=z1-z2.
(ii) The cooperative total value functions are inversely related to pollutant stocks; that is,∂VC/∂P<0.Proof.
(i) From Proposition4, it is easy to get(42)E1C∗-E2C∗=z1+2β1θP+β2θ-z2+2β1θP+β2θ=z1-z2
(ii) Using (41), we obtain ∂VC/∂P=2β1(θ)P+β2(θ)<0.
This ends the proof.


### 4.2. Characterizing the Cooperative Worst-Case Pollution Accumulation Process

Substituting (41) into (34) yields(43)v1C∗=v2C∗=vC∗=-σVPθ=-σ2β1θP+β2θθSubstituting the results of (43) into (9), the pollution dynamics under model misspecification became the following form:(44)dPt=E1C+E2C-2σ2β2θθ-m+4σ2θβ1θPdt+σdB^tThere are two negative effects of model misspecification in (44); one is the additional constant drift term equal to -2σ2β2(θ)/θ>0; this suggests the presence of exogenous sources of pollution beyond those responsible for preindustrial pollution stock P0; another is the term of 4σ2/θβ1(θ)<0; this term shows that the environment’s self-cleaning capacity has been reduced by an amount.Both regions react to this worst-case scenario by adopting an emissions strategyE1C∗ and E2C∗ given by (36a) and (36b), respectively. Therefore, at optimality the worst-case pollution process PC∗ reduces to(45)dPC∗=E1C∗+E2C∗-mPC∗+σv1C∗+v2C∗dt+σdB^Substituting (36a), (36b), and (43) into (45), one gets the stochastic differential equation (46):(46)dPC∗=ξCθμCθ-PC∗dt+σdB^where μC(θ) and ξC(θ) are given by (47)μCθ=z1+z2+2β2θ1-σ2/θ-4β1θ1-σ2/θ-mξCθ=-4β1θ1-σ2θ-mNext, we rely on Proposition6 to show the solutions of the stochastic differential equation (46).Proposition 6.
The expectation, variance, and a stationary distribution of{PC∗(θ,t):t≥0} can be given as(48)EPC∗θ,t=P0e-ξCθt+μCθ1-e-ξCθt(49)VarPC∗θ,t=σ22ξCθ1-e-2ξCθ2(50)NCμCθ,σ22ξCθFrom Proposition6, we find that the expected value and variance of the worst-case pollution levels are decreasing in θ, and when t→∞, we obtain following functions:(51)limt→∞EPC∗θ,t=μCθ=z1+z2+2β2θ1-σ2/θ-4β1θ1-σ2/θ-m(52)limt→∞VarPC∗θ,t=σ22ξCθ=σ2-24β1θ1-σ2/θ-mFrom Proposition6, the entropy of the worst-case model misspecification under cooperative strategy can be given:(53)R1CQ∗θ=R2CQ∗θ=∫0+∞e-rt12ΕQ∗viC∗2dt=σ22θ2∫0+∞e-rt4β12θΕPC∗θ,t2+VarPC∗θ,t+4β1θβ2θΕPC∗θ,t+β22θdt


