A model of strategic sustainable investment

TL;DR

This paper develops a continuous-time, nonzero-sum stochastic differential game between a sustainable investor and a privately owned firm to study irreversible investment and emission abatement. It establishes a general verification theorem for a two-variable VI system and derives an explicit equilibrium in the zero-noise limit, revealing moving boundaries $a(r)$ and $b(r)$ that trigger actions. A policy-iteration–type numerical algorithm is proposed to construct equilibria in the full stochastic setting, with results showing qualitative overlap and incremental activation of the players, driven by the abatement level $R_t$. The study highlights how environmental considerations can shape incentive-compatible investment strategies, offering a framework with direct implications for impact investing and climate finance. The moving-boundary structure provides new insights into how ESG objectives can be aligned with financial performance in dynamic investment contexts.

Abstract

We study a problem of optimal irreversible investment and emission reduction formulated as a nonzero-sum dynamic game between an investor with environmental preferences and a firm. The game is set in continuous time on an infinite-time horizon. The firm generates profits with a stochastic dynamics and may spend part of its revenues towards emission reduction (e.g., renovating the infrastructure). The firm's objective is to maximize the discounted expectation of a function of its profits. The investor participates in the profits, may decide to invest to support the firm's production capacity and uses a profit function which accounts for both financial and environmental factors. Nash equilibria of the game are obtained via a system of variational inequalities. We formulate a general verification theorem for this system in a diffusive setup and construct an explicit solution in the zero-noise limit. Our explicit results and numerical approximations show that both the investor's and the firm's optimal actions are triggered by moving boundaries that increase with the total amount of emission abatement.

Figures (9)

• Figure 1: Illustration of the optimal strategies in the deterministic case with $\mu<0$. Left: $X$ as function of $R$. Right: $X$ (in blue) and $R$ (in red) as function of time. See text for a detailed discussion.
• Figure 2: Illustration of the optimal strategies in the stochastic case. Top graph: production capacity ($X$) and the two boundaries as function of time. Bottom graph: optimal investment $\nu^*$ (thick blue line with left scale) and abatement $R$ (thin red line, right scale) as function of time.
• Figure 3: Comparison between functions $a(r)$ and $b(r)$ for two different values of the parameter $\gamma$ (which determines the sensitivity of investor to the environmental impact of the company). The parameters are: $\rho = \bar{\rho} = 0.3$, $\mu=-0.0741$, $\beta=0.5$, $\gamma=0.35$ (Figure \ref{['figd001']}) and $\gamma=0.55$ (Figure \ref{['figd002']}). The function $a(r)$ is given by \ref{['eq:nua2']}, while the boundary $b(r)$ is described in Proposition \ref{['Proposition_funcb']}. For higher value of $\gamma$, the investors are more concerned about the environmental impact of the company, and both abatement and investment occur earlier (for higher values of the environmental performance.)
• Figure 4: Firm and investor equilibrium expected payoff functions for the deterministic setting discussed in Section \ref{['sec:detmunegative']}, with $\mu=-0.0741$, $\rho=\bar{\rho} = 0.3$, $\beta=0.5$ and $\gamma=0.55$. Figure \ref{['figd01']} shows $w(r,x)$ and Figure \ref{['figd02']} shows $v(r,x)$, for $x > a(r)$.
• Figure 5: Function $\hat{a}(r)$, as in \ref{['eqva0']}, evaluated for $\rho=\bar{\rho} = 0.3$, $\mu=0.0741$, $\sigma=0.3703$, $\beta=0.55$, and $\gamma = \{ 0.1, 0.25, 0.3, 0.45, 0.6, 0.75, 0.9 \}$.

Theorems & Definitions (41)

• Definition 1
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Theorem 1: Equilibrium in the zero noise limit
• Theorem 2: Verification
• proof : Proof of Theorem \ref{['thm:verif']}
• Corollary 1
• Remark 5: Verification with deterministic dynamics