Saddle-Point Structures for Two-Player Zero-Sum SDEs with Lévy Noise

We extend the zero-sum stochastic differential game to a state process driven by Lévy noise. With the generator and standing assumptions of the resolvent paper, the lower and upper values have at most linear growth and are Lipschitz in the state, and under the Isaacs condition they coincide as the unique viscosity solution of at most linear growth of a Hamilton–Jacobi–Isaacs integro-differential equation. Saddle strategies satisfy a first-order condition that is non-local in the attacker’s control: it integrates the value against the jump measure the attacker controls. A central-bank-versus-speculator game on a Kou state has a constant saddle pair and an affine value in closed form, which a numerical evaluation of the Isaacs equation and an exact Monte Carlo at an exponential time both confirm.