NECESSARY CONDITIONS FOR STABLE

CONVERGENCE OF SEMIMARTINGALES

E. Mordecki

Steklov Mathematical Institute, Vavilov 42, 117966 Moscow GSP–1, Russia Universidad de la República, Montevideo, Uruguay

Abstract: We prove an inverse to a theorem on stable convergence of semimartingales due to (Feigin, 1985). As a consequence, it can be stated (under some control in the jumps) that a sequence of martingales X^{n} converges stably to a continuous martingale X with conditionally independent increments if and only if the quadratic variations of X^{n} converge in probability to the quadratic variation of X for each t ∈ R^{+}.

Keywords: Stable convergence, semimartingales.

1.Introduction and Results.

The subject of the present paper is the characterization of the behaviour of a sequence of semimartingales when they converge stably to a continuous semimartingale with conditionally independent increments. In view of the results due to (Lipster and Shiryaev, 1983), and more recently to (Feigin, 1985), it remains only to prove some inverse theorems, and this is done, based on the extension of existing results for weak convergence, and a convenient characterization of stable convergence. The results presented here were announced in (Mordecki, 1993).

For general concepts about semimartingales and their characteristics we refer to (Jacod and Shiryaev, 1987, ch. I and II).

Let us now define the sequence of semimartingales and the limit.

Let h denote a truncation function, and all the characterictics will be considered with respect to h.

(S) Consider for each integer n ∈ N a d-dimensional semimartingale X^{n} defined on a stochastic basis B^{n} = (Ω, F, 𝔽^{n}, P), with 𝔽^{n} = (F_{t}^{n}: t ≥ 0). Denote by (B^{n}, C̃^{n}, ν^{n}) the triplet of modified characteristics of X^{n} under P, and by [X^{n}, X^{n}] the R^{d} ⊗ R^{d} - valued processes whose components are the (co-)quadratic variation [X^{n,j}, X^{n,k}]. Set for simplicity X_{0}^{n} = 0.

As for the limit consider a transition probability Q(ω, dγ) from (Ω, F) into the Skorokhod space

(D(R^{d}), D(R^{d}), (D_{t}: t ≥ 0)),

such that for each ω ∈ Ω the canonical process on D(R^{d}) defined by X_{t}(γ) = γ_{t} is a semimartingale with independent increments, and characteristics (B(ω), C(ω), 0)

Typeset by AMS-TEX

Now we set

Ω̄ = Ω × D(R^{d}),  F̄ = F ⊗ D(R^{d}),

P̄(dω, dγ) = P(dω)Q(ω, dγ),  𝔽̄ = (∩_{s>t}(F ⊗ D_{s}): t ≥ 0)

and every process, variable or σ-field defined on (Ω, F) or (D(R^{d}), D(R^{d})) is naturally extended to (Ω̄, F̄) with the same symbol, e.g., X(ω, γ) = X(γ), X^{n}(ω, γ) = X^{n}(ω), etc.

(L) Let then X be a process defined on (Ω̄, F̄, 𝔽̄, P̄) with (B, C, 0) as characteristics, that is, a process with F-conditionally independent increments. Denote by [X, X] the quadratic variation of X (defined as above), and observe that C̃ = C.

(N) Nesting condition: Assume there exists a sequence of real numbers (α_{n}) decreasing to 0, such that

(1) F_{α_{n}}^{n} ⊂ F_{α_{n+1}}^{n+1},

(2) F = ⋁_{n} F_{α_{n}}^{n}.

Then consider the conditions:

[γ − R^{+}] : C̃_{t}^{n} ⟶^{P} C_{t} ∀t ∈ D

[δ − R^{+}] : 𝕀_{{|x|>ε}} ∗ ν_{t}^{n} ⟶^{P} 0 ∀t ∈ D ∀ε > 0

Denote V ar(B^{n})_{t} = ∑_{i=1}^{d} V ar(B^{n,i})_{t}, where V ar(A)_{t} stands for the total variation of a function A on [0, t].

We will say that the sequence X^{n} converges F-stably to X in D(R^{d}) (denoted by X^{n} ⟶^{d} X (F-stably)), if

E(Y f(X^{n})) → E(Y f(X)) as n → +∞

for all bounded and F-measurable random variables Y and continuous and bounded functions f : D(R^{d}) → R.

Theorem 1. Assume conditions (S) and (L) are fulfilled. If

(i) X^{n} ⟶^{d} X (F-stably),

(ii) (V ar(B^{n})_{t}|P̄) is tight for all t ≥ 0, then

([X^{n}, X^{n}], X^{n}) ⟶^{d} ([X, X], X) (F-stably)

and, in particular,

[X^{n}, X^{n}] ⟶^{d} [X, X] (F-stably)

Corollary 1. Under the assumptions of Theorem 1

[X^{n}, X^{n}]_{t} ⟶^{P} C_{t} ∀t ≥ 0.

