Sharp minimax dependence on the bridge horizon

Determine the sharp minimax dependence of exact multivariate diffusion bridge sampling work on the time horizon, beyond the polynomial upper bound established for the proposed bridge sampler.

Background

The paper constructs an exact query-only sampler for uniformly elliptic multivariate diffusion bridges whose expected work is polynomial in the horizon, up to a logarithmic factor, and proves minimax optimality only with respect to the number of requested observations when the horizon and endpoints are fixed. The authors explicitly identify the sharp dependence on the bridge horizon as unresolved.

References

The sharp minimax dependence on the bridge horizon remains open and is a direction of future research.

— Exact Simulation of Multivariate Diffusions and Bridges  (2610.11330 - Blanchet et al., 8 Oct 2026) in Remark following Theorem 3.2, Section 3.3 (Minimax optimality)

The sharp joint dependence of bridge sampling work on the horizon and the number of requested observations remains open, as does the optimal dependence on the endpoints.

— Exact Simulation of Multivariate Diffusions and Bridges  (2610.11330 - Blanchet et al., 8 Oct 2026) in Section 4, Conclusions and discussions