Research Article | Open Access
Compact Embeddings for Spaces of Forward Rate Curves
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.
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, [2–5] and references therein. The state space, which contains the forward curves, is a separable Hilbert space consisting of functions . In practice, forward curves have the following features.(i)The functions become flat at the long end.(ii)Consequently, the limit exists.The second property is taken into account by choosing the Hilbert space where denotes the weighted Lebesgue space for some constant . 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 for some constant , where the norm is given by 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 we have the compact embedding 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 , applying these operators its solutions can be approximated by a sequence of finite dimensional processes in the larger state space ; 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 , the separable Hilbert spaces and 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 , the limit exists, and the subspace is a closed subspace of ; see . For an open set , we denote by the Sobolev space which, equipped with the inner product is a separable Hilbert space. Here, derivatives are understood as weak derivatives.
For a function , the extension in general, does not belong to . In the present situation, this technical problem can be resolved as follows. Let be a continuous function such that the limit exists. Then, we define the reflection as
Lemma 1. The following statements are true.(1)For each , one has .(2)The mapping , is a bounded linear operator.(3)For each , one has
Proof. This follows from a straightforward calculation following the proof of [8, Theorem 8.6].
Lemma 2. Let be arbitrary. Then, the following statements are true.(1)One has , and (2)One has , and there is a constant such that (3)For each , one has and there is a constant such that
Proof. The first statement is a direct consequence of the representation of the norm on given by (4). Let be arbitrary. By the Cauchy-Schwarz inequality, we obtain proving the second statement. Furthermore, by (12) we have and by estimates (11), (12), we obtain which, together with Lemma 1, concludes the proof.
For , the Fourier transform is defined as Recall that 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 is a continuous linear operator with .
Lemma 4. Let be arbitrary. Then, the following statements are true.(1)For each , one has , and there is a constant such that (2)For each , the mapping is a continuous linear functional.
Proof. We set . Let be arbitrary. By the Cauchy-Schwarz inequality and Lemma 2, we have showing the first statement. Moreover, we have showing that . Let and be arbitrary. By Lemma 2, we have , and hence proving the second statement.
We can also define the Fourier transform on such that is a bijection, and we have the Plancherel isometry Moreover, the two just reviewed definitions of the Fourier transform coincide on . For each , we have
Lemma 5. For every , one has
3. The Embedding Result and Its Proof
In this section, we present the compact embedding result and its proof.
Theorem 6. For all , one has the compact embedding
Proof. Noting that , it suffices to prove the compact embedding . Let be a bounded sequence. Then, there exists a subsequence which converges weakly in . Without loss of generality, we may assume that the original sequence converges weakly in . We will prove that is a Cauchy sequence in . According to Lemma 2, the sequence given by is a bounded sequence in . By Lemma 1 and the Plancherel isometry (24), for all , we get Thus, for every we obtain the estimate By Lemma 5, the sequence is bounded in . Therefore, for an arbitrary there exists a real number such that By Lemma 4, for each the mapping is a continuous linear functional. Consequently, since converges weakly in , for each , the real-valued sequence is convergent. Moreover, by Lemmas 3 and 4, for all , we have the estimate Therefore, the sequence is bounded in . Using Lebesgue’s dominated convergence theorem, we deduce that Combining (31) together with (32) and (35) shows that is a Cauchy sequence in , 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 for an open, bounded subset . Here, denotes the Sobolev space , where is the space of all -functions on with compact support, and where the closure is taken with respect to the topology induced by the inner product . Let us briefly describe the analogies and differences between the two results as follows.(i)In the classical Rellich embedding theorem, the domain is assumed to be bounded, whereas in Theorem 6 we have . Moreover, we consider weighted function spaces with weight functions of the type for some constant . This requires a careful analysis of the results regarding Fourier transforms which we have adapted to the present situation; see Lemma 4.