Compact Embeddings for Spaces of Forward Rate Curves

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Volume 2013, Article ID 709505,6pages http://dx.doi.org/10.1155/2013/709505

Research Article

Compact Embeddings for Spaces of Forward Rate Curves

Stefan Tappe

Institut f¨ur Mathematische Stochastik, Leibniz Universit¨at Hannover, Welfengarten 1, 30167 Hannover, Germany

Correspondence should be addressed to Stefan Tappe; tappe@stochastik.uni-hannover.de Received 19 June 2013; Accepted 2 September 2013

Academic Editor: Alberto Parmeggiani

Copyright © 2013 Stefan Tappe. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The goal of this paper is to prove a compact embedding result for spaces of forward rate curves. As a consequence of this result, we show that any forward rate evolution can be approximated by a sequence of finite dimensional processes in the larger state space.

1. Introduction

The Heath-Jarrow-Morton-Musiela (HJMM) equation is a stochastic partial differential equation that models the evolution of forward rates in a market of zero coupon bonds; we refer to [1] for further details. It has been studied in a series of papers; see, for example, [2–5] and references therein. The state space, which contains the forward curves, is a separable Hilbert space𝐻consisting of functionsℎ : R₊ → R. In practice, forward curves have the following features.

(i) The functionsℎ ∈ 𝐻become flat at the long end.

(ii) Consequently, the limit lim_{𝑥 → ∞}ℎ(𝑥)exists.

The second property is taken into account by choosing the Hilbert space

𝐿²_𝛽⊕R, (1)

where𝐿²_𝛽denotes the weighted Lebesgue space

𝐿²_𝛽:= 𝐿²(R₊, 𝑒^𝛽𝑥𝑑𝑥) , (2) for some constant𝛽 > 0. Such spaces have been used, for example, in [2,3]. As flatness of a function is measured by its derivative, the first property is taken into account by choosing the space

𝐻_𝛾

:={ℎ :R₊󳨀→R: ℎis absolutely continuous with‖ℎ‖_𝛾<∞}, (3)

for some constant𝛾 > 0, where the norm is given by

‖ℎ‖_𝛾:= (|ℎ (0)|²+ ∫

R+󵄨󵄨󵄨󵄨󵄨ℎ^󸀠(𝑥)󵄨󵄨󵄨󵄨󵄨²𝑒^𝛾𝑥𝑑𝑥)^1/2. (4) Such spaces have been introduced in [1] (even with more general weight functions) and further utilized, for example, in [4,5]. Our goal of this paper is to show that for all𝛾 > 𝛽 > 0 we have the compact embedding

𝐻_𝛾⊂⊂ 𝐿²_𝛽⊕R, (5)

that is, the forward curve spaces used in [1] and forthcoming papers are contained in the forward curve spaces used in [2], and the embedding is even compact. Consequently, the embedding operator between these spaces can be approximated by a sequence of finite-rank operators, and hence, when considering the HJMM equation in the state space𝐻_𝛾, applying these operators its solutions can be approximated by a sequence of finite dimensional processes in the larger state space𝐿²_𝛽⊕R; we refer toSection 3for further details.

The remainder of this paper is organized as follows. In Section 2, we provide the required preliminaries. InSection 3, we present the embedding result and its proof, and we outline the described approximation result concerning solutions of the HJMM equation.

2. Preliminaries and Notation

In this section, we provide the required preliminary results and some basic notation. Concerning the upcoming results

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about Sobolev spaces and Fourier transforms, we refer to any textbook about functional analysis, such as [6] or [7].

As noted in the introduction, for positive real numbers 𝛽, 𝛾 > 0, the separable Hilbert spaces𝐿²_𝛽⊕Rand𝐻_𝛾are given by (2) and (3), respectively. These spaces and the forthcoming Sobolev spaces will be regarded as spaces of complex-valued functions. For everyℎ ∈ 𝐻_𝛾, the limitℎ(∞) :=lim_{𝑥 → ∞}ℎ(𝑥) exists, and the subspace

𝐻_𝛾⁰:= {ℎ ∈ 𝐻_𝛾 : ℎ (∞) = 0} (6) is a closed subspace of𝐻_𝛾; see [1]. For an open setΩ ⊂R, we denote by𝑊¹(Ω)the Sobolev space

𝑊¹(Ω) := {𝑓 ∈ 𝐿²(Ω) : 𝑓^󸀠∈ 𝐿²(Ω)exists} , (7) which, equipped with the inner product

⟨𝑓, 𝑔⟩_𝑊1(Ω)= ⟨𝑓, 𝑔⟩_𝐿2(Ω)+ ⟨𝑓^󸀠, 𝑔^󸀠⟩_𝐿2(Ω), (8) is a separable Hilbert space. Here, derivatives are understood as weak derivatives.

For a functionℎ ∈ 𝑊¹((0, ∞)), the extensionℎ1_(0,∞) : R → Cin general, does not belong to𝑊¹(R). In the present situation, this technical problem can be resolved as follows.

Letℎ : (0, ∞) → Cbe a continuous function such that the limit ℎ(0) := lim_{𝑥 → 0}ℎ(𝑥) exists. Then, we define the reflectionℎ^∗:R → Cas

ℎ^∗(𝑥) := {ℎ (𝑥) , if𝑥 ≥ 0,

ℎ (−𝑥) , if𝑥 < 0. (9) Lemma 1. The following statements are true.

