Abstract
This work presents a novel policy iteration algorithm to tackle nonzero-sum stochastic impulse games arising naturally in many applications. Despite the obvious impact of solving such problems, there are no suitable numerical methods available, to the best of our knowledge. Our method relies on the recently introduced characterisation of the value functions and Nash equilibrium via a system of quasi-variational inequalities. While our algorithm is heuristic and we do not provide a convergence analysis, numerical tests show that it performs convincingly in a wide range of situations, including the only analytically solvable example available in the literature at the time of writing.
Highlights
Stochastic impulse games (SIGs) are at the intersection between differential game theory and stochastic impulse control
In the case of one sole player, they reduce to stochastic impulse optimisation problems where an agent seeks to control an underlying—otherwise governed by a stochastic differential equation—in order to maximize the expected value of some target functional
A rigorous framework has been put in place—combining Howard’s algorithm [3,4,5, 9] with viscosity-capturing finite difference schemes—which allows for robust numerical approximations [7, 8, 10]
Summary
Stochastic impulse games (SIGs) are at the intersection between differential game theory and stochastic impulse control. The control-theoretical problem typically boils down to determining an optimal strategy as a function of the current value of the underlying. Optimality can naturally be characterized via the notion of Nash equilibrium: intuitively, the pair of value functions displays the best expected outcome in the sense that if a player changes her strategy while her opponent does not, the former can only be worse off. We make the first attempt (to the best of our knowledge) at numerically solving NZSSIGs. In a nutshell, the iterative algorithm we put forward treats the NZSSIG at each iteration as a combination of a fixed point problem and a slowly relaxing one-player SIG.
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