AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 709505 10.1155/2013/709505 709505 Research Article Compact Embeddings for Spaces of Forward Rate Curves http://orcid.org/0000-0003-3523-0618 Tappe Stefan Parmeggiani Alberto Institut für Mathematische Stochastik Leibniz Universität Hannover Welfengarten 1, 30167 Hannover Germany uni-hannover.de 2013 3 10 2013 2013 19 06 2013 02 09 2013 2013 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  for further details. It has been studied in a series of papers; see, for example,  and references therein. The state space, which contains the forward curves, is a separable Hilbert space H consisting of functions h:+. In practice, forward curves have the following features.

The functions hH become flat at the long end.

Consequently, the limit limxh(x) exists.

The second property is taken into account by choosing the Hilbert space (1)Lβ2, where Lβ2 denotes the weighted Lebesgue space (2)Lβ2:=L2(+,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:+:his  absolutely  continuous  with  hγ<}, for some constant γ>0, where the norm is given by (4)hγ=(|h(0)|2++|h(x)|2eγxdx)1/2. Such spaces have been introduced in  (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)HγLβ2, that is, the forward curve spaces used in  and forthcoming papers are contained in the forward curve spaces used in , 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; 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  or .

As noted in the introduction, for positive real numbers β,γ>0, the separable Hilbert spaces Lβ2 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 hHγ, the limit h():=limxh(x) exists, and the subspace (6)Hγ0={hHγ:h()=0} is a closed subspace of Hγ; see . For an open set Ω, we denote by W1(Ω) the Sobolev space (7)W1(Ω)={fL2(Ω):fL2(Ω)exists}, which, equipped with the inner product (8)f,gW1(Ω)=f,gL2(Ω)+f,gL2(Ω), is a separable Hilbert space. Here, derivatives are understood as weak derivatives.

For a function hW1((0,)), the extension h1(0,): in general, does not belong to W1(). In the present situation, this technical problem can be resolved as follows. Let h:(0,) be a continuous function such that the limit h(0):=limx0h(x) exists. Then, we define the reflection h*: as (9)h*(x)={h(x),if  x0,h(-x),if  x<0.

Lemma 1.

The following statements are true.

For each hW1((0,)), one has h*W1().

The mapping W1((0,))W1(), hh* is a bounded linear operator.

For each hW1((0,)), one has (10)hW1((0,))h*W1()2hW1((0,)),hL2((0,))h*L2()2hL2((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 Hγ0Hβ0, and (11)hβhγhHγ0.

One has Hγ0Lβ2, and there is a constant C1=C1(β,γ)>0 such that (12)hLβ2C1hγhHγ0.

For each hHγ0, one has (13)he(β/2)|(0,)W1((0,)),(he(β/2)|(0,))*W1(), and there is a constant C2=C2(β,γ)>0 such that (14)(he(β/2)|(0,))*W1()C2hγhHγ0.

Proof.

The first statement is a direct consequence of the representation of the norm on Hγ0 given by (4). Let hHγ0 be arbitrary. By the Cauchy-Schwarz inequality, we obtain (15)hLβ22=+|h(x)|2eβxdx=+(xh(η)e(γ/2)ηe-(γ/2)ηdη)2eβxdx+(x|h(η)|2eγηdη)(xe-γηdη)eβxdx+(+|h(η)|2eγηdη)1γe-γxeβxdx1γ(+e-(γ-β)xdx)hγ2=1γ(γ-β)hγ2, proving the second statement. Furthermore, by (12) we have (16)he(β/2)|(0,)L2((0,))2=+|h(x)e(β/2)x|2dx=+|h(x)|2eβxdx=hLβ22C12hγ2, and by estimates (11), (12), we obtain (17)(ddx)(he(β/2)|(0,))L2((0,))2=+|ddx(h(x)e(β/2)x)|2dx=+|h(x)e(β/2)x+β2h(x)e(β/2)x|2dx2(+|h(x)|2eβxdx+β24+|h(x)|2eβxdx)2hβ2+β22hLβ2(2+β2C122)hγ2, which, together with Lemma 1, concludes the proof.

For hL1(), the Fourier transform h: is defined as (18)(h)(ξ)=12πh(x)e-iξxdx,ξ. Recall that C0() 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 :L1()C0() is a continuous linear operator with 1/2π.

Lemma 4.

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

For each hHγ0, one has (he(β/2)|(0,))*L1(), and there is a constant C3=C3(β,γ)>0 such that (19)(he(β/2)|(0,))*L1()C3hγhHγ0.

