Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale

Abstract

Let for each 𝑛∈ℕ𝑋𝑛 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).