### 4.3. Individually Rational and Time-Consistent Imputation and Payment Distribution Mechanism

An agreed upon optimality principle must be sought to allocate the cooperative payoff. In a dynamic framework individual rationality has to be maintained at every instant of time within the cooperative durationt0,∞ along the cooperative trajectory (46). Let Xt∗ denote the set of realizable values of P∗(t) at time t generated by (46). The term Pt∗ is used to denote an element in the set Xt∗. For τ∈t0,∞, let vector υi(Pt∗)=υ1(Pt∗),υ2(Pt∗) denote the solution imputation (payoff under cooperation) over the period [τ,∞) to Region i∈1,2 given that the state is Pτ∗∈Xτ∗. Individual rationality along the cooperative trajectory requires(54)υτiτ,Pτ∗≥Vτiτ,Pτ∗,fori∈1,2where V(τ)i(τ,Pτ∗) denote the payoff to region i under noncooperation over the period. Let G(s)=G1(s),G2(s) denote the instantaneous payoff of the cooperative game at time s∈[t0,∞) for the cooperative game ΓC(Pt0∗). We apply Proposition 7 to investigate the time-consistent imputation.Proposition 7.
An instantaneous payment at timeτ∈t0,∞ equaling(55)Giτ=rξiPτ∗-ξpτ∗ipτ∗∈Xτ∗P˙τ∗pt∗t≥t0for pτ∗∈Xτ∗ and i∈1,2 yields a subgame consistent solution to the cooperative game Γc(xt0∗).Proof.
Along the cooperative trajectoryP(t)∗t≥t0, we define(56)Υiτ,τ,Pτ∗=υiPτ∗=∫τ∞Gise-rs-τds;Υiτ,t,Pt∗=∫t∞Gise-rs-τdsfort≥τNote that(57)Υiτ,t,Pt∗=e-rt-τ∫t∞Gise-rs-tds=e-rt-τΥit,t,Pt∗
Expression (57) means that the extension of the solution policy to a situation with a later starting time and along the optimal trajectory remains optimal, so condition in (57) guarantees time consistency of the solution imputations.
SinceΥk(t;t,xt∗) is continuously differentiable in t and Pt∗, we get(58)Υkτ;τ,Pτ∗=∫ττ+ΔtGise-rs-τds+e-rΔtΥiτ+Δt;τ+Δt,Pτ∗+ΔPτ∗,forτ∈t0,Tandi∈1,2where ΔPτ∗=P˙τ∗Δt+ο(Δt) and ο(Δt)/Δt→0 as Δt→0.
SinceΔPτ∗=Pτ+Δt∗, from (58), one can obtain(59)Υkτ;τ+Δt,Pτ+Δt∗=e-rΔtυiPτ+Δt∗=e-rΔtΥiτ+Δt;τ+Δt,Pτ∗+ΔPτ∗
Therefore, (58) can be expressed as(60)Υkτ;τ,Pτ∗=∫ττ+ΔtGise-rs-τds+Υiτ;τ+Δt,Pτ+Δt∗
There is a relationship as follows:(61)∫ττ+ΔtGise-rs-τds=Υiτ;τ,Pτ∗-Υiτ;τ+Δt,Pτ+Δt∗
WhenΔt→0, (61) can be expressed as(62)GiτΔt=-Υiτ;t,Pt∗t=τΔt-Υiτ;t,Pt∗t=τP˙τ∗Δt-οΔt
Dividing (62) throughout by Δt, with Δt→0, we have(63)Giτ=-Υiτ;t,Pt∗t=τ-Υiτ;t,Pt∗t=τP˙τ∗
Appling (57), we obtain Υi(τ;t,Pt∗)=e-r(t-τ)υi(Pτ∗), and Υi(τ;τ,Pt∗)=υi(Pτ∗). Then (63) can be converted to (55). Hence Proposition 7 follows.
Next, we consider time consistent solutions under specific optimality principles. From Yeung and Petrosyan [1], let us use “the principle of equality” to build a payment distribution mechanism under which both regions’ expected gain from cooperation is shared proportionally to the regions’ relative sizes of expected noncooperative payoffs. In accordance with the principle of equality, the imputation scheme has to satisfy the following.
In the gameΓC(P0), at time t0 and at time τ∈t0,∞, an imputation in shown as (64) and (65), respectively:(64)υiP0=ViNP0+12VCP0-∑j=12VjNP0(65)υiPτ∗=ViNPτ∗+12VCPτ∗-∑j=12VjNPτ∗where i∈1,2. Appling Proposition 7, we obtain an instantaneous payoff of the cooperative game at time τ∈t0,∞:(66)Giτ=12rViNPτ∗-ViNPτ∗Pτ∗P˙τ∗+12rVCPτ∗-VCPτ∗Pτ∗P˙τ∗-12rVjNPτ∗-VjNPτ∗Pτ∗P˙τ∗
The specific payment imputation in the gameΓC(P0) can be given by using the distribution mechanism (66) and the value functions (17a), (17b), and (39) at time t0, respectively:(67a)NV1=12β1θPC2+β2θPC+β3+α11θPN2+α12θPN+α13θ-α21θPN2-α22θPN-α23θ(67b)NV2=12β1θPC2+β2θPC+β3+α21θPN2+α22θPN+α23θ-α11θPN2-α12θPN-α13θ
Having obtained the cooperative and noncooperative strategy and cooperative residual distribution mechanism applied to satisfy individual rational condition, next, some numerical examples are wielded to investigate the results of model analysis.


