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It is shown that if a diffusion process, ₜ: 0 ≤ t ≤ 1\, on Rᵈ satisfies dXₜ = b(t, Xₜ) dt + σ (t, Xₜ) dwₜ then the reversed process, ̄ₜ: 0 ≤ t ≤ 1\ where X̄ₜ = X1 - t, is again a diffusion with drift b̄ and diffusion coefficient σ̄, provided some mild conditions on b, σ, and p₀, the density of the law of X₀, hold. Moreover b̄ and σ are identified.
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Haussmann et al. (1986) studied this question.
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