Magnus Perninge: BSDEs with constrained jumps, double obstacle PDEs and games of impulse control

Stochastic differential games in which both players implement impulse controls have important applications in areas such as finance, energy, and engineering. Although this area has attracted considerable attention in recent years, comprehensive theoretical results remain largely confined to zero-sum settings (where one player’s loss equals the other’s gain) and often rely on restrictive assumptions on the model coefficients.

In this talk, we approach the aforementioned problem by first demonstrating a link between optimal stopping for backward stochastic differential equations (BSDEs) with jumps and viscosity solutions of double obstacle partial differential equations (PDEs). To obtain the second obstacle, we consider BSDEs whose jumps are constrained by an upper barrier. Using a control randomization argument, we show that the solution to the corresponding optimal stopping problem coincides with both the upper and lower value functions of a zero-sum game of impulse control in a general setting, thereby providing a solution to the game.

The proposed framework offers a new probabilistic approach to stochastic differential games and appears promising for extension to the non-zero-sum case. Moreover, the representation we obtain naturally connects with recent machine learning approaches, opening the door to efficient numerical approximations even in high-dimensional problems.