## 4.1. Group Optimality and Cooperative State Trajectory

To secure group optimality, the participating two regions would seek to maximize their joint expected payoff by solving following stochastic control problem:(32)VP,θ=maxE1≥0,E2≥0E∫0+∞e-rtz1E1+z2E2-12E12-12E22-g1+g22P2+θv12+v222dts.t.dP=E1+E2-mP+σv1+v2dt+σdB^P0=p0,P≥0The Bellman-Isaacs condition of the robust control problem (32) is given by(33)rV=maxE1≥0,E2≥0minv1,v2z1E1+z2E2-12E12-12E22-g1+g22P2+θv12+v222+VPE1+E2-mP+σv1+v2+σ22VPPwhere VP=∂V/∂P, VPP=∂2V/∂P2.Minimizing with respect tov1 and v2 of the Bellman-Isaacs condition (33), respectively, we obtain(34)v1∗=v2∗=-σVPθSubstituting the results of (34) into (33), it takes the following form:(35)rV=maxE1≥0,E2≥0z1E1+z2E2-12E12-12E22-g1+g22P2+σ2VP2θ+VPE1+E2-mP-2σ2VPθ+σ22VPPNext, we will apply Proposition4 to investigate the cooperative strategy of both regions and identify the game equilibrium results with subscript “C”.Proposition 4.
If both regions use cooperative strategy, their optimal instantaneous emissions can be given as(36a)E1C∗=z1+2β1θP+β2θ(36b)E2C∗=z2+2β1θP+β2θwhere β1(θ), β2(θ), and β3(θ) are undetermined parameters related to θ.Proof.
Solving the first-order partial derivative of (35) with respect to E1 and E2 and setting it equal to zero, respectively, we get(37a)∂rV∂E1=z1-E1+VP=0⇒E1C∗=z1+VP(37b)∂rV∂E2=z2-E2+VP=0⇒E2C∗=z2+VP
Then identify following second-order partial derivatives:∂2(rV)/∂(E1)2=-1<0; ∂2(rV)/∂(E2)2=-1<0; ∂2(rV)/∂(E1)∂(E2)=0; ∂2(rV)/∂(E1)2∂2(rV)/∂(E2)2-∂2(rV)/∂(E1)∂(E2)>0. According to these second-order conditions, one can conclude that E1C∗ and E2C∗ maximize the cooperative value V; in other words, E1C∗ and E2C∗ are optimal for the partners.
Substituting (37a) and (37b) into (35), we get(38)rV=z1+z2-mPVP+1-σ2θVP2+σ22VPP-g1+g22P2+z12+z222
In order to solve the value functions (38), we make the following assumption:(39)VC=β1θP2+β2θP+β3θwhere undetermined parameters β1(θ), β2(θ), and β3(θ) can be given by(40a)β1θ=m+r-m+r2+81-σ2/θg1+g281-σ2/θ≤0(40b)β2θ=2β1θz1+z241-σ2/θβ1θ-m-r≤0(40c)β3θ=1rz12+z222+z1+z2β2θ+1-σ2θβ22θ+σ2β1θ
Inspecting (40a)-(40c), we find that the value function is well-defined for θ>σ2 and diverges for θ=σ2. Hence the breakpoint is equal to θ_=σ2 and we from now on only consider θ>θ_=σ2.
Differentiating (39) with respect to P yields(41)VCP=2β1θP+β2θ
Substituting the results of (41) into (37a) and (37b), Proposition 4 is obtained.
This ends the proof.Investigating the relationship between the value functions and optimal emissions of both regions under cooperative strategy, we obtain the following Lemma.Lemma 5.
(i) The difference between both regions’ optimal emissions under cooperative strategy is equal to the difference between both regions’ utility parameter; that is,E1C∗-E2C∗=z1-z2.
(ii) The cooperative total value functions are inversely related to pollutant stocks; that is,∂VC/∂P<0.Proof.
(i) From Proposition4, it is easy to get(42)E1C∗-E2C∗=z1+2β1θP+β2θ-z2+2β1θP+β2θ=z1-z2
(ii) Using (41), we obtain ∂VC/∂P=2β1(θ)P+β2(θ)<0.
This ends the proof.


