Testing for One-Factor Models versus Stochastic Volatility Models

computation of S_n² (X_i/_n) and RVm,r will become clear in the next subsection.

In the sequel we shall need the following assumption.

Assumption 1.

(a) σ (∙) and μ (∙), defined in (1), satisfy local Lipschitz and growth conditions. Therefore, for
any compact subsets M (under the null hypothesis) and J (under the alternative hypothesis)
of the range of the process Xt, there exist constants K₁^M, K₂^M, K₃^M, K₄^M, K₁^J and K₂^J, such
that, ∀(x,y) ∈ M and ∀(x',y') ∈ J,

∣^σ(χ) - ^σ(y)I ≤ KM ^lx ^- y^l,

∣σ(x)1² ≤ KM(1 + ∣x∣²),

∣μ^(χ) - μ⁽y)I ≤ KMIx ^- y∣^,   ∣μ^(χ) - μ⁽y)∣ ≤ KJIx ^- y^,∖

and

xμ(x) ≤ KM(1 + ∣x∣²),   Xμ(X) ≤ KJ (1 + ∣x^,∣2).

(b) σJ (∙) and b (∙), defined in (2), satisfy local Lipschitz and growth conditions. Therefore, for
any compact subset L of the range of the process ft, there exist constants K_J^L, K₂^L, K₃^L and
K₄^L , such that, ∀(p, q) ∈ L,

∣σJ(p) - σJ(q)∣ ≤ K_J^L ∣p- q∣ ,

∣σJ(p)∣² ≤ K2^L(1 + ∣p∣²),

∣b(p) - b(q)∣ ≤ K₃^L ∣p - q∣

and

pb(p) ≤ K₄^L(1 + ∣p∣²).

(c) μ(∙), σ(∙) and g (∙) are continuously differentiable.

Assumption 1(a) states local Lipschitz and growth conditions for the drift term under both hy-
potheses and for the variance term under the null hypothesis. Assumption 1(b) states local Lipschitz
and growth conditions for the variance term under the alternative. Assumptions 1(a)(b) ensure the
existence of a unique strong solution under both hypotheses (see e.g. Chung and Williams, 1990,
p.229). Since we are studying the diffusion processes over a fixed time span, we do not need to
impose more demanding assumptions, such as stationarity and ergodicity.⁹

⁹Note that Bandi and Phillips (2001, 2003) allow the time span to approach infinity, and then require the diffusion
to be null Harris recurrent.

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