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 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 limx→∞h(x) exists.

The second property is taken into account by choosing the Hilbert space
(1)Lβ2⊕ℝ,
where Lβ2 denotes the weighted Lebesgue space
(2)Lβ2:=L2(ℝ+,eβxdx),
for some constant β>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
(3)Hγ∶={h:ℝ+⟶ℝ:hisabsolutelycontinuouswith∥h∥γ<∞},
for some constant γ>0, where the norm is given by
(4)∥h∥γ∶=(|h(0)|2+∫ℝ+|h′(x)|2eγxdx)1/2.
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 all γ>β>0 we have the compact embedding
(5)Hγ⊂⊂Lβ2⊕ℝ,
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 space Hγ, applying these operators its solutions can be approximated by a sequence of finite dimensional processes in the larger state space Lβ2⊕ℝ; we refer to Section 3 for further details.

The remainder of this paper 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.

2. 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 [6] or [7].

As noted in the introduction, for positive real numbers β,γ>0, the separable Hilbert spaces Lβ2⊕ℝ and Hγ are given by (2) and (3), respectively. These spaces and the forthcoming Sobolev spaces will be regarded as spaces of complex-valued functions. For every h∈Hγ, the limit h(∞):=limx→∞h(x) exists, and the subspace
(6)Hγ0∶={h∈Hγ:h(∞)=0}
is a closed subspace of Hγ; see [1]. For an open set Ω⊂ℝ, we denote by W1(Ω) the Sobolev space
(7)W1(Ω)∶={f∈L2(Ω):f′∈L2(Ω)exists},
which, equipped with the inner product
(8)〈f,g〉W1(Ω)=〈f,g〉L2(Ω)+〈f′,g′〉L2(Ω),
is a separable Hilbert space. Here, derivatives are understood as weak derivatives.

For a function h∈W1((0,∞)), the extension h1(0,∞):ℝ→ℂ in general, does not belong to W1(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):=limx→0h(x) exists. Then, we define the reflection h*:ℝ→ℂ as
(9)h*(x)∶={h(x),ifx≥0,h(-x),ifx<0.

Lemma 1.

The following statements are true.

For each h∈W1((0,∞)), one has h*∈W1(ℝ).

The mapping W1((0,∞))→W1(ℝ), h↦h* is a bounded linear operator.

For each h∈W1((0,∞)), one has
(10)∥h∥W1((0,∞))≤∥h*∥W1(ℝ)≤2∥h∥W1((0,∞)),∥h∥L2((0,∞))≤∥h*∥L2(ℝ)≤2∥h∥L2((0,∞)).

Proof.

This follows from a straightforward calculation following the proof of [8, Theorem 8.6].

Lemma 2.

Let γ>β>0 be arbitrary. Then, the following statements are true.

One has Hγ0⊂Hβ0, and
(11)∥h∥β≤∥h∥γ∀h∈Hγ0.

One has Hγ0⊂Lβ2, and there is a constant C1=C1(β,γ)>0 such that
(12)∥h∥Lβ2≤C1∥h∥γ∀h∈Hγ0.

For each h∈Hγ0, one has
(13)he(β/2)•|(0,∞)∈W1((0,∞)),(he(β/2)•|(0,∞))*∈W1(ℝ),
and there is a constant C2=C2(β,γ)>0 such that
(14)∥(he(β/2)•|(0,∞))*∥W1(ℝ)≤C2∥h∥γ∀h∈Hγ0.

Proof.

