Densities for scalar-valued BSDEs via unique continuation and backward uniqueness

Abstract

We give sufficient conditions ensuring that, at every fixed positive time, the scalar backward component of a Markovian forward-backward stochastic differential equation with multidimensional forward process admits a density with respect to Lebesgue measure. Existing density criteria for BSDEs often obtain Malliavin non-degeneracy through sign or monotonicity assumptions. We develop a different route for a scalar backward component with an arbitrary-dimensional forward state. Under regularity assumptions and a structural compatibility condition on the generator, the terminal condition is only required to be non-constant. The key idea is to deduce Malliavin non-degeneracy from deterministic rigidity of the critical set of the decoupling field. We combine Malliavin calculus with unique continuation and backward uniqueness for the associated semilinear parabolic equation. Unique continuation precludes the spatial gradient of the decoupling field from vanishing on a set of positive measure unless it vanishes identically on that time slice, while backward uniqueness propagates such vanishing to the terminal time. Along the way, we establish a unique continuation property from sets of positive measure and a backward uniqueness result on the whole space for the linear parabolic systems arising from differentiated semilinear equations.