(1)For eachℎ ∈ 𝑊¹((0, ∞)), one hasℎ^∗∈ 𝑊¹(R).

(2)The mapping𝑊¹((0, ∞)) → 𝑊¹(R),ℎ 󳨃→ ℎ^∗ is a bounded linear operator.

(3)For eachℎ ∈ 𝑊¹((0, ∞)), one has

‖ℎ‖_𝑊¹_((0,∞))≤ 󵄩󵄩󵄩󵄩ℎ^∗󵄩󵄩󵄩󵄩𝑊¹(R)≤ √2‖ℎ‖_𝑊¹_((0,∞)),

‖ℎ‖_𝐿²_((0,∞)) ≤ 󵄩󵄩󵄩󵄩ℎ^∗󵄩󵄩󵄩󵄩𝐿²(R)≤ √2‖ℎ‖_𝐿²_((0,∞)). (10) Proof. This follows from a straightforward calculation following the proof of [8, Theorem 8.6].

Lemma 2. Let𝛾 > 𝛽 > 0be arbitrary. Then, the following statements are true.

(1)One has𝐻_𝛾⁰⊂ 𝐻_𝛽⁰, and

‖ℎ‖_𝛽≤ ‖ℎ‖_𝛾 ∀ℎ ∈ 𝐻_𝛾⁰. (11) (2)One has𝐻_𝛾⁰ ⊂ 𝐿²_𝛽, and there is a constant𝐶₁= 𝐶₁(𝛽,

𝛾) > 0such that

‖ℎ‖_𝐿²_𝛽 ≤ 𝐶₁‖ℎ‖_𝛾 ∀ℎ ∈ 𝐻_𝛾⁰. (12)

(3)For eachℎ ∈ 𝐻_𝛾⁰, one has

ℎ𝑒^(𝛽/2)∙󵄨󵄨󵄨󵄨󵄨(0,∞)∈ 𝑊¹((0, ∞)) , (ℎ𝑒^(𝛽/2)∙󵄨󵄨󵄨󵄨󵄨(0,∞))^∗ ∈ 𝑊¹(R) , (13) and there is a constant𝐶₂= 𝐶₂(𝛽, 𝛾) > 0such that

󵄩󵄩󵄩󵄩󵄩󵄩󵄩(ℎ𝑒^(𝛽/2)∙󵄨󵄨󵄨󵄨󵄨(0,∞))^∗󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑊¹(R)≤ 𝐶₂‖ℎ‖_𝛾 ∀ℎ ∈ 𝐻_𝛾⁰. (14) Proof. The first statement is a direct consequence of the repre- sentation of the norm on𝐻_𝛾⁰given by (4). Letℎ ∈ 𝐻_𝛾⁰ be arbitrary. By the Cauchy-Schwarz inequality, we obtain

‖ℎ‖²_𝐿²

𝛽 = ∫

R+

|ℎ (𝑥)|²𝑒^𝛽𝑥𝑑𝑥

= ∫R+

(∫^∞

𝑥 ℎ^󸀠(𝜂) 𝑒^(𝛾/2)𝜂𝑒^{−(𝛾/2)𝜂}𝑑𝜂)²𝑒^𝛽𝑥𝑑𝑥

≤ ∫R+

(∫^∞

𝑥 󵄨󵄨󵄨󵄨󵄨ℎ^󸀠(𝜂)󵄨󵄨󵄨󵄨󵄨²𝑒^𝛾𝜂𝑑𝜂) (∫^∞

𝑥 𝑒^−𝛾𝜂𝑑𝜂) 𝑒^𝛽𝑥𝑑𝑥

≤ ∫R+

(∫R+󵄨󵄨󵄨󵄨󵄨ℎ^󸀠(𝜂)󵄨󵄨󵄨󵄨󵄨²𝑒^𝛾𝜂𝑑𝜂)1

𝛾𝑒^−𝛾𝑥𝑒^𝛽𝑥𝑑𝑥

≤ 1 𝛾(∫

R+

𝑒^{−(𝛾−𝛽)𝑥}𝑑𝑥) ‖ℎ‖²_𝛾= 1

𝛾 (𝛾 − 𝛽)‖ℎ‖²_𝛾,

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proving the second statement. Furthermore, by (12) we have

󵄩󵄩󵄩󵄩󵄩󵄩ℎ𝑒^(𝛽/2)∙󵄨󵄨󵄨󵄨󵄨(0,∞)󵄩󵄩󵄩󵄩󵄩󵄩²𝐿²((0,∞))

= ∫R+󵄨󵄨󵄨󵄨󵄨ℎ (𝑥) 𝑒^(𝛽/2)𝑥󵄨󵄨󵄨󵄨󵄨²𝑑𝑥 = ∫

R+

|ℎ (𝑥)|²𝑒^𝛽𝑥𝑑𝑥

= ‖ℎ‖²_𝐿²_𝛽 ≤ 𝐶²₁‖ℎ‖²_𝛾,

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and by estimates (11), (12), we obtain

󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩( 𝑑

𝑑𝑥) (ℎ𝑒^(𝛽/2)∙󵄨󵄨󵄨󵄨󵄨(0,∞))󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩²𝐿²((0,∞))

