A model of strategic sustainable investment

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.

💡 Research Summary

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The paper develops a continuous‑time, infinite‑horizon, non‑zero‑sum stochastic game that captures the strategic interaction between a sustainable investor and a privately‑owned firm. The firm produces a single good and its instantaneous profit equals its production capacity (X_t). The firm can also allocate a flow of resources to emission‑reduction activities, measured by the cumulative expenditure (R_t). The investor can inject capital directly into the firm, increasing its production capacity, and evaluates the firm’s performance through a profit function that depends on both the financial output (X_t) and the environmental outcome (R_t).

State dynamics
Without any control the capacity follows a geometric Brownian motion:
(dX_t = \mu X_t dt + \sigma X_t dB_t).
The investor’s control (\nu_t) is a non‑decreasing, càdlàg, singular (lumpy) process; each increment raises the capacity instantly. The firm’s control (\eta_t) is a regular bounded rate ( (0\le \eta_t\le \eta_{\max}) ) that reduces the capacity because greener technologies or clean‑energy mixes are costlier. The controlled dynamics become
(dX_t = \mu X_t dt + \sigma X_t dB_t + d\nu_t - \eta_t dt,)
(dR_t = \eta_t dt.)

Objectives
The firm maximises the discounted expected utility of its profit:
(J_F = \mathbb{E}\big