(ii) and are different kinds of spaces. While the norm on given by (8) involves the -norms of a function and its derivative , the norm (4) on only involves the -norm of the derivative and a point evaluation. Therefore, the embedding follows right away, whereas we require the assumption for the embedding ; see Lemma 2.(iii)The classical Rellich embedding theorem does not need to be true with being replaced by . The reason behind this is that, in general, it is not possible to extend a function to a function , which, however, is crucial in order to apply the results about Fourier transforms. Usually, one assumes that satisfies a so-called cone condition; see, for example,  for further details. In our situation, we have to ensure that every function can be extended to a function , 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 on some separable Hilbert space with denoting the generator of some strongly continuous semigroup on , driven by a Wiener process and a homogeneous Poisson random measure with compensator on some mark space . We assume that the standard Lipschitz and linear growth conditions are satisfied which ensure for each initial condition the existence of a unique weak solution to (36); that is, for each , we have almost surely see, for example,  for further details. Let be a larger separable Hilbert space with compact embedding . By virtue of Theorem 6, this is in particular satisfied for the forward curve spaces and for . If, furthermore, is the differential operator, which is generated by the translation semigroup given by , and is given by the so-called HJM drift condition 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 , there exist orthonormal systems of and of , and a decreasing sequence with such that see, for example, [7, Theorem VI.3.6]. The numbers are the singular numbers of the identity operator . Defining the sequence of finite-rank operators where , we even have with respect to the operator norm see, for example, [7, Corollary VI.3.7]. Consequently, denoting by the weak solution to the SPDE (36) for some initial condition , the sequence is a sequence of -valued stochastic processes, and we have almost surely showing that the weak solution —when considered on the larger state space —can be approximated by the sequence of finite dimensional processes with distance between and estimated in terms of the operator norm , as shown in (42). However, the sequence does not need to be a sequence of Itô processes. This issue is addressed by the following result.
Proposition 8. Let be an arbitrary decreasing sequence with . Then, for every initial condition , there exists a sequence of -valued Itô processes such that almost surely where denotes the weak solution to (36).
Proof. According to [6, Theorems 13.35.c and 13.12], the domain is dense in . Therefore, for each , there exist elements such that where we use the convention for . We define the sequence of finite-rank operators as By the geometric series, for all , we have For each , let be the -valued Itô process with parameters given by Since is a weak solution to (36), we obtain almost surely which finishes the proof.
We will conclude this section with further consequences regarding the speed of convergence of the approximations provided by Proposition 8. Let be an arbitrary initial condition and denote by the weak solution to (36). Furthermore, let be a finite time horizon. Since see, for example, [10, Corollary 10.3], by (43) there exists a constant such that providing a uniform estimate for the distance of and in the mean-square sense. Moreover, considering the pure diffusion case the sample paths of are continuous; for every constant the stopping time is strictly positive, and by (43) for the stopped processes we obtain almost surely that is, locally the solution stays in a bounded subset of and we obtain the uniform convergence (54).
The author is grateful to an anonymous referee for valuable comments and suggestions.
- D. Filipović, Consistency Problems for Heath-Jarrow-Morton Interest Rate Models, vol. 1760 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2001.
- A. Rusinek, “Mean reversion for HJMM forward rate models,” Advances in Applied Probability, vol. 42, no. 2, pp. 371–391, 2010.
- M. Barski and J. Zabczyk, “Heath-Jarrow-Morton-Musiela equation with Lévy perturbation,” Journal of Differential Equations, vol. 253, no. 9, pp. 2657–2697, 2012.
- D. Filipović and S. Tappe, “Existence of Lévy term structure models,” Finance and Stochastics, vol. 12, no. 1, pp. 83–115, 2008.
- D. Filipović, 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.
- W. Rudin, Functional Analysis, International Series in Pure and Applied Mathematics, McGraw-Hill, New York, NY, USA, 2nd edition, 1991.
- D. Werner, Funktionalanalysis, Springer, Berlin, Germany, 2007.
- H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, NY, USA, 2011.
- R. A. Adams and J. J. F. Fournier, Sobolev Spaces, vol. 140 of Pure and Applied Mathematics, Elsevier/Academic Press, Amsterdam, The Netherlands, 2nd edition, 2003.
- D. Filipović, S. Tappe, and J. Teichmann, “Jump-diffusions in Hilbert spaces: existence, stability and numerics,” Stochastics, vol. 82, no. 5, pp. 475–520, 2010.
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.