An Impulsive Differential Game Arising in Finance with Interesting Singularities

We investigate a differential game motivated by a problem in mathematical finance. This game displays two interesting features. On the one hand, one of the players,

P

ursuer say, may, and will, use infinitely large controls, i.e., impulses, producing “jumps” in the state variables. Standard optimal trajectories are made of such a jump followed by a “coasting period” where

P

exerts no control. This leads to barriers of a somewhat new type. But because the cost of jumps is only proportional to their amplitude, some singular optimal trajectories arise where

P

uses an intermediary control, nonzero but finite. (In classical impulse control, there is a minimum positive cost to any use of the control, forbidding such a mixed situation.)

On the other hand, the complete solution of the game exhibits a type of singularity, the existence of which had long been conjectured (noticeably by Arik Melikyan in discussions with the first author) but, as far as we know, never shown in actual examples: a two-dimensional focal manifold traversed by noncollinear optimal fields depending on the control used by

E

vader. It is on this manifold that intermediary controls for

P

arise.

Finally, we show that the Isaacs equation of a discrete-time version of the problem provides a discretization scheme that converges to the value function of the differential game. This is done through the investigation of a (degenerate) quasi-variational inequality and its viscosity solution, with the help of an equivalent, but nonimpulsive, differential game—a method of interest per se that we credit to Joshua—to which we apply essentially the classical method of Capuzzo Dolcetta extended to differential games by Pourtallier and Tidball, with some technical adaptations.