Pricing American-style Derivatives under the Heston Model Dynamics: A Fast Fourier Transformation in the Geske–Johnson Scheme

4.2. Mathematical Issues
4.2.1. Multivariate Ch.F. and Inversion Theorem

Let X = (Xi, ...,X_p) be a p x 1 random vector with c.d.f. Fχ and consider arbitrary ζ ∈ K^p. The ch.f. of

X is the Fourier transform of F_χ∙

Φχ (C) = Л--/' e^it*'^χdFχ.
J    √KP

Now, suppose that X has a density function fχ. If Ψχ is Lebesgue integrable (Φχ ∈ L¹ (K^p)), 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 K^p, fχ ∈ Lⁱ (K^p). Then, by the multivariate extension of the Riemann-Lebesgue
lemma in Rudin (1991, theorem 7.5), its Fourier transform, Φχ, belongs to Cq (K^p), where Cq (K^p) is the
supremum-normed Banach space of all complex continuous functions on K^p that vanish at infinity.

A precise statement of this important fact is that for any e > 0 there exists a compact subset K_e in the
domain d such that ∣Φχ (C)∣ < e for any ζ ∈ ^r∂ ∖ K_e. In particular, Φχ ∈ Cq (K^p) 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∣∣∞ ≤ ll^fχ∣lι^{, where} IHI_r = {(Ю P ʃ∙∙∙ ‰ K^dχ}^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, Φ_χ ∈ L∞ (K^p), which,
unfortunately, does not imply that Φχ ∈ Lⁱ (K^p).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, s_t,v_t,τ, a, β,y, p), that guarantee absolute continuity of the random vector (s_T ,v_τ ), 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, m_p, to preserve the symmetry of the Fourier forward
and inverse transformations.

⁵Relationship L^r (^^p) C L^s (^^p) when 0 < s ≤ r ≤ ∞ does not hold for m_p, since m_p (^^p) = +∞.

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