= ∫R+

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑑

𝑑𝑥(ℎ (𝑥) 𝑒^(𝛽/2)𝑥)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨²𝑑𝑥

= ∫R+

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨ℎ^󸀠(𝑥) 𝑒^(𝛽/2)𝑥+𝛽

2ℎ (𝑥) 𝑒^(𝛽/2)𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨²𝑑𝑥

≤ 2 (∫

R+󵄨󵄨󵄨󵄨󵄨ℎ^󸀠(𝑥)󵄨󵄨󵄨󵄨󵄨²𝑒^𝛽𝑥𝑑𝑥 +𝛽² 4 ∫

R+

|ℎ (𝑥)|²𝑒^𝛽𝑥𝑑𝑥)

≤ 2‖ℎ‖²_𝛽+𝛽²

2‖ℎ‖_𝐿²_𝛽 ≤ (2 + 𝛽²𝐶²₁ 2 ) ‖ℎ‖²_𝛾,

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which, together withLemma 1, concludes the proof.

Forℎ ∈ 𝐿¹(R), theFourier transformFℎ : R → Cis defined as

(Fℎ) (𝜉) := 1

√2𝜋∫

Rℎ (𝑥) 𝑒^{−𝑖𝜉𝑥}𝑑𝑥, 𝜉 ∈R. (18)

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Recall that𝐶₀(R)denotes the space of all continuous functions vanishing at infinity, which, equipped with the supre- mum norm, is a Banach space. We have the following result.

Lemma 3. The Fourier transformF : 𝐿¹(R) → 𝐶₀(R)is a continuous linear operator with‖F‖ ≤ 1/√2𝜋.

Lemma 4. Let𝛾 > 𝛽 > 0be arbitrary. Then, the following statements are true.

(1)For eachℎ ∈ 𝐻_𝛾⁰, one has(ℎ𝑒^(𝛽/2)∙|_(0,∞))^∗∈ 𝐿¹(R), and there is a constant𝐶₃= 𝐶₃(𝛽, 𝛾) > 0such that

󵄩󵄩󵄩󵄩󵄩󵄩󵄩(ℎ𝑒^(𝛽/2)∙󵄨󵄨󵄨󵄨󵄨(0,∞))^∗󵄩󵄩󵄩󵄩󵄩󵄩󵄩^𝐿¹^(R)≤ 𝐶₃‖ℎ‖_𝛾 ∀ℎ ∈ 𝐻_𝛾⁰. (19)

(2)For each𝜉 ∈R, the mapping

𝐻_𝛾⁰󳨀→R, ℎ 󳨃󳨀→F(ℎ𝑒^(𝛽/2)∙󵄨󵄨󵄨󵄨󵄨(0,∞))^∗(𝜉) (20)

is a continuous linear functional.

Proof. We set𝛿 := (1/2)(𝛽 + 𝛾) ∈ (𝛽, 𝛾). Letℎ ∈ 𝐻_𝛾⁰ be arbitrary. By the Cauchy-Schwarz inequality andLemma 2, we have

󵄩󵄩󵄩󵄩󵄩󵄩󵄩(ℎ𝑒^(𝛽/2)∙󵄨󵄨󵄨󵄨󵄨(0,∞))^∗󵄩󵄩󵄩󵄩󵄩󵄩󵄩^𝐿¹^(R)

= 2󵄩󵄩󵄩󵄩󵄩ℎ𝑒^(𝛽/2)∙󵄩󵄩󵄩󵄩󵄩𝐿¹(R+)= 2 ∫

R+󵄨󵄨󵄨󵄨󵄨ℎ (𝑥) 𝑒^(𝛽/2)𝑥󵄨󵄨󵄨󵄨󵄨 𝑑𝑥

= 2 ∫

R₊|ℎ (𝑥)| 𝑒^(𝛿/2)𝑥𝑒−((𝛿−𝛽)/2)𝑥𝑑𝑥

≤ 2(∫

R+

|ℎ (𝑥)|²𝑒^𝛿𝑥𝑑𝑥)^1/2(∫

R+

𝑒^{−(𝛿−𝛽)𝑥}𝑑𝑥)^1/2

= 2√ 1

𝛿 − 𝛽‖ℎ‖_𝐿²_𝛿 ≤ 2𝐶₁(𝛿, 𝛾) √ 1 𝛿 − 𝛽‖ℎ‖_𝛾,

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showing the first statement. Moreover, we have

󵄩󵄩󵄩󵄩󵄩𝑒^{((𝛽/2)−𝛿)∙}󵄩󵄩󵄩󵄩󵄩²𝐿²_𝛿 = ∫

R+

𝑒2((𝛽/2)−𝛿)𝑥𝑒^𝛿𝑥𝑑𝑥

= ∫R+

𝑒^{−(𝛿−𝛽)𝑥}𝑑𝑥 = 1 𝛿 − 𝛽,

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showing that𝑒^{((𝛽/2)−𝛿)∙} ∈ 𝐿²_𝛿. Letℎ ∈ 𝐻_𝛾⁰ and 𝜉 ∈ Rbe arbitrary. ByLemma 2, we haveℎ ∈ 𝐿²_𝛿, and hence

F(ℎ𝑒^(𝛽/2)∙󵄨󵄨󵄨󵄨󵄨(0,∞))^∗(𝜉)

