An existence theorem for weak solutions of stochastic differential equations with discontinuous right-hand sides and with reflection at the boundary

Abstract

We consider the stochastic differential equation x(t) = x0 + t∫ 0 f (τ, x(τ ))dτ + t∫ 0 g(τ, x(τ ))dW (τ ) + K(t) (1) in a domain D with reflection at the boundary. Here x0 ∈ ¯D, x(t) is the reflecting process on ¯D, K is a bounded variation process with variation |K| increasing only when x(t) ∈ ∂D, W is a Brownian motion process, and f : R+ × Rd → Rd and g : R+ × Rd → Rd×d are measurable bounded functions. It was proved in [1] that Eq. (1) is solvable if D satisfies the Lions–Sznitman conditions [conditions (A) and (B) below] and if the relations g(t, x) = 0 and f (t, x) = 0, (t, x) ∈ H, are valid on the set H of points where the mapping g is in some sense degenerate and at least one of the mappings f and g is discontinuous. We prove an existence theorem under the only assumption that f and g are bounded measurable functions. Note, however, that by a weak solution of Eq. (1) we understand a solution of some stochastic differential inclusion associated with Eq.