Compact embeddings for spaces of forward rate curves

The goal of this note 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.


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 [4] for further details. It has been studied in a series of papers, see, e.g. [9,2], [5,7] and references therein. The state space, which contains the forward curves, is a separable Hilbert space H consisting of functions h : R + → R. In practice, forward curves have the following features: • The functions h ∈ H become flat at the long end.
• Consequently, the limit lim x→∞ h(x) exists. The second property is taken into account by choosing the Hilbert space β denotes the weighted Lebesgue space L 2 β := L 2 (R + , e βx dx) (1) for some constant β > 0. Such spaces have been used, e.g., in [9,2]. 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 : h is absolutely continuous with h γ < ∞} (2) for some constant γ > 0, where the norm is given by Such spaces have been introduced in [4] (even with more general weight functions) and further utilized, e.g., in [5,7]. Our goal of this note is to show that for all γ > β > 0 we have the compact embedding H γ ⊂⊂ L 2 β ⊕ R, that is, the forward curve spaces used in [4] and forthcoming papers are contained in the forward curve spaces used in [9], 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 space H γ , applying these operators its solutions can be approximated by a sequence of finite dimensional processes in the larger state space L 2 β ⊕ R; we refer to Section 3 for further details.
The remainder of this note is organized as follows. In Section 2 we provide the required preliminaries. In Section 3 we present the embedding result and its proof, and we outline the described approximation result concerning solutions of the HJMM equation.

Preliminaries and notation
In this section, we provide the required preliminary results and some basic notation. Concerning the upcoming results about Sobolev spaces and Fourier transforms, we refer to any textbook about functional analysis, such as [8] or [10].
As noted in the introduction, for positive real numbers β, γ > 0 the separable Hilbert spaces L 2 β ⊕ R and H γ are given by (1) and (2), respectively. These spaces and the forthcoming Sobolev spaces will be regarded as spaces of complex-valued functions. For every h ∈ H γ the limit h(∞) := lim x→∞ h(x) exists and the subspace is a closed subspace of H γ , see [4]. For an open set Ω ⊂ R we denote by W 1 (Ω) the Sobolev space which, equipped with the inner product is a separable Hilbert space. Here, derivatives are understood as weak derivatives.
For a function h ∈ W 1 ((0, ∞)) the extension h½ (0,∞) : R → C does, in general, not belong to W 1 (R). In the present situation, this technical problem can be resolved as follows. Let h : (0, ∞) → C be a continuous function such that the limit h(0) := lim x→0 h(x) exists. Then we define the reflection h * : R → C as Lemma 2.1. The following statements are true: Proof. This follows from a straightforward calculation following the proof of [3,Theorem 8.6].
Lemma 2.2. Let γ > β > 0 be arbitrary. Then the following statements are true: (2) We have H 0 γ ⊂ L 2 β and there is a constant C 1 = C 1 (β, γ) > 0 such that and there is a constant C 2 = C 2 (β, γ) > 0 such that The first statement is a direct consequence of the representation of the norm on H 0 γ given by (3). Let h ∈ H 0 γ be arbitrary. By the Cauchy-Schwarz inequality we obtain proving the second statement. Furthermore, by (6) we have and by estimates (5), (6) we obtain which, together with Lemma 2.1, concludes the proof.
Recall that C 0 (R) denotes the space of all continuous functions vanishing at infinity, which, equipped with the supremum norm, is a Banach space. We have the following result: Lemma 2.4. Let γ > β > 0 be arbitrary. Then the following statements are true: (2) For each ξ ∈ R the mapping Proof. We set δ := 1 2 (β + γ) ∈ (β, γ). Let h ∈ H 0 γ be arbitrary. By the Cauchy-Schwarz inequality and Lemma 2.2 we have showing the first statement. Moreover, we have showing that e ((β/2)−δ)• ∈ L 2 δ . Let h ∈ H 0 γ and ξ ∈ R be arbitrary. By Lemma 2.2 we have h ∈ L 2 δ , and hence proving the second statement.
We can also define the Fourier transform on L 2 (R) such that F : L 2 (R) → L 2 (R) is a bijection and we have the Plancherel isometry Moreover, the two just reviewed definitions of the Fourier transform coincide on Proof. Let h ∈ W 1 (R) be arbitrary. By identity (9) and the Plancherel isometry (8) we have finishing the proof.

