MODERATE DEVIATIONS FOR MARTINGALES WITH BOUNDED JUMPS

We prove that the Moderate Deviation Principle (MDP) holds for the trajectory of a locally square integrable martingale with bounded jumps as soon as its quadratic covariation, properly scaled, converges in probability at an exponential rate. A consequence of this MDP is the tightness of the method of bounded martingale di(cid:11)erences in the regime of moderate deviations.

Recall that a family of random variables {Z k ; k > 0} with values in a topological vector space X equipped with σ-field B satisfies the Large Deviation Principle (LDP) with speed a k ↓ 0 and good rate function I(·) if the level sets {x; I(x) ≤ α} are compact for all α < ∞ and for (where Γ o and Γ denote the interior and closure of Γ, respectively). The family of random variables {Z k ; k > 0} satisfies the Moderate Deviation Principle with good rate function I(·) and critical speed 1/h k if for every speed a k ↓ 0 such that h k a k → ∞, the random variables √ a k Z k satisfy the LDP with the good rate function I(·). Suppose that X ∈ M 2 loc,0 is a locally square integrable martingale with bounded jumps |∆X| ≤ a (and X 0 = 0). We denote by (A, C, ν) the triplet predictable characteristics of X, where here A = 0, C = (C t ) t≥0 is the F-predictable quadratic variation process of the continuous part of X and ν = ν(ds, dx) is the F-compensator of the measure of jumps of X. Without loss of generality we may assume that and for all s < t, The predictable quadratic characteristic (covariation) of X is the process where x denotes the transpose of x ∈ IR d , and A = sup |λ|=1 |λ Aλ| for any d × d symmetric matrix A.
Our main result is as follows.
Proposition 1 Suppose the symmetric positive-semi-definite d × d matrix Q and the regularly varying function h t of index α > 0 are such that for all δ > 0: Then h , B) (equipped with the locally uniform topology) with critical speed 1/h k and the good rate function where , and AC 0 = {φ : IR + → IR d with φ(0) = 0 and absolutely continuous coordinates }.
Remark 1 Note that both (5) and the MDP are invariant to replacing h t by g t such that h t /g t → c ∈ (0, ∞) and taking cQ instead of Q. Thus, if Q = 0 we may take h t = median X t , and in general we may assume with no loss of generality that h t ∈ D(IR + ) is strictly increasing of bounded jumps. Remark 2 If X is a locally square integrable martingale with independent increments, then X is a deterministic process, hence suffices that h −1 t X t → Q for (5) to hold.
As stated in the next corollary, less is needed if only X k (or sup s≤k X s ) is of interest. Remark 3 For d = 1, discrete-time martingales, and assuming that h k = X k is non-random, strong Normal approximation for the law of h −1/2 k X k is proved in [9] for the range of values corresponding to a 3 k h k → ∞. Remark 4 The difference between Proposition 1 and Corollary 1 is best demonstrated when considering X t = B ht , with B s the standard Brownian motion. The MDP for h −1/2 t B ht in IR then trivially holds, whereas the MDP for h , and thus holds only when h t is regularly varying of index α > 0. Remark 5 When d = 1 and Q = 0, the rate function for the MDP of part (a) of Corollary 1 is (5). Consequently, for such values of x, y the inequality (2) is tight for k → ∞ (see also Remark 9 below for non-asymptotic results). Remark 6 In contrast with Corollary 1 we note that the LDP with speed m −1 may fail for m −1 X m even when X is a real valued discrete-parameter martingale with bounded independent increments such that X m = m.

Proof of Proposition 1
The cumulant G(λ) = (G t (λ)) t≥0 associated with X is The stochastic (or the Doléans-Dade) exponential of G(λ), denoted E(G(λ)) is given by where The next lemma which is of independent interest, is key to the proof of Proposition 1.
(c.f. Remark 8). The inequality (2) then follows by Chebycheff's inequality and optimization over λ ≥ 0. For the special case of a real-valued discrete-parameter martingale X m also and we can even replace w(|λ|a) in (12) The required bound then follows from (7) since To establish the corresponding lower bound, note that since ∆G s (λ) ≥ 0 (see (9)) and log(1 + x) − x ≥ −x 2 /2 for all x ≥ 0, we have that Moreover, again by (9) we see that Hence, and the required lower bound follows by noting that To prove Proposition 1 we need the following immediate consequence of Lemma 1.
Lemma 2 Suppose there exists q ∈ C[0, ∞), a positive-semi-definite matrix Q and an unbounded function h : Then, for every λ ∈ IR d and a k → 0 such that h k a k → ∞, lim sup k→∞ a k log IP sup Proof: Use (10), noting that a k = 1 hk (a k h k ) with a k h k → ∞, and that lim The next lemma is a simple application of the results of [8], relating (14) with the LDP (with speed a k ) of ak hk X k· .
Lemma 3 When (14) holds, the processes ak hk X k· , k > 0 satisfy the LDP in (D(IR d ), B) with speed a k and the good rate function (where q ∈ M + (IR + ) is the continuous locally finite measure on (IR + , B IR+ ) such that q([0, t]) = q(t)).
Proof: For each sequence k n → ∞ we shall apply [8,Theorem 2.2] for the local martingales a kn /h kn X knt replacing 1 n throughout by a kn . Cramér's condition [8, (2.6)] is trivially holding in the current setting, while for G t (λ) = 1 2 q(t)λ Qλ the condition (sup E) of [8, Theorem 2.2] is merely (14). Moreover, for this G t (λ) the condition [8, (G)] is easily shown to hold (as H s,t (·) is then a positive-definite quadratic form on the linear subspace domH s,t for all s < t). Thus, the LDP in Skorohod topology follows from [8, Theorem 2.2] and the explicit form (15) of the rate function follows from [8, (2.4)] taking there g t (λ) = 1 2 λ Qλ. Suppose I(φ) < ∞. Then, φ q and since q ∈ C[0, ∞) it follows that φ ∈ C(IR d ). Hence, by [8, Theorem C] we may replace the Skorohod topology by the stronger locally uniform topology on D(IR d ).
Proposition 1 follows by combining Lemmas 2 and 3 with the next lemma.

Proof of Corollary 1
(a) Assume first that h t is regularly varying of index 1. Given Proposition 1, this case is easily settled by applying the contraction principle for the continuous mapping φ → φ(1) : D[IR d ] → IR d . In the general case, we take without loss of generality h t ∈ D(IR + ) strictly increasing of bounded jumps (see Remark 1). Let σ s = inf{t ≥ 0 : h t ≥ s} and g s = h σs . Note that g s − s is bounded, while (5) holds for the locally square integrable martingale Y s = X σs of bounded jumps and the regularly varying function g s of index 1. Consequently, {g −1/2 s Y s } satisfies the MDP with the critical speed 1/g s and the good rate function Λ * (·). Since h t is strictly increasing and unbounded it follows that σ(IR + ) = IR + . Hence, this MDP is equivalent to the MDP for {h −1/2 k X k }. (b) As in part (a) above suffices to prove the stated MDP for h t regularly varying of index 1. Applying the contraction principle for the continuous mapping φ → sup s≤1 φ(s) we deduce the stated MDP from Proposition 1. Since Λ * (v) = v 2 /(2Q), the good rate function for this MDP is (c.f. (6)) Clearly, φ(0) = 0 implies that I(z) = ∞ for z < 0, while taking φ(s) = (s ∧ 1)z we conclude that I(z) = z 2 /(2Q) for z ≥ 0.