Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale

Let for each n ∈ N Xn be an Rd-valued locally square integrable martingale w.r.t. a filtration (Fn(t), t ∈ R+) (probability spaces may be different for different n). It is assumed that the discontinuities of Xn are in a sense asymptotically small as n → ∞ and the relation E (f(⟨zXn⟩(t))|Fn(s)) − f (⟨zXn⟩(t)) −P→ 0 holds for all t > s > 0, row vectors z and bounded uniformly continuous functions f. Under these two principal assumptions and a number of technical ones it is proved that the Xn’s are asymptotically conditionally Gaussian processes with conditionally independent increments. If, moreover, the compound processes (Xn(0), ⟨Xn⟩) converge in distribution to some ( ˚X,H ) , then a sequence (Xn) converges in distribution to a continuous local martingale X with initial value ˚X and quadratic characteristic H, whose finite-dimensional distributions are explicitly expressed via those of ( ˚X,H ) .