For each ξ, the mapping (20)Hγ0,h(he(β/2)|(0,))*(ξ) is a continuous linear functional.

Proof.

We set δ:=(1/2)(β+γ)(β,γ). Let hHγ0 be arbitrary. By the Cauchy-Schwarz inequality and Lemma 2, we have (21)(he(β/2)|(0,))*L1()=2he(β/2)L1(+)=2+|h(x)e(β/2)x|dx=2+|h(x)|e(δ/2)xe-((δ-β)/2)xdx2(+|h(x)|2eδxdx)1/2(+e-(δ-β)xdx)1/2=21δ-βhLδ22C1(δ,γ)1δ-βhγ, showing the first statement. Moreover, we have (22)e((β/2)-δ)Lδ22=+e2((β/2)-δ)xeδxdx=+e-(δ-β)xdx=1δ-β, showing that e((β/2)-δ)Lδ2. Let hHγ0 and ξ be arbitrary. By Lemma 2, we have hLδ2, and hence (23)(he(β/2)|(0,))*(ξ)=12π(0h(x)e(β/2)xe-iξxdx+-0h(-x)e-(β/2)xe-iξxdx)=12π(0h(x)e(β/2)xe-iξxdx+0h(x)e(β/2)xeiξxdx)=12πh,e((β/2)-δ)(e-iξ+eiξ)Lδ2, proving the second statement.

We can also define the Fourier transform on L2() such that :L2()L2() is a bijection, and we have the Plancherel isometry (24)f,gL2()=f,gL2()f,gL2(). Moreover, the two just reviewed definitions of the Fourier transform coincide on L1()L2(). For each hW1(), we have (25)(h)(ξ)=iξ(h)(ξ),ξ.

Lemma 5.

For every hW1(), one has (26)hL2()hW1().

Proof.

Let hW1() be arbitrary. By identity (25) and the Plancherel isometry (24), we have (27)hL2()=hL2()=hL2()hW1(), 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)HγLβ2.

Proof.

Noting that HγHγ0, it suffices to prove the compact embedding Hγ0Lβ2. Let (hj)jHγ0 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 converges weakly in Hγ0. We will prove that (hj)j is a Cauchy sequence in Lβ2. According to Lemma 2, the sequence (gj)j given by (29)gj=(hje(β/2)|(0,))*,j, is a bounded sequence in W1(). By Lemma 1 and the Plancherel isometry (24), for all j,k, we get (30)hk-hjLβ22=hke(β/2)-hje(β/2)L2(+)2gk-gjL2()2=gk-gjL2()2=|(gk)(x)-(gj)(x)|2dx. Thus, for every R>0 we obtain the estimate (31)hk-hjLβ22{|x|R}|(gk)(x)-(gj)(x)|2dx+{|x|>R}|(gk)(x)-(gj)(x)|2dx. By Lemma 5, the sequence (gj)j is bounded in L2(). Therefore, for an arbitrary ϵ>0 there exists a real number R>0 such that (32){|x|>R}|(gk)(x)-(gj)(x)|2dx1R2{|x|>R}|x|2|(gk)(x)-(gj)(x)|2dx<ϵhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhj,k. By Lemma 4, for each ξ the mapping (33)Hγ0,h(he(β/2)|(0,))*(ξ) is a continuous linear functional. Consequently, since (hj)j converges weakly in Hγ0, for each ξ, the real-valued sequence ((gj)(ξ))j is convergent. Moreover, by Lemmas 3 and 4, for all hHγ0, we have the estimate (34)((he(β/2)|(0,))*)C0()12π(he(β/2)|(0,))*L1()C32πhγ. Therefore, the sequence (gj)j is bounded in C0(). Using Lebesgue’s dominated convergence theorem, we deduce that (35){|x|R}|(gk)(x)-(gj)(x)|2dx0for  j,k. Combining (31) together with (32) and (35) shows that (hj)j 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 Ωn. 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=+. 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 Hγ0Lβ2; 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 hW1(Ω) to a function h~W1(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, for example,  for further details. In our situation, we have to ensure that every function hHγ0 can be extended to a function h~W1(), 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 dtν(dξ) on some mark space E. We assume that the standard Lipschitz and linear growth conditions are satisfied which ensure for each initial condition h0H1 the existence of a unique weak solution r to (36); that is, for each ζ𝒟(A*), we have almost surely (37)ζ,rt=ζ,h0H1+0t(A*ζ,rsH1+ζ,α(s,rs)H1)ds+0tζ,σ(s,rs)H1dWs+0tEζ,γ(s,rs-,ξ)H1(𝔭(ds,dξ)-ν(dξ)ds)hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhht0; see, for example,  for further details. Let H2 be a larger separable Hilbert space with compact embedding H1H2. By virtue of Theorem 6, this is in particular satisfied for the forward curve spaces H1=Hγ and H2=Lβ2 for γ>β>0. If, furthermore, A=d/dx is the differential operator, which is generated by the translation semigroup (St)t0 given by Sth=h(t+), and α=αHJM is given by the so-called HJM drift condition (38)αHJM(t,h)=jσj(t,h)0σj(t,h)(η)dη-Eγ(t,h,ξ)[exp(-0γ(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  for further details.