The first statement is a direct consequence of the representation of the norm on Hγ0 given by (4). Let h∈Hγ0 be arbitrary. By the Cauchy-Schwarz inequality, we obtain
(15)∥h∥Lβ22=∫ℝ+|h(x)|2eβxdx=∫ℝ+(∫x∞h′(η)e(γ/2)ηe-(γ/2)ηdη)2eβxdx≤∫ℝ+(∫x∞|h′(η)|2eγηdη)(∫x∞e-γηdη)eβxdx≤∫ℝ+(∫ℝ+|h′(η)|2eγηdη)1γe-γxeβxdx≤1γ(∫ℝ+e-(γ-β)xdx)∥h∥γ2=1γ(γ-β)∥h∥γ2,
proving the second statement. Furthermore, by (12) we have
(16)∥he(β/2)•|(0,∞)∥L2((0,∞))2=∫ℝ+|h(x)e(β/2)x|2dx=∫ℝ+|h(x)|2eβxdx=∥h∥Lβ22≤C12∥h∥γ2,
and by estimates (11), (12), we obtain
(17)∥(ddx)(he(β/2)•|(0,∞))∥L2((0,∞))2=∫ℝ+|ddx(h(x)e(β/2)x)|2dx=∫ℝ+|h′(x)e(β/2)x+β2h(x)e(β/2)x|2dx≤2(∫ℝ+|h′(x)|2eβxdx+β24∫ℝ+|h(x)|2eβxdx)≤2∥h∥β2+β22∥h∥Lβ2≤(2+β2C122)∥h∥γ2,
which, together with Lemma 1, concludes the proof.

For h∈L1(ℝ), the Fourier transform ℱh:ℝ→ℂ is defined as
(18)(ℱh)(ξ)∶=12π∫ℝh(x)e-iξxdx,ξ∈ℝ.
Recall that C0(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 3.

The Fourier transform ℱ:L1(ℝ)→C0(ℝ) is a continuous linear operator with ∥ℱ∥≤1/2π.

Lemma 4.

Let γ>β>0 be arbitrary. Then, the following statements are true.

For each h∈Hγ0, one has (he(β/2)•|(0,∞))*∈L1(ℝ), and there is a constant C3=C3(β,γ)>0 such that
(19)∥(he(β/2)•|(0,∞))*∥L1(ℝ)≤C3∥h∥γ∀h∈Hγ0.

For each ξ∈ℝ, the mapping
(20)Hγ0⟶ℝ,h⟼ℱ(he(β/2)•|(0,∞))*(ξ)
is a continuous linear functional.

Proof.

We set δ:=(1/2)(β+γ)∈(β,γ). Let h∈Hγ0 be arbitrary. By the Cauchy-Schwarz inequality and Lemma 2, we have
(21)∥(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)|2eδxdx)1/2(∫ℝ+e-(δ-β)xdx)1/2=21δ-β∥h∥Lδ2≤2C1(δ,γ)1δ-β∥h∥γ,
showing the first statement. Moreover, we have
(22)∥e((β/2)-δ)•∥Lδ22=∫ℝ+e2((β/2)-δ)xeδxdx=∫ℝ+e-(δ-β)xdx=1δ-β,
showing that e((β/2)-δ)•∈Lδ2. Let h∈Hγ0 and ξ∈ℝ be arbitrary. By Lemma 2, we have h∈Lδ2, and hence
(23)ℱ(he(β/2)•|(0,∞))*(ξ)=12π(∫0∞h(x)e(β/2)xe-iξxdx+∫-∞0h(-x)e-(β/2)xe-iξxdx)=12π(∫0∞h(x)e(β/2)xe-iξxdx+∫0∞h(x)e(β/2)xeiξxdx)=12π〈h,e((β/2)-δ)•(e-iξ•+eiξ•)〉Lδ2,
proving the second statement.

We can also define the Fourier transform on L2(ℝ) such that ℱ:L2(ℝ)→L2(ℝ) is a bijection, and we have the Plancherel isometry
(24)〈ℱf,ℱg〉L2(ℝ)=〈f,g〉L2(ℝ)∀f,g∈L2(ℝ).
Moreover, the two just reviewed definitions of the Fourier transform coincide on L1(ℝ)∩L2(ℝ). For each h∈W1(ℝ), we have
(25)(ℱh′)(ξ)=iξ(ℱh)(ξ),ξ∈ℝ.

Lemma 5.

For every h∈W1(ℝ), one has
(26)∥•ℱh∥L2(ℝ)≤∥h∥W1(ℝ).

Proof.

Let h∈W1(ℝ) be arbitrary. By identity (25) and the Plancherel isometry (24), we have
(27)∥•ℱh∥L2(ℝ)=∥ℱh′∥L2(ℝ)=∥h′∥L2(ℝ)≤∥h∥W1(ℝ),
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 all γ>β>0, one has the compact embedding
(28)Hγ⊂⊂Lβ2⊕ℝ.