## 4.2. Characterizing the Cooperative Worst-Case Pollution Accumulation Process

Substituting (41) into (34) yields(43)v1C∗=v2C∗=vC∗=-σVPθ=-σ2β1θP+β2θθSubstituting the results of (43) into (9), the pollution dynamics under model misspecification became the following form:(44)dPt=E1C+E2C-2σ2β2θθ-m+4σ2θβ1θPdt+σdB^tThere are two negative effects of model misspecification in (44); one is the additional constant drift term equal to -2σ2β2(θ)/θ>0; this suggests the presence of exogenous sources of pollution beyond those responsible for preindustrial pollution stock P0; another is the term of 4σ2/θβ1(θ)<0; this term shows that the environment’s self-cleaning capacity has been reduced by an amount.Both regions react to this worst-case scenario by adopting an emissions strategyE1C∗ and E2C∗ given by (36a) and (36b), respectively. Therefore, at optimality the worst-case pollution process PC∗ reduces to(45)dPC∗=E1C∗+E2C∗-mPC∗+σv1C∗+v2C∗dt+σdB^Substituting (36a), (36b), and (43) into (45), one gets the stochastic differential equation (46):(46)dPC∗=ξCθμCθ-PC∗dt+σdB^where μC(θ) and ξC(θ) are given by (47)μCθ=z1+z2+2β2θ1-σ2/θ-4β1θ1-σ2/θ-mξCθ=-4β1θ1-σ2θ-mNext, we rely on Proposition6 to show the solutions of the stochastic differential equation (46).Proposition 6.
The expectation, variance, and a stationary distribution of{PC∗(θ,t):t≥0} can be given as(48)EPC∗θ,t=P0e-ξCθt+μCθ1-e-ξCθt(49)VarPC∗θ,t=σ22ξCθ1-e-2ξCθ2(50)NCμCθ,σ22ξCθFrom Proposition6, we find that the expected value and variance of the worst-case pollution levels are decreasing in θ, and when t→∞, we obtain following functions:(51)limt→∞EPC∗θ,t=μCθ=z1+z2+2β2θ1-σ2/θ-4β1θ1-σ2/θ-m(52)limt→∞VarPC∗θ,t=σ22ξCθ=σ2-24β1θ1-σ2/θ-mFrom Proposition6, the entropy of the worst-case model misspecification under cooperative strategy can be given:(53)R1CQ∗θ=R2CQ∗θ=∫0+∞e-rt12ΕQ∗viC∗2dt=σ22θ2∫0+∞e-rt4β12θΕPC∗θ,t2+VarPC∗θ,t+4β1θβ2θΕPC∗θ,t+β22θdt


