Stochastic differential equations for sticky Brownian motions

We study (i) the SDE system dXt = I(Xt 6=0) dBt I(Xt=0) dt = 1µ d` 0 t (X) for Brownian motion X in IR sticky at 0, and (ii) the SDE system dXt = 12 d` 0 t (X) + I(Xt>0) dBt I(Xt=0) dt = 12µ d` 0 t (X) for reflecting Brownian motion X in IR+ sticky at 0, where X starts at x in the state space, µ ∈ (0,∞) is a given constant, `0(X) is a local time of X at 0, and B is a standard Brownian motion. We prove that both systems (i) have a jointly unique weak solution and (ii) have no strong solution. The latter fact verifies Skorokhod’s conjecture on sticky Brownian motion and provides alternative arguments to those given in the literature. 1