Proof.

Noting that Hγ≅Hγ0⊕ℝ, it suffices to prove the compact embedding Hγ0⊂⊂Lβ2. Let (hj)j∈ℕ⊂Hγ0 be a bounded sequence. Then, there exists a subsequence which converges weakly in Hγ0. Without loss of generality, we may assume that the original sequence (hj)j∈ℕ converges weakly in Hγ0. We will prove that (hj)j∈ℕ is a Cauchy sequence in Lβ2. According to Lemma 2, the sequence (gj)j∈ℕ given by
(29)gj∶=(hje(β/2)•|(0,∞))*,j∈ℕ,
is a bounded sequence in W1(ℝ). By Lemma 1 and the Plancherel isometry (24), for all j,k∈ℕ, we get
(30)∥hk-hj∥Lβ22=∥hke(β/2)•-hje(β/2)•∥L2(ℝ+)2≤∥gk-gj∥L2(ℝ)2=∥ℱgk-ℱgj∥L2(ℝ)2=∫ℝ|(ℱgk)(x)-(ℱgj)(x)|2dx.
Thus, for every R>0 we obtain the estimate
(31)∥hk-hj∥Lβ22≤∫{|x|≤R}|(ℱgk)(x)-ℱ(gj)(x)|2dx+∫{|x|>R}|(ℱgk)(x)-ℱ(gj)(x)|2dx.
By Lemma 5, the sequence (•ℱgj)j∈ℕ is bounded in L2(ℝ). Therefore, for an arbitrary ϵ>0 there exists a real number R>0 such that
(32)∫{|x|>R}|(ℱgk)(x)-(ℱgj)(x)|2dx≤1R2∫{|x|>R}|x|2|(ℱgk)(x)-(ℱgj)(x)|2dx<ϵhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh∀j,k∈ℕ.
By Lemma 4, for each ξ∈ℝ the mapping
(33)Hγ0⟶ℝ,h⟼ℱ(he(β/2)•|(0,∞))*(ξ)
is a continuous linear functional. Consequently, since (hj)j∈ℕ converges weakly in Hγ0, for each ξ∈ℝ, the real-valued sequence ((ℱgj)(ξ))j∈ℕ is convergent. Moreover, by Lemmas 3 and 4, for all h∈Hγ0, we have the estimate
(34)∥ℱ((he(β/2)•|(0,∞))*)∥C0(ℝ)≤12π∥(he(β/2)•|(0,∞))*∥L1(ℝ)≤C32π∥h∥γ.
Therefore, the sequence (ℱgj)j∈ℕ is bounded in C0(ℝ). Using Lebesgue’s dominated convergence theorem, we deduce that
(35)∫{|x|≤R}|(ℱgk)(x)-(ℱgj)(x)|2dx⟶0forj,k⟶∞.
Combining (31) together with (32) and (35) shows that (hj)j∈ℕ is a Cauchy sequence in Lβ2, 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 embedding H01(Ω)⊂⊂L2(Ω) for an open, bounded subset Ω⊂ℝn. Here, H01(Ω) denotes the Sobolev space H01(Ω)=𝒟(Ω)¯, where 𝒟(Ω) is the space of all C∞-functions on Ω with 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.

In the classical Rellich embedding theorem, the domain Ω is assumed to be bounded, whereas in Theorem 6 we have Ω=ℝ+. Moreover, we consider weighted function spaces with weight functions of the type w(x)=eβx for some constant β>0. This requires a careful analysis of the results regarding Fourier transforms which we have adapted to the present situation; see Lemma 4.

Hγ and H01(Ω) are different kinds of spaces. While the norm on H01(Ω) given by (8) involves the L2-norms of a function h and its derivative h′, the norm (4) on Hγ only involves the L2-norm of the derivative h′ and a point evaluation. Therefore, the embedding H01(Ω)⊂L2(Ω) follows right away, whereas we require the assumption β<γ for the embedding Hγ0⊂Lβ2; see Lemma 2.