## 4.3. Individually Rational and Time-Consistent Imputation and Payment Distribution Mechanism

An agreed upon optimality principle must be sought to allocate the cooperative payoff. In a dynamic framework individual rationality has to be maintained at every instant of time within the cooperative durationt0,∞ along the cooperative trajectory (46). Let Xt∗ denote the set of realizable values of P∗(t) at time t generated by (46). The term Pt∗ is used to denote an element in the set Xt∗. For τ∈t0,∞, let vector υi(Pt∗)=υ1(Pt∗),υ2(Pt∗) denote the solution imputation (payoff under cooperation) over the period [τ,∞) to Region i∈1,2 given that the state is Pτ∗∈Xτ∗. Individual rationality along the cooperative trajectory requires(54)υτiτ,Pτ∗≥Vτiτ,Pτ∗,fori∈1,2where V(τ)i(τ,Pτ∗) denote the payoff to region i under noncooperation over the period. Let G(s)=G1(s),G2(s) denote the instantaneous payoff of the cooperative game at time s∈[t0,∞) for the cooperative game ΓC(Pt0∗). We apply Proposition 7 to investigate the time-consistent imputation.Proposition 7.
An instantaneous payment at timeτ∈t0,∞ equaling(55)Giτ=rξiPτ∗-ξpτ∗ipτ∗∈Xτ∗P˙τ∗pt∗t≥t0for pτ∗∈Xτ∗ and i∈1,2 yields a subgame consistent solution to the cooperative game Γc(xt0∗).Proof.
Along the cooperative trajectoryP(t)∗t≥t0, we define(56)Υiτ,τ,Pτ∗=υiPτ∗=∫τ∞Gise-rs-τds;Υiτ,t,Pt∗=∫t∞Gise-rs-τdsfort≥τNote that(57)Υiτ,t,Pt∗=e-rt-τ∫t∞Gise-rs-tds=e-rt-τΥit,t,Pt∗
Expression (57) means that the extension of the solution policy to a situation with a later starting time and along the optimal trajectory remains optimal, so condition in (57) guarantees time consistency of the solution imputations.
SinceΥk(t;t,xt∗) is continuously differentiable in t and Pt∗, we get(58)Υkτ;τ,Pτ∗=∫ττ+ΔtGise-rs-τds+e-rΔtΥiτ+Δt;τ+Δt,Pτ∗+ΔPτ∗,forτ∈t0,Tandi∈1,2where ΔPτ∗=P˙τ∗Δt+ο(Δt) and ο(Δt)/Δt→0 as Δt→0.
SinceΔPτ∗=Pτ+Δt∗, from (58), one can obtain(59)Υkτ;τ+Δt,Pτ+Δt∗=e-rΔtυiPτ+Δt∗=e-rΔtΥiτ+Δt;τ+Δt,Pτ∗+ΔPτ∗
Therefore, (58) can be expressed as(60)Υkτ;τ,Pτ∗=∫ττ+ΔtGise-rs-τds+Υiτ;τ+Δt,Pτ+Δt∗
There is a relationship as follows:(61)∫ττ+ΔtGise-rs-τds=Υiτ;τ,Pτ∗-Υiτ;τ+Δt,Pτ+Δt∗
WhenΔt→0, (61) can be expressed as(62)GiτΔt=-Υiτ;t,Pt∗t=τΔt-Υiτ;t,Pt∗t=τP˙τ∗Δt-οΔt
Dividing (62) throughout by Δt, with Δt→0, we have(63)Giτ=-Υiτ;t,Pt∗t=τ-Υiτ;t,Pt∗t=τP˙τ∗
Appling (57), we obtain Υi(τ;t,Pt∗)=e-r(t-τ)υi(Pτ∗), and Υi(τ;τ,Pt∗)=υi(Pτ∗). Then (63) can be converted to (55). Hence Proposition 7 follows.
Next, we consider time consistent solutions under specific optimality principles. From Yeung and Petrosyan [1], let us use “the principle of equality” to build a payment distribution mechanism under which both regions’ expected gain from cooperation is shared proportionally to the regions’ relative sizes of expected noncooperative payoffs. In accordance with the principle of equality, the imputation scheme has to satisfy the following.
In the gameΓC(P0), at time t0 and at time τ∈t0,∞, an imputation in shown as (64) and (65), respectively:(64)υiP0=ViNP0+12VCP0-∑j=12VjNP0(65)υiPτ∗=ViNPτ∗+12VCPτ∗-∑j=12VjNPτ∗where i∈1,2. Appling Proposition 7, we obtain an instantaneous payoff of the cooperative game at time τ∈t0,∞:(66)Giτ=12rViNPτ∗-ViNPτ∗Pτ∗P˙τ∗+12rVCPτ∗-VCPτ∗Pτ∗P˙τ∗-12rVjNPτ∗-VjNPτ∗Pτ∗P˙τ∗
The specific payment imputation in the gameΓC(P0) can be given by using the distribution mechanism (66) and the value functions (17a), (17b), and (39) at time t0, respectively:(67a)NV1=12β1θPC2+β2θPC+β3+α11θPN2+α12θPN+α13θ-α21θPN2-α22θPN-α23θ(67b)NV2=12β1θPC2+β2θPC+β3+α21θPN2+α22θPN+α23θ-α11θPN2-α12θPN-α13θ
Having obtained the cooperative and noncooperative strategy and cooperative residual distribution mechanism applied to satisfy individual rational condition, next, some numerical examples are wielded to investigate the results of model analysis.