= 1

√2𝜋(∫^∞

0 ℎ (𝑥) 𝑒^(𝛽/2)𝑥𝑒^{−𝑖𝜉𝑥}𝑑𝑥 + ∫⁰

−∞ℎ (−𝑥) 𝑒^{−(𝛽/2)𝑥}𝑒^{−𝑖𝜉𝑥}𝑑𝑥)

= 1

√2𝜋(∫^∞

0 ℎ (𝑥) 𝑒^(𝛽/2)𝑥𝑒^{−𝑖𝜉𝑥}𝑑𝑥 +∫^∞

0 ℎ (𝑥) 𝑒^(𝛽/2)𝑥𝑒^𝑖𝜉𝑥𝑑𝑥)

= 1

√2𝜋⟨ℎ, 𝑒^{((𝛽/2)−𝛿)∙}(𝑒^{−𝑖𝜉∙}+ 𝑒^𝑖𝜉∙)⟩_𝐿₂

𝛿,

(23) proving the second statement.

We can also define the Fourier transform on𝐿²(R)such thatF : 𝐿²(R) → 𝐿²(R)is a bijection, and we have the Plancherel isometry

⟨F𝑓,F𝑔⟩_𝐿2(R)= ⟨𝑓, 𝑔⟩_𝐿2(R) ∀𝑓, 𝑔 ∈ 𝐿²(R) . (24) Moreover, the two just reviewed definitions of the Fourier transform coincide on𝐿¹(R) ∩ 𝐿²(R). For eachℎ ∈ 𝑊¹(R), we have

(Fℎ^󸀠) (𝜉) = 𝑖𝜉 (Fℎ) (𝜉) , 𝜉 ∈R. (25) Lemma 5. For everyℎ ∈ 𝑊¹(R), one has

‖∙Fℎ‖_𝐿²_(R)≤ ‖ℎ‖_𝑊¹_(R). (26) Proof. Letℎ ∈ 𝑊¹(R)be arbitrary. By identity (25) and the Plancherel isometry (24), we have

‖∙Fℎ‖_𝐿²_(R)= 󵄩󵄩󵄩󵄩󵄩Fℎ^󸀠󵄩󵄩󵄩󵄩󵄩𝐿²(R)= 󵄩󵄩󵄩󵄩󵄩ℎ^󸀠󵄩󵄩󵄩󵄩󵄩𝐿²(R)≤ ‖ℎ‖_𝑊¹_(R), (27) finishing the proof.

3. The Embedding Result and Its Proof

In this section, we present the compact embedding result and its proof.

Theorem 6. For all𝛾 > 𝛽 > 0, one has the compact embedding

𝐻_𝛾⊂⊂ 𝐿²_𝛽⊕R. (28)

Proof. Noting that𝐻_𝛾 ≅ 𝐻_𝛾⁰ ⊕R, it suffices to prove the compact embedding𝐻_𝛾⁰ ⊂⊂ 𝐿²_𝛽. Let (ℎ_𝑗)_𝑗∈N ⊂ 𝐻_𝛾⁰ be a bounded sequence. Then, there exists a subsequence which converges weakly in𝐻_𝛾⁰. Without loss of generality, we may assume that the original sequence(ℎ_𝑗)_𝑗∈N converges weakly

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in𝐻_𝛾⁰. We will prove that(ℎ_𝑗)_𝑗∈Nis a Cauchy sequence in𝐿²_𝛽. According toLemma 2, the sequence(𝑔_𝑗)_𝑗∈Ngiven by

𝑔_𝑗:= (ℎ_𝑗𝑒^(𝛽/2)∙󵄨󵄨󵄨󵄨󵄨(0,∞))^∗, 𝑗 ∈N, (29) is a bounded sequence in 𝑊¹(R). By Lemma 1 and the Plancherel isometry (24), for all𝑗, 𝑘 ∈N, we get

󵄩󵄩󵄩󵄩󵄩ℎ^𝑘− ℎ_𝑗󵄩󵄩󵄩󵄩󵄩²𝐿²_𝛽 = 󵄩󵄩󵄩󵄩󵄩ℎ^𝑘𝑒^(𝛽/2)∙− ℎ_𝑗𝑒^(𝛽/2)∙󵄩󵄩󵄩󵄩󵄩²𝐿²(R+)

≤ 󵄩󵄩󵄩󵄩󵄩𝑔^𝑘− 𝑔_𝑗󵄩󵄩󵄩󵄩󵄩²𝐿²(R)= 󵄩󵄩󵄩󵄩󵄩F𝑔_𝑘−F𝑔_𝑗󵄩󵄩󵄩󵄩󵄩²𝐿²(R)

= ∫R󵄨󵄨󵄨󵄨󵄨(F𝑔_𝑘) (𝑥) − (F𝑔_𝑗) (𝑥)󵄨󵄨󵄨󵄨󵄨²𝑑𝑥.

(30)

Thus, for every𝑅 > 0we obtain the estimate

󵄩󵄩󵄩󵄩󵄩ℎ^𝑘− ℎ_𝑗󵄩󵄩󵄩󵄩󵄩²𝐿²_𝛽 ≤ ∫

{|𝑥|≤𝑅}󵄨󵄨󵄨󵄨󵄨(F𝑔_𝑘) (𝑥) −F(𝑔_𝑗) (𝑥)󵄨󵄨󵄨󵄨󵄨²𝑑𝑥 + ∫{|𝑥|>𝑅}󵄨󵄨󵄨󵄨󵄨(F𝑔_𝑘) (𝑥) −F(𝑔_𝑗) (𝑥)󵄨󵄨󵄨󵄨󵄨²𝑑𝑥.

