Weak uniqueness for the one-dimensional SDE dZ_t = δ dt + Z_t^γ dB_t with γ∈(0,1/2)

Determine whether weak uniqueness (uniqueness in law) holds for non-negative solutions to the stochastic differential equation dZ_t = δ dt + Z_t^γ dB_t with γ ∈ (0,1/2) and Z_0 ≥ 0.

Background

The paper analyzes zero sets of class M_{γ,δ} semimartingales and derives consequences for solutions of dZ_t = δ dt + Z_tγ dB_t, but it does not settle uniqueness in law for this SDE when γ∈(0,1/2).

While pathwise uniqueness is known to hold for non-negative solutions of the driftless Girsanov SDE at zero and to fail for certain signed solutions, the weak uniqueness question for this particular non-negative, non-Lipschitz SDE remains unresolved.

References

We also remark, pursuant to our discussion of uniqueness in the Introduction, that as far as we are aware it is unknown if weak uniqueness holds for eq_Zsde when $\gamma \in (0,1/2)$.

— A compact support property for infinite-dimensional SDEs with Hölder continuous coefficients  (2603.29442 - Hughes et al., 31 Mar 2026) in Introduction (discussion around Theorem on the zero set of solutions to dZ_t = δ dt + Z_t^γ dB_t)