Theorem 2. Assume conditions (S) and (L) are fulfilled. If

(i) X^{n} ⟶^{d} X (F-stably),

(ii) (V ar(B^{n})_{t}|P̄) is tight; then

(1) [δ − R^{+}] : 𝕀_{{|x|>ε}} ∗ ν_{t}^{n} ⟶^{PP} 0 ∀t ∈ R^{+} ∀ε > 0,

(2) [γ − R^{+}] : C̃_{t}^{n} ⟶^{P} C_{t} ∀t ∈ R^{+}.

Corollary 2. Assume conditions (S), (L) and (N) are fulfilled. Suppose that X^{n}’s are local martingales satisfying |ΔX_{t}^{n}| ≤ c for some constant c, and X is a continuous F-conditionally Gaussian martingale (for each fixed ω, X is a Gaussian martingale).

Then the following statements are equivalent:

(1) X^{n} ⟶^{d} X (F-stably),

(2) [X^{n}, X^{n}]_{t} ⟶^{P} [X, X]_{t} ∀t ∈ R^{+},

(3) [γ − R^{+}] + [δ − R^{+}].

2. Proofs.

Let us first characterize stable convergence in a convenient way. Let G be a sub-σ-field of F. Denote by P(G) the set of probability measures P̃ on (Ω, F) such that dP̃ = Y dP for any G-measurable, positive and bounded random variable Y. Given X a random variable; let σ(X) denote the σ-field generated by X.

Lemma 1. Suppose on a probability space (Ω, F, P) is given a sequence of random elements X^{n} and X with values in a metric space (E, δ). Let G ⊂ F. Then (1) and (2) are equivalent:

(1) X^{n} ⟶^{d} X (G-stably),

(2) X^{n} ⟶^{L(P̃)} X ∀P̃ ∈ P(G)

If E = R^{d}, δ is the Euclidean metric in R^{d}, and σ(X) ⊂ G, then (1) or (2) are equivalent to (3)

(3) X^{n} ⟶^{P} X

Proof. Equivalence (1) ⇔ (2) follows from linearity and normalization.

(1) ⇒ (3). We know that E(exp{i(λX +μX^{n})}) → E(exp{i(λX +μX)}) for the real and continuous functions cos(λx), cos(μx), sin(λx), sin(μx) and the bounded and G-measurable random variables cos(λX), cos(μX), sin(λX), sin(μX).

From this fact it follows that

(X, X^{n}) ⟶^{L(P)} (X, X) and X^{n} − X ⟶^{P} 0.

(3) ⇒ (2). If P̃ ∈ P then P ∼ P̃. So X^{n} ⟶^{P̃} X and X^{n} ⟶^{L(P̃)} X.

Proof of Theorem 1. According to Lemma 1 it is suffices to verify

([X^{n}, X^{n}], X^{n}) ⟶^{L(P̃)} ([X, X], X),

where P̃ ∈ P.

We know that the quadratic variation is invariant under an absolutely continuous change of measure, so it is natural to apply Theorem VI.6.1 of (Jacod and Shiryaev, 1987) under P̃. Let us verify its hypothesis. We know

X^{n} ⟶^{L(P̃)} X

((1) ⇒ (2) in Lemma 1). In order to prove (V ar(B^{n})_{t}|P̃) is tight, let us fix an ε > 0. As P̃ ∼ P̄, there exists a δ > 0 such that

P̄(A) < δ ⇒ P̃(A) < ε, A ∈ F.

Let K_{ε} be such that

P̄(V ar(B^{n})_{t} > K_{ε}) < δ ∀n ∈ ℕ

then

P̃(V ar(B^{n})_{t} > K_{ε}) < ε ∀n ∈ ℕ

Now the application of Theorem VI.6.1 of (Jacod and Shiryaev, 1987) completes the proof

([X^{n}, X^{n}], X^{n}) ⟶^{L(P̃)} ([X, X], X)

Proof of Corollary 1. In view of Theorem 1 and implication (1) ⇒ (3) in Lemma 1 the result follows from the fact that C_{t} is F-measurable.

Proof of Theorem 2. (1) As X is continuous,

sup_{s≤t} |ΔX_{s}^{n}| ⟶^{P} 0  (2.1)

which is equivalent to [δ − R^{+}] in virtue of the Lenglart-Rebolledo inequalities (see (Jacod and Shiryaev, 1987) section VI.4.22).