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ByLemma 5, the sequence(∙F𝑔_𝑗)_𝑗∈Nis bounded in𝐿²(R).

Therefore, for an arbitrary𝜖 > 0there exists a real number 𝑅 > 0such that

∫{|𝑥|>𝑅}󵄨󵄨󵄨󵄨󵄨(F𝑔_𝑘) (𝑥) − (F𝑔𝑗) (𝑥)󵄨󵄨󵄨󵄨󵄨²𝑑𝑥

≤ 1 𝑅²∫

{|𝑥|>𝑅}|𝑥|²󵄨󵄨󵄨󵄨󵄨(F𝑔_𝑘) (𝑥) − (F𝑔_𝑗) (𝑥)󵄨󵄨󵄨󵄨󵄨²𝑑𝑥 < 𝜖

∀𝑗, 𝑘 ∈N.

(32)

ByLemma 4, for each𝜉 ∈Rthe mapping

𝐻_𝛾⁰󳨀→R, ℎ 󳨃󳨀→F(ℎ𝑒^(𝛽/2)∙󵄨󵄨󵄨󵄨󵄨(0,∞))^∗(𝜉) (33) is a continuous linear functional. Consequently, since(ℎ_𝑗)_𝑗∈N converges weakly in 𝐻_𝛾⁰, for each 𝜉 ∈ R, the real-valued sequence((F𝑔_𝑗)(𝜉))_𝑗∈Nis convergent. Moreover, by Lemmas 3and4, for allℎ ∈ 𝐻_𝛾⁰, we have the estimate

󵄩󵄩󵄩󵄩󵄩󵄩󵄩F((ℎ𝑒^(𝛽/2)∙󵄨󵄨󵄨󵄨󵄨(0,∞))^∗)󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝐶0(R)

≤ 1

√2𝜋󵄩󵄩󵄩󵄩󵄩󵄩󵄩(ℎ𝑒^(𝛽/2)∙󵄨󵄨󵄨󵄨󵄨(0,∞))^∗󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝐿¹(R)≤ 𝐶₃

√2𝜋‖ℎ‖_𝛾. (34)

Therefore, the sequence(F𝑔_𝑗)_𝑗∈Nis bounded in𝐶₀(R). Using Lebesgue’s dominated convergence theorem, we deduce that

∫{|𝑥|≤𝑅}󵄨󵄨󵄨󵄨󵄨(F𝑔_𝑘) (𝑥) − (F𝑔_𝑗) (𝑥)󵄨󵄨󵄨󵄨󵄨²𝑑𝑥 󳨀→ 0 for𝑗, 𝑘 󳨀→ ∞.

(35) Combining (31) together with (32) and (35) shows that(ℎ_𝑗)_𝑗∈N is a Cauchy sequence in𝐿²_𝛽, completing the proof.

Remark 7. Note that the proof of Theorem 6 has certain analogies to the proof of the classical Rellich embedding theorem (see, e.g., [7, Theorem V.2.13]), which states the compact embedding𝐻₀¹(Ω) ⊂⊂ 𝐿²(Ω)for an open, bounded subset Ω ⊂ R^𝑛. Here, 𝐻₀¹(Ω) denotes the Sobolev space 𝐻₀¹(Ω) =D(Ω), whereD(Ω)is the space of all𝐶^∞-functions onΩwith compact support, and where the closure is taken with respect to the topology induced by the inner product

⟨⋅, ⋅⟩_𝑊¹. Let us briefly describe the analogies and differences between the two results as follows.

(i) In the classical Rellich embedding theorem, the domain Ω is assumed to be bounded, whereas in Theorem 6we haveΩ = R₊. Moreover, we consider weighted function spaces with weight functions of the type 𝑤(𝑥) = 𝑒^𝛽𝑥 for some constant 𝛽 > 0.

This requires a careful analysis of the results regarding Fourier transforms which we have adapted to the present situation; seeLemma 4.

(ii)𝐻_𝛾and𝐻₀¹(Ω)are different kinds of spaces. While the norm on𝐻₀¹(Ω)given by (8) involves the𝐿²-norms of a functionℎand its derivativeℎ^󸀠, the norm (4) on𝐻_𝛾 only involves the𝐿²-norm of the derivativeℎ^󸀠and a point evaluation. Therefore, the embedding𝐻₀¹(Ω) ⊂ 𝐿²(Ω) follows right away, whereas we require the assumption𝛽 < 𝛾for the embedding𝐻_𝛾⁰ ⊂ 𝐿²_𝛽; see Lemma 2.

(iii) The classical Rellich embedding theorem does not need to be true with𝐻₀¹(Ω)being replaced by𝑊¹(Ω).

The reason behind this is that, in general, it is not possible to extend a function ℎ ∈ 𝑊¹(Ω) to a functioñℎ ∈ 𝑊¹(R^𝑛), which, however, is crucial in order to apply the results about Fourier transforms.

Usually, one assumes thatΩsatisfies a so-called cone condition; see, for example, [9] for further details. In our situation, we have to ensure that every function ℎ ∈ 𝐻_𝛾⁰ can be extended to a functioñℎ ∈ 𝑊¹(R), and this is provided byLemma 2.

