  This paper addresses a continuous-time risk-minimizing two-player zero-sum
stochastic differential game (SDG), in which each player aims to minimize its
probability of failure. Failure occurs in the event when the state of the game
enters into predefined undesirable domains, and one player's failure is the
other's success. We derive a sufficient condition for this game to have a
saddle-point equilibrium and show that it can be solved via a
Hamilton-Jacobi-Isaacs (HJI) partial differential equation (PDE) with Dirichlet
boundary condition. Under certain assumptions on the system dynamics and cost
function, we establish the existence and uniqueness of the saddle-point of the
game. We provide explicit expressions for the saddle-point policies which can
be numerically evaluated using path integral control. This allows us to solve
the game online via Monte Carlo sampling of system trajectories. We implement
our control synthesis framework on two classes of risk-minimizing zero-sum
SDGs: a disturbance attenuation problem and a pursuit-evasion game. Simulation
studies are presented to validate the proposed control synthesis framework.