(2) Let

X^{n} = M^{n} + B^{n} + X̌^{n}(h)

be the semimartingale decomposition of X^{n}:

X̌^{n}(h) = ∑_{s≤t} ΔX_{s}^{n} − h(ΔX_{s}^{n})

M^{n} is a local martingale with bounded jumps and M_{0}^{n} = 0,

B^{n} is a predictable process with finite variation in intervals.

By (2.1) we know P(sup_{s≤t} |X̌^{n}(h)_{s}| > 0) → 0

Furthermore on the set {sup_{s≤t} |ΔX̌_{s}^{n}| = 0} we have

[X^{n}, X^{n}] = [M^{n}, M^{n}] + [B^{n}, B^{n}] + 2[M^{n}, B^{n}]

and

sup_{s≤t} |[X^{n}, X^{n}] − [M^{n}, M^{n}]| ≤ V ar(B^{n})_{t} sup_{s≤t}(|ΔB_{s}^{n}| + 2|ΔM_{s}^{n}|).

We know that V ar(B^{n})_{t} is tight. Let us first check that

sup_{s≤t} |ΔB_{s}^{n}| ⟶^{P} 0.  (2.2)

Observe that

|ΔB_{s}^{n}| = h(x) ∗ ν^{n}({s}, dx) ≤ ε + Δ𝕀_{{|x|>ε}} ∗ ν_{s}^{n} ≤ ε + 𝕀_{{|x|>ε}} ∗ ν_{t}^{n}

(assuming h(x) = x on the set {|x| < ε} and |h(x)| ≤ 1). So

sup_{s≤t} |ΔB_{s}^{n}| ≤ ε + 𝕀_{{|x|>ε}} ∗ ν_{t}^{n}

wich yields (2.2) in virtue of [δ − R^{+}].

Let us then prove

sup_{s≤t} |ΔM_{s}^{n}| ⟶^{P} 0  (2.3)

We have

ΔM_{s}^{n} = ΔX_{s}^{n} − ΔB_{s}^{n} − ΔX̌(h)_{s} = h(ΔX_{s}^{n}) − ΔB_{s}^{n}

so, as |h(x)| ≤ |x|,

|ΔM_{s}^{n}| ≤ |ΔX_{s}^{n}| + |ΔB_{s}^{n}|

and (2.3) follows.

Thus, we have then proved that:

sup_{s≤t} |[X^{n}, X^{n}] − [M^{n}, M^{n}]| ⟶^{P} 0

and as Corollary 1 gives [X^{n}, X^{n}]_{t} ⟶^{P} C_{t} we obtain [M^{n}, M^{n}]_{t} ⟶^{P} C_{t}. Now [γ − R^{+}] follows from a standard argument given in (Rebolledo, 1980) that establishes, that under (2.1)

sup_{s≤t} | < M^{n}, M^{n} > −[M^{n}, M^{n}]| ⟶^{P} 0.  (2.4)

Proof of Corollary 2. Let us first take a truncation function h that satisfies h(x) = x for |x| ≤ c: we then have B^{n}(h) = 0.

(1) ⇒ (2) is a particular case of Corollary 1.

(3) ⇒ (1) is a particular case of Feigin’s Theorem (1985), see also section VIII.5.42 in (Jacod and Shiryaev, 1987).

(2) ⇒ (3). Since (2.1) ∀t ∈ R^{+} is equivalent to [δ − R^{+}] and under (2.1) convergence (2.4) takes place, it is enough to prove that (2) ⇒ (2.1).

Now it follows from (2) that

sup_{s≤t} |Δ[X^{n}, X^{n}]_{s}| ⟶^{P} 0

Since |ΔX^{n}|^{2} = ∑_{i=1}^{d} Δ[X^{n,i}, X^{n,i}], (2.1) holds.

Aknowledgements:. The author is indebted to A.N. Shiryaev, D.O. Kramkov and A.A. Gushchin for helpful discussions.

References

Feigin P.D. (1985), Stable convergence of semimartingales, Stochastic Process. Appl. 19, 125–134. Jacod J. and Shiryaev A.N. (1987), Limit theorems for stochastic processes, Springer-Verlag,

Berlin, Heidelberg, New York. Liptser R.S. and Shiryaev A.N. (1983), On weak convergence of semimartingales to stochasti-

cally continuous processes with independent and conditionally independent increments, Math. USSR Sbornik 44, 299–323.

Mordecki E. (1993), About necessary conditions for stable convergence of semimartingales, Us-

pekhi Mat. Nauk 48, 193–194. Rebolledo R. (1980), Central limit theorems for local martingales, Z. Wahrscheinlichkeitstheor.

verw. Geb. 51, 269-286.