4.2. Mathematical Issues
4.2.1. Multivariate Ch.F. and Inversion Theorem
Let X = (Xi, ...,Xp) be a p x 1 random vector with c.d.f. Fχ and consider arbitrary ζ ∈ Kp. The ch.f. of
X is the Fourier transform of Fχ∙
Φχ (C) = Л--/' eit*'χdFχ.
J √KP
Now, suppose that X has a density function fχ. If Ψχ is Lebesgue integrable (Φχ ∈ L1 (Kp)), then, by
the inversion theorem∙
∕χ (X) = -ɪp /.../ e-≈<'xφχ (C ) dζ.
(2P J ⅛P
Further extensions are considered in Shephard (1991a) and Shephard (1991b).
4.2.2. Riemann-Lebesgue Lemma
Suppose that X has a density function fχ. Since the density function is nonnegative, then, fχ = ∣fχ∣, and
since it integrates to 1 on Kp, fχ ∈ Li (Kp). Then, by the multivariate extension of the Riemann-Lebesgue
lemma in Rudin (1991, theorem 7.5), its Fourier transform, Φχ, belongs to Cq (Kp), where Cq (Kp) is the
supremum-normed Banach space of all complex continuous functions on Kp that vanish at infinity.
A precise statement of this important fact is that for any e > 0 there exists a compact subset Ke in the
domain d such that ∣Φχ (C)∣ < e for any ζ ∈ r∂ ∖ Ke. In particular, Φχ ∈ Cq (Kp) implies∙
Iim Φχ (C)=0 + г - 0, (8)
any ζ→∞
as a sequence of vectors with at least one exploding component cannot be contained in any compact set.
Moreover, ∣∣Φχ∣∣∞ ≤ llfχ∣lι, where IHIr = {(Ю P ʃ∙∙∙ ‰ Kdχ}r, 1 ≤ r < ∞ (and for r = ∞ it is
the essential supremum of ∣g∣ x (2π) 2 ).4 ∣∣fχ∣∣i is a finite number, (2π) 2. Then, Φχ ∈ L∞ (Kp), which,
unfortunately, does not imply that Φχ ∈ Li (Kp).5
4.3. Joint P.D.F.
It is beyond the scope of this paper to establish restrictions on the underlying parameter vector,
c = (r,δ, st,vt,τ, a, β,y, p), that guarantee absolute continuity of the random vector (sT ,vτ ), that is, exis-
tence of the density per se. My best guess is that a rigorous proof would appeal to conditions under which
Ito processes are continuous semimartingales and, certainly, to the condition under which {vi}t<l<T ≥ 0,
4Rudin integrates with respect to the normalized Lebesgue measure, mp, to preserve the symmetry of the Fourier forward
and inverse transformations.
5Relationship Lr (^p) C Ls (^p) when 0 < s ≤ r ≤ ∞ does not hold for mp, since mp (^p) = +∞.