Testing for One-Factor Models versus Stochastic Volatility Models

A Proofs

Before proving Theorem 1, we need the following Lemmas.

Lemma 1. Let Assumption 1 hold. Then

sup ∣μ(Xs) | = Oa.s. (n^ε/4),
s∈[0,1]

sup ∣σ²(Xs)∣ = Oa.s.(n^ε/2),
s∈[0,1]

sup ∣g( fs) | = Oa.s. (n^ε/²),
s∈[0,1]

for any ε > 0, arbitrarily small.

A.1 Proof of Lemma 1

We start from the case when Xt follows (1). Define Rl = {inf t : |Xt| >l}. Thus, Rl is an

Ft-measurable stopping time. Let

min(t,R_l)

=         μ ( Xs )d s + /

00

Obviously, for all t ≤ Rl, X_min(_{t r}) = X_t. Now let Ω_l = {ω : Rl > 1} and l = l_n = n^ε/4. Thus, given
the growth conditions in Assumption 1(a), X_t is a non-explosive diffusion, and so Pr(Ωl_n → 1) = 1.
By a similar argument, given Assumptions 1(a), 1(b), the same holds when the volatility process
follows (2). Therefore, the statement follows.                                                     ■

Lemma 2. Let Assumption 1 hold. Under H0, if, as n →∞, nξn →∞, nξ_n² → 0 and, for any
ε > 0 arbitrarily small, m/n^{1 -ε} → 0, then, pointwise in r,

A.2 Proof of Lemma 2

By Ito’s formula

√    ^l ( ^n-1)^rj

m £  ( S²n ( Xi/n ) — σ ² ( Xi/n ))

i=1

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An,m,r

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