The embedding result and its proof
In this section, we present the compact embedding result and its proof.
Proof. Noting that H γ ∼ = H 0 γ ⊕ R, it suffices to prove the compact embedding H 0 γ ⊂⊂ L 2 β . Let (h j ) j∈N ⊂ H 0 γ be a bounded sequence. Then there exists a subsequence which converges weakly in H 0 γ . Without loss of generality, we may assume that the original sequence (h j ) j∈N converges weakly in H 0 γ . We shall prove that (h j ) j∈N is a Cauchy sequence in L 2 β . According to Lemma 2.2, the sequence (g j ) j∈N given by is a bounded sequence in W 1 (R). By Lemma 2.1 and the Plancherel isometry (8), for all j, k ∈ N we get Thus, for every R > 0 we obtain the estimate By Lemma 2.5, the sequence (•F g j ) j∈N is bounded in L 2 (R). Therefore, for an arbitrary ǫ > 0 there exists a real number R > 0 such that By Lemma 2.4, for each ξ ∈ R the mapping is a continuous linear functional. Consequently, since (h j ) j∈N converges weakly in H 0 γ , for each ξ ∈ R the real-valued sequence ((F g j )(ξ)) j∈N is convergent. Moreover, by Lemmas 2.3 and 2.4, for all h ∈ H 0 γ we have the estimate Therefore, the sequence (F g j ) j∈N is bounded in C 0 (R). Using Lebesgue's dominated convergence theorem, we deduce that Combining (10) together with (11) and (12) shows that (h j ) j∈N is a Cauchy sequence in L 2 β , completing the proof.  (3) on H γ only involves the L 2 -norm of the derivative h ′ and a point evaluation. Therefore, the embedding H 1 0 (Ω) ⊂ L 2 (Ω) follows right away, whereas we require the assumption β < γ for the embedding H 0 γ ⊂ L 2 β ; see Lemma 2.2.
• The classical Rellich embedding theorem does not need to be true with H 1 0 (Ω) being replaced by W 1 (Ω). The reason behind it is that, in general, it is not possible to extend a function h ∈ W 1 (Ω) to a functionh ∈ W 1 (R n ), which, however, is crucial in order to apply the results about Fourier transforms. Usually, one assumes that Ω satisfies a so-called cone condition, see, e.g., [1] for further details. In our situation, we have to ensure that every function h ∈ H 0 γ can be extended to a functionh ∈ W 1 (R), and this is provided by Lemma 2.2.
For the rest of this section, we shall 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 + E γ(t, r t− , ξ)(p(dt, dξ) − ν(dξ)dt) on some separable Hilbert space H 1 with A denoting the generator of some strongly continuous semigroup on H 1 , driven by a Wiener process W and a homogeneous Poisson random measure p with compensator dt⊗ ν(dξ) on some mark space E. We assume that the standard Lipschitz and linear growth conditions are satisfied which ensure for each initial condition h 0 ∈ H 1 the existence of a unique weak solution r to (13), that is, for each ζ ∈ D(A * ) we have almost surely ζ, γ(s, r s− , ξ) H1 (p(ds, dξ) − ν(dξ)ds) for all t ≥ 0, see, e.g., [6] for further details. Let H 2 be a larger separable Hilbert space with compact embedding H 1 ⊂⊂ H 2 . By virtue of Theorem 3.1, this is in particular satisfied for the forward curve spaces H 1 = H γ and H 2 = L 2 β ⊕ R for γ > β > 0. If, furthermore, A = d/dx is the differential operator, which is generated by the translation semigroup (S t ) t≥0 given by S t h = h(t + •), and α = α HJM is given by the so-called HJM drift condition then the SPDE (13), which in this case becomes the mentioned HJMM equation, describes the evolution of interest rates in an arbitrage free bond market; we refer to [7] for further details.
By virtue of the compact embedding H 1 ⊂⊂ H 2 , there exist orthonormal systems (e k ) k∈N of H 1 and (f k ) k∈N of H 2 , and a decreasing sequence see, e.g., [10,Satz VI.3.6]. The numbers s k are the singular numbers of the identity operator Id : H 1 → H 2 . Defining the sequence (T n ) n∈N of finite-rank operators where F n := f 1 , . . . , f n , we even have T n → Id with respect to the operator norm see, e.g., [10,Korollar VI.3.7]. Consequently, denoting by r the weak solution to the SPDE (13) for some initial condition h 0 ∈ H 1 , the sequence (T n (r)) n∈N is a sequence of F n -valued stochastic processes, and we have almost surely showing that the weak solution r -when considered on the larger state space H 2 -can be approximated by the sequence of finite dimensional processes (T n (r)) n∈N with distance between T n (r) and r estimated in terms of the operator norm T n − Id , as shown in (14). However, the sequence (T n (r)) n∈N does not need to be a sequence of Itô processes. This issue is addressed by the following result: Proposition 3.3. Let (ǫ n ) n∈N ⊂ (0, ∞) be an arbitrary decreasing sequence with ǫ n → 0. Then, for every initial condition h 0 ∈ H 1 there exists a sequence (r (n) ) n∈N of F n -valued Itô processes such that almost surely where r denotes the weak solution to (13).
Proof. According to [8,Theorems 13.35.c and 13.12], the domain D(A * ) is dense in H 1 . Therefore, for each n ∈ N there exist elements ζ where we use the convention x 0 := ∞ for x > 0. We define the sequence (S n ) n∈N of finite-rank operators as By the geometric series, for all n ∈ N we have For each n ∈ N let r (n) be the F n -valued Itô process with parameters given by Since r is a weak solution to (13), we obtain almost surely k , r s H1 + ζ We shall conclude this section with further consequences regarding the speed of convergence of the approximations (r (n) ) n∈N provided by Proposition 3.3. Let h 0 ∈ H 1 be an arbitrary initial condition and denote by r the weak solution to (13). Furthermore, let T > 0 be a finite time horizon. Since 1/2 ≤ K( T n − Id + ǫ n ) → 0, providing a uniform estimate for the distance of r (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 r 0 = h 0 , the sample paths of r are continuous, for every constant K > h 0 H1 the stopping time τ := inf{t ≥ 0 : r t ≥ K} is strictly positive, and by (15) for the stopped processes we obtain almost surely sup t∈R+ r (n) t∧τ − r t∧τ H2 ≤ K( T n − Id + ǫ n ) → 0, i.e., locally the solution r stays in a bounded subset of H γ and we obtain the uniform convergence (16).