## 5. Numerical Examples

In this section, we will apply the solutions of model reached at Sections3 and 4 to perform a numerical exercise that provides some context for the theoretical results. The parameters used in the numerical examples are presented in Table 1; in particular, we assume the robust control parameter θ equal to 0.9, 1.9, 2.9, and 3.9, respectively. In theory, the greater the value of robust control parameter θ, the more cautious the region. We use version 7.0 of the Wolfram Mathematica Matlab to obtain the numerical solutions.Table 1
The parameters used in the numerical example.
r m z 1 z 2 g 1 g 2 P 0 σ θ 0.08 0.06 11.8 11.3 0.02 0.022 80 0.23 0.9;1.9; 2.9; 3.9First, through Figures1 and 2, we investigate the effects of robust control parameter θ on the v1N∗ and v2N∗.Figure 1
The optimal paths of the time evolutions ofv1N∗.Figure 2
The optimal paths of the time evolutions ofv2N∗.From Figures1 and 2, we find that v1N∗ and v2N∗ are convex and decreasing with time t and θ has significant influence on v1N∗ and v2N∗. The smaller the value of θ, the greater the changes in time evolution of v1N∗ and v2N∗, which reflects that the more cautious the region, the smaller the worst-case misspecification.Next, the effects of robust control parameterθ on optimal emission strategy of both regions are shown in Figures 3 and 4.Figure 3
The optimal paths of the time evolutions ofE1N∗.Figure 4
The optimal paths of the time evolutions ofE2N∗.From Figures3 and 4, we find that E1N∗ and E2N∗ are concave and decreasing with time t and θ has significant influence on E1N∗ and E2N∗. The smaller the value of θ, the greater the change in the time evolution of E1N∗ and E2N∗ which reflects that the more cautious the region, the smaller the amount of optimal emission of both regions.Next, in Figure5 the optimal time evolution paths of the pollution stock are shown under different robust control parameters θ.Figure 5
The optimal paths of the time evolutions ofPN∗.Figure5 shows that optimal pollution stock is concave and decreasing with time t and θ has significant influence on it. The smaller the value of θ, the greater the change in the time evolution of the optimal pollution stock which reflects that the more cautious the region, the smaller the amount of the optimal pollution stock; that is to say, the precautious participant tends to reduce pollution emissions.The noncooperative emissions strategy and gaming results have been shown in Figures3–5. Next, we use Figures 6–8 to carry out numerical analysis of cooperative emissions strategy and gaming results.Figure 6
The optimal paths of the time evolutions ofE1C∗.Figure 7
The optimal paths of the time evolutions ofE2C∗.Figure 8
The optimal paths of the time evolutions ofPC∗.Figures6-7 show that both regions’ optimal emission levels are concave and decreasing with time t under cooperation and the optimal emission levels tend to reduce with the increasing of robust control parameters θ. Figure 8 shows that the optimal pollution stock is convex and decreasing with time t.From Figures1–8, one can see the effects of robust control parameters θ on the behavior of Region i, i∈1,2, in cooperation and noncooperation fully. In order to contrast the cooperative strategy and noncooperative strategy, we perform several analyses shown in Figures 9–12, in which the robust control parameters θ are fixed at 0.9.Figure 9
The optimal paths of the time evolution ofE1N∗ and E1C∗ when θ=0.9.Figure 10
The optimal paths of the time evolution ofE2N∗ and E2C∗ when θ=0.9.Figure 11
The optimal paths of the time evolution ofV1N∗ and NV1∗ when θ=0.9.Figure 12
The optimal paths of the time evolution ofV2N∗ and NV2∗ when θ=0.9.Figures9-10 show that, compared to noncooperative strategy, both regions’ optimal emission levels are lower when both regions play a cooperative game.Figures11-12 display that at any point in time, compared to noncooperative strategy, both regions earn higher net benefits when they play a cooperative game. This means that the individual rationality condition and group rationality condition are satisfied simultaneously. Therefore, our cooperative residual distribution mechanism given in Section 4 can ensure a subgame-consistent solution.


## 6. Conclusion

In this paper, we apply the robust control framework of Hansen and Sargent [8] to investigate a stochastic differential game of transboundary pollution between two regions under Knightian uncertainty of stock dynamics. Both regions are presumed to play a Markov Nash equilibrium strategy and a cooperative strategy and the worst-case pollution accumulation processes for discrete robustness parameters are characterized. The results show the following.(i) Under Knightian uncertainty of stock dynamics, when the regions get more cautious, the model miscalculation reduces and the time path of each region’s optimal emission levels is significantly decreasing.(ii) Compared to noncooperative strategy, both regions’ optimal emission levels are lower when both regions play a cooperative game.(iii) Compared to noncooperative strategy, both regions earn higher net benefits when they play a cooperative game.(iv) Under the mechanism that the expected gain from cooperation is shared proportionally to the regions’ relative sizes of expected noncooperative payoffs, the individual rationality condition and group rationality condition can be satisfied simultaneously which ensures a subgame-consistent solution.



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