For the rest of this section, we will describe the announced application regarding the approximation of solutions to semilinear stochastic partial differential equations (SPDEs), which in particular applies to the modeling of interest rates. Consider a SPDE of the form

𝑑𝑟_𝑡= (𝐴𝑟_𝑡+ 𝛼 (𝑡, 𝑟_𝑡)) 𝑑𝑡 + 𝜎 (𝑡, 𝑟_𝑡) 𝑑𝑊_𝑡 + ∫𝐸𝛾 (𝑡, 𝑟_𝑡−, 𝜉) (p(𝑑𝑡, 𝑑𝜉) −](𝑑𝜉) 𝑑𝑡)

𝑟₀= ℎ₀,

(36)

on some separable Hilbert space 𝐻₁ with 𝐴 denoting the generator of some strongly continuous semigroup on 𝐻₁, driven by a Wiener process𝑊and a homogeneous Poisson random measure pwith compensator𝑑𝑡 ⊗ ](𝑑𝜉)on some mark space 𝐸. We assume that the standard Lipschitz and linear growth conditions are satisfied which ensure for each

(5)

initial conditionℎ₀ ∈ 𝐻₁ the existence of a unique weak solution𝑟to (36); that is, for each𝜁 ∈D(𝐴^∗), we have almost surely

⟨𝜁, 𝑟_𝑡⟩ = ⟨𝜁, ℎ₀⟩_𝐻₁+ ∫^𝑡

0(⟨𝐴^∗𝜁, 𝑟_𝑠⟩_𝐻₁+ ⟨𝜁, 𝛼 (𝑠, 𝑟_𝑠)⟩_𝐻₁) 𝑑𝑠 + ∫^𝑡

0⟨𝜁, 𝜎 (𝑠, 𝑟_𝑠)⟩_𝐻

1𝑑𝑊_𝑠 + ∫^𝑡

0∫

𝐸⟨𝜁, 𝛾 (𝑠, 𝑟_𝑠−, 𝜉)⟩_𝐻₁(p(𝑑𝑠, 𝑑𝜉) −](𝑑𝜉) 𝑑𝑠)

∀𝑡 ≥ 0;

(37) see, for example, [10] for further details. Let𝐻₂be a larger separable Hilbert space with compact embedding𝐻₁⊂⊂ 𝐻₂. By virtue ofTheorem 6, this is in particular satisfied for the forward curve spaces𝐻₁ = 𝐻_𝛾 and𝐻₂ = 𝐿²_𝛽⊕Rfor𝛾 >

𝛽 > 0. If, furthermore,𝐴 = 𝑑/𝑑𝑥is the differential operator, which is generated by the translation semigroup(𝑆_𝑡)_𝑡≥0given by𝑆_𝑡ℎ = ℎ(𝑡 + ∙), and𝛼 = 𝛼_HJMis given by the so-called HJM drift condition

𝛼HJM(𝑡, ℎ)

= ∑

𝑗 𝜎^𝑗(𝑡, ℎ) ∫^∙

0𝜎^𝑗(𝑡, ℎ) (𝜂) 𝑑𝜂

− ∫𝐸𝛾 (𝑡, ℎ, 𝜉) [exp(− ∫^∙

0𝛾 (𝑡, ℎ, 𝜉) (𝜂) 𝑑𝜂) − 1]](𝑑𝜉) , (38) then the SPDE (36), which in this case becomes the men- tioned HJMM equation, describes the evolution of interest rates in an arbitrage free bond market; we refer to [5] for further details.

By virtue of the compact embedding𝐻₁ ⊂⊂ 𝐻₂, there exist orthonormal systems(𝑒_𝑘)_𝑘∈N of𝐻₁and(𝑓_𝑘)_𝑘∈N of𝐻₂, and a decreasing sequence(𝑠_𝑘)_𝑘∈N ⊂R₊ with𝑠_𝑘 → 0such that

ℎ =∑^∞

𝑘=1

𝑠_𝑘⟨ℎ, 𝑒_𝑘⟩_𝐻

1𝑓_𝑘 ∀ℎ ∈ 𝐻₁; (39) see, for example, [7, Theorem VI.3.6]. The numbers𝑠_𝑘are the singular numbers of the identity operator Id : 𝐻₁ → 𝐻₂. Defining the sequence(𝑇_𝑛)_𝑛∈Nof finite-rank operators

𝑇_𝑛: 𝐻₁󳨀→ 𝐹_𝑛, 𝑇_𝑛ℎ := ∑^𝑛

𝑘=1

𝑠_𝑘⟨ℎ, 𝑒_𝑘⟩_𝐻

1𝑓_𝑘, (40)

where𝐹_𝑛:= ⟨𝑓₁, . . . , 𝑓_𝑛⟩, we even have𝑇_𝑛 → Id with respect to the operator norm

‖𝑇‖ := sup

‖ℎ‖_𝐻1≤1‖𝑇ℎ‖_𝐻₂; (41)

see, for example, [7, Corollary VI.3.7]. Consequently, denoting by𝑟the weak solution to the SPDE (36) for some initial

conditionℎ₀ ∈ 𝐻₁, the sequence(𝑇_𝑛(𝑟))_𝑛∈Nis a sequence of 𝐹_𝑛-valued stochastic processes, and we have almost surely

