Likelihood and Asymptotic Efficiency

Let (Ω, F) be a measurable space and X = (X ₁, X ₂) be an RCLL process on the interval [0, ∞). Let F = {F _s , s ≥0} be the filtration generated by X. For simplicity, assume that F ₀ = and suppose that X1 and X ₂ are real-valued processes.