[Paper Review] Nonzero-sum stochastic differential games with impulse controls : a verification theorem with applications

René Aïd, Matteo Basei+3 more|arXiv (Cornell University)|Apr 29, 2016

Stochastic processes and financial applications23 references46 citations

TL;DR

This paper develops a verification theorem for nonzero-sum stochastic differential games with impulse controls, establishing a system of quasi-variational inequalities (QVIs) that characterize Nash equilibria. It applies this framework to a one-dimensional impulse game with linear running payoffs, explicitly solving for a family of Nash equilibria and deriving closed-form expressions for equilibrium strategies and payoffs, with asymptotic analysis of intervention costs and numerical solutions for non-symmetric cases including cubic payoffs.

ABSTRACT

We consider a general nonzero-sum impulse game with two players. The main mathematical contribution of the paper is a verification theorem which provides, under some regularity conditions, a suitable system of quasi-variational inequalities for the value functions and the optimal strategies of the two players. As an application, we study an impulse game with a one-dimensional state variable, following a real-valued scaled Brownian motion, and two players with linear and symmetric running payoffs. We fully characterize a Nash equilibrium and provide explicit expressions for the optimal strategies and the value functions. We also prove some asymptotic results with respect to the intervention costs. Finally, we consider two further non-symmetric examples where a Nash equilibrium is found numerically.

Motivation & Objective

• To develop a general verification theorem for nonzero-sum stochastic differential games with impulse controls.
• To characterize Nash equilibria via a system of quasi-variational inequalities (QVIs) under regularity and growth conditions.
• To solve explicitly a symmetric one-dimensional impulse game with linear running payoffs and impulse costs.
• To analyze asymptotic behavior of equilibria with respect to intervention costs.
• To extend the framework to non-symmetric cases with cubic and mixed linear-cubic payoffs, solved numerically.

Proposed method

• Formulate a two-player nonzero-sum impulse game with state dynamics governed by a scaled Brownian motion.
• Define player payoffs as discounted integrals of running gains/losses, plus discrete impulse rewards and penalties, with priority rules for simultaneous interventions.
• Derive a system of coupled QVIs involving infinitesimal generator A, intervention operators Mi and Hi, and payoff functions Vi.
• Establish a verification theorem: if Vi solve the QVI system and satisfy regularity and growth conditions, they represent equilibrium payoffs.
• Apply the theorem to a symmetric linear case by constructing candidate solutions and verifying QVI conditions via analytical and monotonicity arguments.
• For non-symmetric cases (cubic, mixed), solve the 8-equation QVI system numerically and verify sufficient conditions for equilibrium.

Experimental results

Research questions

• RQ1 Under what conditions can a system of quasi-variational inequalities characterize a Nash equilibrium in a nonzero-sum impulse game with two players?
• RQ2 What explicit form do equilibrium strategies and payoffs take in a one-dimensional symmetric impulse game with linear running payoffs?
• RQ3 How do intervention costs affect the structure and existence of Nash equilibria in such games?
• RQ4 Can the verification theorem be applied to non-symmetric payoff structures, such as cubic or mixed linear-cubic payoffs?
• RQ5 What numerical methods are effective for solving the resulting system of QVIs when analytical solutions are not available?

Key findings

• A family of Nash equilibria exists in the symmetric linear case, characterized by player 1 intervening when X < ¯x1 and moving it to x∗1, and player 2 when X > ¯x2, moving it to x∗2.
• Explicit expressions are derived for ¯x1, ¯x2, x∗1, x∗2, and the equilibrium payoffs V1 and V2 in terms of model parameters including ρ, σ, c, ˜c, λ, ˜λ, s1, s2.
• When c = ˜c and λ = ˜λ, no admissible Nash equilibrium exists, indicating a critical threshold in cost symmetry.
• The continuation region ]¯x1, ¯x2[ shifts rightward as ˜s increases, and V2(x) is decreasing in ˜s with limits V+∞2 (x) = −∞ and V−∞2 (x) = +∞.
• For cubic payoffs, a numerical solution to the 8-equation QVI system yields ¯x1 = −0.732, ¯x2 = 0.464, x∗1 = 0.186, x∗2 = −0.453 under specific parameter values.
• In the mixed linear-cubic case, the continuation region ]¯x1, ¯x2[ is closer to s1 than to s2, reflecting higher incentives for player 1 to intervene due to steeper payoff curvature.