The non-anticipative semimartingale representation is a foundational decomposition of Lévy random bridge dynamics into a predictable drift and a martingale component. For a Φ-initialized random bridge {ξ_t^x} to Ψ driven by a Lévy process with mutually independent coordinates and finite first moments, the representation is:
ξ_t^x = ∫₀ᵗ (E[Y | ξ_s^x] - ξ_s^x)/(T-s) ds + M_t
where M_t is a martingale with M₀ = x. This holds for all t ∈ [0, T).
The key property is that the drift term (E[Y | ξ_t^x] - ξ_t^x)/(T-t) is non-anticipative — it depends only on the current state ξ_t^x and the conditional expectation of the target Y, not on future values. This drift has the identical functional form to the optimal rectified flow velocity field v*(z,t) = (E[Y|R_t=z] - z)/(T-t), establishing the fundamental connection between stochastic bridges and deterministic flow-based generative models.
The representation holds universally across all Lévy driving processes (Brownian motion, Poisson process, stable processes, etc.), making it a unifying result. The specific form of the martingale M_t changes with the driver — for Brownian bridges M_t = σW_t, for Poisson bridges M_t is a compensated counting process — but the drift structure remains invariant.
Key Details
- The conditional expectation form E[dξ_t^x | ξ_t^x] = (E[Y|ξ_t^x] - ξ_t^x)/(T-t) dt follows from the martingale property of M_t
- Temporal consistency: the velocity can be expressed via any future time u ∈ (t,T] as (E[ξ_u|ξ_t] - ξ_t)/(u-t)
- For Brownian bridges, the drift decomposes further into source-push + score: (ξ_t - x)/t + σ²∇log p_t(ξ_t)
- For Poisson bridges, the drift equals the intensity process λ_{tT}
- This representation was derived using results from Hoyle, Hughston, and Macrina (2011) on Lévy random bridges