By virtue of the compact embedding H1H2, there exist orthonormal systems (ek)k of H1 and (fk)k of H2, and a decreasing sequence (sk)k+ with sk0 such that (39)h=k=1skh,ekH1fk        hH1; see, for example, [7, Theorem VI.3.6]. The numbers sk are the singular numbers of the identity operator Id:H1H2. Defining the sequence (Tn)n of finite-rank operators (40)Tn:H1Fn,Tnh=k=1nskh,ekH1fk, where Fn:=f1,,fn, we even have TnId with respect to the operator norm (41)T=suphH11ThH2; see, for example, [7, Corollary VI.3.7]. Consequently, denoting by r the weak solution to the SPDE (36) for some initial condition h0H1, the sequence (Tn(r))n is a sequence of Fn-valued stochastic processes, and we have almost surely (42)Tn(rt)-rtH2Tn-IdrtH10t0, 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 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 does not need to be a sequence of Itô processes. This issue is addressed by the following result.

Proposition 8.

Let (ϵn)n(0,) be an arbitrary decreasing sequence with ϵn0. Then, for every initial condition h0H1, there exists a sequence (r(n))n of Fn-valued Itô processes such that almost surely (43)rt(n)-rtH2(Tn-Id+ϵn)rtH10t0, 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, there exist elements ζ1(n),,ζn(n)𝒟(A*) such that (44)ζk(n)-ekH1<ϵn2k·sk        k=1,,n, where we use the convention x/0:= for x>0. We define the sequence (Sn)n of finite-rank operators as (45)Sn:H1Fn,Snh=k=1nskh,ζk(n)H1fk. By the geometric series, for all n, we have (46)Sn-IdSn-Tn+Tn-Idk=1nsk,ζk(n)-ekH1+Tn-Idϵnk=1n12k+Tn-Idϵn+Tn-Id. For each n, let r(n) be the Fn-valued Itô process (47)rt(n)=h0(n)+0tαs(n)ds+0tσs(n)dWs+0tEδs(n)(ξ)(𝔭(ds,dξ)-ν(dξ,ds)), with parameters given by (48)h0(n)=k=1nskζk(n),h0H1fk,αt(n)=k=1nsk(A*ζk(n),rtH1+ζk(n),α(t,rt)H1)fk,σt(n)=k=1nskζk(n),σ(t,rt)H1fk,        δt(n)(ξ)=k=1nskζk(n),δ(t,rt-,ξ)H1fk. Since r is a weak solution to (36), we obtain almost surely (49)Sn(rt)=k=1nskζk(n),rtH1fk=k=1nsk(0tζk(n),h0H1hhhhhhhh+0t(A*ζk(n),rsH1+ζk(n),α(s,rs)H1)dshhhhhhhh+0tζk(n),σ(s,rs)H1dWshhhhhhhh+0tEζk(n),δ(s,rs-,ξ)H1hhhhhhhhhhh×(𝔭(ds,dξ)-ν(dξ,ds))0t)fk=h0(n)+0tαs(n)ds+0tσs(n)dWs+0tEδs(n)(ξ)(𝔭(ds,dξ)-ν(dξ,ds))=rt(n)t0, which finishes the proof.

We will conclude this section with further consequences regarding the speed of convergence of the approximations (r(n))n provided by Proposition 8. Let h0H1 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]rtH12]<, see, for example, [10, Corollary  10.3], by (43) there exists a constant K>0 such that (51)𝔼[supt[0,T]rt(n)-rtH22]1/2K(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>h0H1 the stopping time (53)τ=inf{t0:rtK} is strictly positive, and by (43) for the stopped processes we obtain almost surely (54)supt+rtτ(n)-rtτH2K(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.

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