󵄩󵄩󵄩󵄩𝑇^𝑛(𝑟_𝑡) − 𝑟_𝑡󵄩󵄩󵄩󵄩𝐻₂≤ 󵄩󵄩󵄩󵄩𝑇^𝑛−Id󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑟^𝑡󵄩󵄩󵄩󵄩𝐻₁󳨀→ 0 ∀𝑡 ≥ 0, (42) showing that the weak solution𝑟—when considered on the larger state space𝐻₂—can be approximated by the sequence of finite dimensional processes (𝑇_𝑛(𝑟))_𝑛∈N with distance between𝑇_𝑛(𝑟)and𝑟estimated in terms of the operator norm

‖𝑇_𝑛−Id‖, as shown in (42). However, the sequence(𝑇_𝑛(𝑟))_𝑛∈N does not need to be a sequence of Itˆo processes. This issue is addressed by the following result.

Proposition 8. Let(𝜖_𝑛)_𝑛∈N ⊂ (0, ∞)be an arbitrary decreas- ing sequence with𝜖_𝑛 → 0. Then, for every initial condition ℎ₀∈ 𝐻₁, there exists a sequence(𝑟^(𝑛))_𝑛∈Nof𝐹_𝑛-valued Itˆo pro- cesses such that almost surely

󵄩󵄩󵄩󵄩󵄩𝑟^(𝑛)^𝑡 − 𝑟_𝑡󵄩󵄩󵄩󵄩󵄩𝐻₂≤ (󵄩󵄩󵄩󵄩𝑇^𝑛−Id󵄩󵄩󵄩󵄩 + 𝜖^𝑛) 󵄩󵄩󵄩󵄩𝑟^𝑡󵄩󵄩󵄩󵄩𝐻1 󳨀→ 0 ∀𝑡 ≥ 0, (43) where𝑟denotes the weak solution to(36).

Proof. According to [6, Theorems 13.35.c and 13.12], the domainD(𝐴^∗)is dense in𝐻₁. Therefore, for each𝑛 ∈N, there exist elements𝜁₁^(𝑛), . . . , 𝜁_𝑛^(𝑛)∈D(𝐴^∗)such that

󵄩󵄩󵄩󵄩󵄩𝜁^𝑘^(𝑛)− 𝑒_𝑘󵄩󵄩󵄩󵄩󵄩𝐻1< 𝜖_𝑛

2^𝑘⋅ 𝑠_𝑘 ∀𝑘 = 1, . . . , 𝑛, (44) where we use the convention𝑥/0 := ∞for𝑥 > 0. We define the sequence(𝑆_𝑛)_𝑛∈Nof finite-rank operators as

𝑆_𝑛: 𝐻₁󳨀→ 𝐹_𝑛, 𝑆_𝑛ℎ :=∑^𝑛

𝑘=1

𝑠_𝑘⟨ℎ, 𝜁_𝑘^(𝑛)⟩_𝐻

1𝑓_𝑘. (45) By the geometric series, for all𝑛 ∈N, we have

󵄩󵄩󵄩󵄩𝑆^𝑛−Id󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩󵄩𝑆^𝑛− 𝑇_𝑛󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑇^𝑛−Id󵄩󵄩󵄩󵄩

≤∑^𝑛

𝑘=1

𝑠_𝑘󵄩󵄩󵄩󵄩󵄩󵄩⟨∙, 𝜁^𝑘^(𝑛)− 𝑒_𝑘⟩_𝐻

1󵄩󵄩󵄩󵄩󵄩󵄩 +󵄩󵄩󵄩󵄩𝑇^𝑛−Id󵄩󵄩󵄩󵄩

≤ 𝜖_𝑛∑^𝑛

𝑘=1

1

2^𝑘 + 󵄩󵄩󵄩󵄩𝑇^𝑛−Id󵄩󵄩󵄩󵄩 ≤ 𝜖^𝑛+ 󵄩󵄩󵄩󵄩𝑇^𝑛−Id󵄩󵄩󵄩󵄩.

(46)

For each𝑛 ∈N, let𝑟^(𝑛)be the𝐹_𝑛-valued Itˆo process 𝑟^(𝑛)_𝑡 = ℎ^(𝑛)₀ + ∫^𝑡

0𝛼^(𝑛)_𝑠 𝑑𝑠 + ∫^𝑡

0𝜎^(𝑛)_𝑠 𝑑𝑊_𝑠 + ∫^𝑡

0∫

𝐸𝛿_𝑠^(𝑛)(𝜉) (p(𝑑𝑠, 𝑑𝜉) −](𝑑𝜉, 𝑑𝑠)) ,

(47)

(6)

with parameters given by ℎ₀^(𝑛)= ∑^𝑛

𝑘=1

𝑠_𝑘⟨𝜁_𝑘^(𝑛), ℎ₀⟩_𝐻

1𝑓_𝑘, 𝛼_𝑡^(𝑛)= ∑^𝑛

𝑘=1

𝑠_𝑘(⟨𝐴^∗𝜁_𝑘^(𝑛), 𝑟_𝑡⟩_𝐻

1+ ⟨𝜁_𝑘^(𝑛), 𝛼 (𝑡, 𝑟_𝑡)⟩_𝐻

1) 𝑓_𝑘, 𝜎_𝑡^(𝑛)= ∑^𝑛

𝑘=1

𝑠_𝑘⟨𝜁_𝑘^(𝑛), 𝜎 (𝑡, 𝑟_𝑡)⟩_𝐻

1𝑓_𝑘, 𝛿^(𝑛)_𝑡 (𝜉) = ∑^𝑛

𝑘=1

𝑠_𝑘⟨𝜁_𝑘^(𝑛), 𝛿 (𝑡, 𝑟_𝑡−, 𝜉)⟩_𝐻

1𝑓_𝑘.

