On moderate deviations for martingales

Abstract

Let Xn = (Xnt, ℱnt)0≤t≤1 be square integrable martingales with the quadratic characteristics (Xn), n = 1, 2, . . . . We prove that the large deviations relation P(Xn1 ≥ r)/(1 - Φ(r)) → 1 holds true for r growing to infinity with some rate depending on Ln2δ = E ∑0≤t≤1 |ΔXnt|2+2δ and Nn2δ = E|(Xn)1 - 1|1+δ, where δ > 0 and Ln2δ → 0, Nn2δ → 0 as n → ∞. The exact bound for the remainder is also obtained.