The classical Rellich embedding theorem does not need to be true with H01(Ω) being replaced by W1(Ω). The reason behind this is that, in general, it is not possible to extend a function h∈W1(Ω) to a function h~∈W1(ℝn), 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, for example, [9] for further details. In our situation, we have to ensure that every function h∈Hγ0 can be extended to a function h~∈W1(ℝ), and this is provided by Lemma 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
(36)drt=(Art+α(t,rt))dt+σ(t,rt)dWt+∫Eγ(t,rt-,ξ)(𝔭(dt,dξ)-ν(dξ)dt)r0=h0,
on some separable Hilbert space H1 with A denoting the generator of some strongly continuous semigroup on H1, 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 h0∈H1 the existence of a unique weak solution r to (36); that is, for each ζ∈𝒟(A*), we have almost surely
(37)〈ζ,rt〉=〈ζ,h0〉H1+∫0t(〈A*ζ,rs〉H1+〈ζ,α(s,rs)〉H1)ds+∫0t〈ζ,σ(s,rs)〉H1dWs+∫0t∫E〈ζ,γ(s,rs-,ξ)〉H1(𝔭(ds,dξ)-ν(dξ)ds)hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh∀t≥0;
see, for example, [10] for further details. Let H2 be a larger separable Hilbert space with compact embedding H1⊂⊂H2. By virtue of Theorem 6, this is in particular satisfied for the forward curve spaces H1=Hγ and H2=Lβ2⊕ℝ for γ>β>0. If, furthermore, A=d/dx is the differential operator, which is generated by the translation semigroup (St)t≥0 given by Sth=h(t+•), and α=αHJM is given by the so-called HJM drift condition
(38)αHJM(t,h)=∑jσj(t,h)∫0•σj(t,h)(η)dη-∫Eγ(t,h,ξ)[exp(-∫0•γ(t,h,ξ)(η)dη)-1]ν(dξ),
then the SPDE (36), which in this case becomes the mentioned 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 embedding H1⊂⊂H2, there exist orthonormal systems (ek)k∈ℕ of H1 and (fk)k∈ℕ of H2, and a decreasing sequence (sk)k∈ℕ⊂ℝ+ with sk→0 such that
(39)h=∑k=1∞sk〈h,ek〉H1fk∀h∈H1;
see, for example, [7, Theorem VI.3.6]. The numbers sk are the singular numbers of the identity operator Id:H1→H2. Defining the sequence (Tn)n∈ℕ of finite-rank operators
(40)Tn:H1⟶Fn,Tnh∶=∑k=1nsk〈h,ek〉H1fk,
where Fn:=〈f1,…,fn〉, we even have Tn→Id with respect to the operator norm
(41)∥T∥∶=sup∥h∥H1≤1∥Th∥H2;
see, for example, [7, Corollary VI.3.7]. Consequently, denoting by r the weak solution to the SPDE (36) for some initial condition h0∈H1, the sequence (Tn(r))n∈ℕ is a sequence of Fn-valued stochastic processes, and we have almost surely
(42)∥Tn(rt)-rt∥H2≤∥Tn-Id∥∥rt∥H1⟶0∀t≥0,
showing that the weak solution r—when considered on the larger state space H2—can be approximated by the sequence of finite dimensional processes (Tn(r))n∈ℕ with distance between Tn(r) and r estimated in terms of the operator norm ∥Tn-Id∥, as shown in (42). However, the sequence (Tn(r))n∈ℕ does not need to be a sequence of Itô processes. This issue is addressed by the following result.

Proposition 8.

Let (ϵn)n∈ℕ⊂(0,∞) be an arbitrary decreasing sequence with ϵn→0. Then, for every initial condition h0∈H1, there exists a sequence (r(n))n∈ℕ of Fn-valued Itô processes such that almost surely
(43)∥rt(n)-rt∥H2≤(∥Tn-Id∥+ϵn)∥rt∥H1⟶0∀t≥0,
where r denotes the weak solution to (36).

Proof.