(48) Since𝑟is a weak solution to (36), we obtain almost surely 𝑆_𝑛(𝑟_𝑡) =∑^𝑛

𝑘=1

𝑠_𝑘⟨𝜁_𝑘^(𝑛), 𝑟_𝑡⟩_𝐻

1𝑓_𝑘

=∑^𝑛

𝑘=1

𝑠_𝑘(⟨𝜁_𝑘^(𝑛), ℎ₀⟩_𝐻

1

+ ∫^𝑡

0(⟨𝐴^∗𝜁_𝑘^(𝑛), 𝑟_𝑠⟩_𝐻

1+ ⟨𝜁^(𝑛)_𝑘 , 𝛼 (𝑠, 𝑟_𝑠)⟩_𝐻

1) 𝑑𝑠 + ∫^𝑡

0⟨𝜁^(𝑛)_𝑘 , 𝜎 (𝑠, 𝑟_𝑠)⟩_𝐻

1𝑑𝑊_𝑠 + ∫^𝑡

0∫

𝐸⟨𝜁^(𝑛)_𝑘 , 𝛿 (𝑠, 𝑟_𝑠−, 𝜉)⟩_𝐻

1

× (p(𝑑𝑠, 𝑑𝜉) −](𝑑𝜉, 𝑑𝑠))) 𝑓_𝑘

= ℎ^(𝑛)₀ + ∫^𝑡

0𝛼^(𝑛)_𝑠 𝑑𝑠 + ∫^𝑡

0𝜎_𝑠^(𝑛)𝑑𝑊_𝑠 + ∫^𝑡

0∫

𝐸𝛿^(𝑛)_𝑠 (𝜉) (p(𝑑𝑠, 𝑑𝜉) −](𝑑𝜉, 𝑑𝑠))

= 𝑟_𝑡^(𝑛) ∀𝑡 ≥ 0,

(49) which finishes the proof.

We will conclude this section with further consequences regarding the speed of convergence of the approximations (𝑟^(𝑛))_𝑛∈N provided by Proposition 8. Let ℎ₀ ∈ 𝐻₁ be an arbitrary initial condition and denote by𝑟the weak solution to (36). Furthermore, let𝑇 > 0be a finite time horizon. Since

E[sup

𝑡∈[0,𝑇]󵄩󵄩󵄩󵄩𝑟^𝑡󵄩󵄩󵄩󵄩²𝐻₁] < ∞, (50) see, for example, [10, Corollary 10.3], by (43) there exists a constant𝐾 > 0such that

E[sup

𝑡∈[0,𝑇]󵄩󵄩󵄩󵄩󵄩𝑟^𝑡^(𝑛)− 𝑟_𝑡󵄩󵄩󵄩󵄩󵄩²𝐻2]

1/2

≤ 𝐾 (󵄩󵄩󵄩󵄩𝑇^𝑛−Id󵄩󵄩󵄩󵄩 + 𝜖^𝑛) 󳨀→ 0, (51)

providing a uniform estimate for the distance of𝑟^(𝑛)and 𝑟 in the mean-square sense. Moreover, considering the pure diffusion case

𝑑𝑟_𝑡= (𝐴𝑟_𝑡+ 𝛼 (𝑡, 𝑟_𝑡)) 𝑑𝑡 + 𝜎 (𝑡, 𝑟_𝑡) 𝑑𝑊_𝑡

𝑟₀= ℎ₀, (52)

the sample paths of𝑟are continuous; for every constant𝐾 >

‖ℎ₀‖_𝐻₁the stopping time

𝜏 :=inf{𝑡 ≥ 0 : 󵄩󵄩󵄩󵄩𝑟^𝑡󵄩󵄩󵄩󵄩 ≥ 𝐾} (53) is strictly positive, and by (43) for the stopped processes we obtain almost surely

𝑡∈Rsup+󵄩󵄩󵄩󵄩󵄩𝑟^(𝑛)^𝑡∧𝜏− 𝑟_𝑡∧𝜏󵄩󵄩󵄩󵄩󵄩𝐻2 ≤ 𝐾 (󵄩󵄩󵄩󵄩𝑇^𝑛−Id󵄩󵄩󵄩󵄩 + 𝜖^𝑛) 󳨀→ 0; (54) that is, locally the solution𝑟stays in a bounded subset of𝐻_𝛾 and we obtain the uniform convergence (54).

Acknowledgment

The author is grateful to an anonymous referee for valuable comments and suggestions.

References

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[4] D. Filipovi´c and S. Tappe, “Existence of L´evy term structure models,”Finance and Stochastics, vol. 12, no. 1, pp. 83–115, 2008.

[5] D. Filipovi´c, S. Tappe, and J. Teichmann, “Term structure models driven by Wiener processes and Poisson measures: existence and positivity,”SIAM Journal on Financial Mathematics, vol. 1, no. 1, pp. 523–554, 2010.

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Compact Embeddings for Spaces of Forward Rate Curves

Research Article