No abstract

We formulate a stochastic differential game in continuous time that represents the unique viscosity solution to a terminal value problem for a parabolic partial differential equation involving the normalized [Formula: see text]-Laplace operator. Our game is formulated in a way that covers the full range [Formula: see text]. Furthermore, we prove the uniqueness of viscosity solutions to our equation in the whole space under suitable assumptions.

Abstract. We show that a uniform measure density condition implies game regularity for all 2 < p < ∞ in a stochastic game called 'tug-of-war with noise'. The proof utilizes suitable choices of strategies combined with estimates for the associated stopping times and density estimates for the sum of independent and identically distributed random vectors.