According to [6, Theorems 13.35.c and 13.12], the domain 𝒟(A*) is dense in H1. Therefore, for each n∈ℕ, there exist elements ζ1(n),…,ζn(n)∈𝒟(A*) such that
(44)∥ζk(n)-ek∥H1<ϵn2k·sk∀k=1,…,n,
where we use the convention x/0:=∞ for x>0. We define the sequence (Sn)n∈ℕ of finite-rank operators as
(45)Sn:H1⟶Fn,Snh∶=∑k=1nsk〈h,ζk(n)〉H1fk.
By the geometric series, for all n∈ℕ, we have
(46)∥Sn-Id∥≤∥Sn-Tn∥+∥Tn-Id∥≤∑k=1nsk∥〈•,ζk(n)-ek〉H1∥+∥Tn-Id∥≤ϵn∑k=1n12k+∥Tn-Id∥≤ϵn+∥Tn-Id∥.
For each n∈ℕ, let r(n) be the Fn-valued Itô process
(47)rt(n)=h0(n)+∫0tαs(n)ds+∫0tσs(n)dWs+∫0t∫Eδs(n)(ξ)(𝔭(ds,dξ)-ν(dξ,ds)),
with parameters given by
(48)h0(n)=∑k=1nsk〈ζk(n),h0〉H1fk,αt(n)=∑k=1nsk(〈A*ζk(n),rt〉H1+〈ζk(n),α(t,rt)〉H1)fk,σt(n)=∑k=1nsk〈ζk(n),σ(t,rt)〉H1fk,δt(n)(ξ)=∑k=1nsk〈ζk(n),δ(t,rt-,ξ)〉H1fk.
Since r is a weak solution to (36), we obtain almost surely
(49)Sn(rt)=∑k=1nsk〈ζk(n),rt〉H1fk=∑k=1nsk(∫0t〈ζk(n),h0〉H1hhhhhhhh+∫0t(〈A*ζk(n),rs〉H1+〈ζk(n),α(s,rs)〉H1)dshhhhhhhh+∫0t〈ζk(n),σ(s,rs)〉H1dWshhhhhhhh+∫0t∫E〈ζk(n),δ(s,rs-,ξ)〉H1hhhhhhhhhhh×(𝔭(ds,dξ)-ν(dξ,ds))∫0t)fk=h0(n)+∫0tαs(n)ds+∫0tσs(n)dWs+∫0t∫Eδs(n)(ξ)(𝔭(ds,dξ)-ν(dξ,ds))=rt(n)∀t≥0,
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 by r the weak solution to (36). Furthermore, let T>0 be a finite time horizon. Since
(50)𝔼[supt∈[0,T]∥rt∥H12]<∞,
see, for example, [10, Corollary 10.3], by (43) there exists a constant K>0 such that
(51)𝔼[supt∈[0,T]∥rt(n)-rt∥H22]1/2≤K(∥Tn-Id∥+ϵn)⟶0,
providing a uniform estimate for the distance of r(n) and r in the mean-square sense. Moreover, considering the pure diffusion case
(52)drt=(Art+α(t,rt))dt+σ(t,rt)dWtr0=h0,
the sample paths of r are continuous; for every constant K>∥h0∥H1 the stopping time
(53)τ∶=inf{t≥0:∥rt∥≥K}
is strictly positive, and by (43) for the stopped processes we obtain almost surely
(54)supt∈ℝ+∥rt∧τ(n)-rt∧τ∥H2≤K(∥Tn-Id∥+ϵn)⟶0;
that is, locally the solution r stays in a bounded subset of Hγ and we obtain the uniform convergence (54).

Acknowledgment

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

FilipovićD.RusinekA.Mean reversion for HJMM forward rate modelsBarskiM.ZabczykJ.Heath-Jarrow-Morton-Musiela equation with Lévy perturbationFilipovićD.TappeS.Existence of Lévy term structure modelsFilipovićD.TappeS.TeichmannJ.Term structure models driven by Wiener processes and Poisson measures: existence and positivityRudinW.WernerD.BrezisH.AdamsR. A.FournierJ. J. F.FilipovićD.TappeS.TeichmannJ.Jump-diffusions in Hilbert spaces: existence, stability and numerics