Compact Embeddings for Spaces of Forward Rate Curves

Abstract

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.(i)The functions ℎ∈𝐻 become 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) where 𝐿²_𝛽 denotes the weighted Lebesgue space 𝐿²_𝛽∶=𝐿²ℝ₊,𝑒^𝛽𝑥𝑑𝑥, (2) 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 𝐻_𝛾∶=ℎ∶ℝ₊⟶ℝ∶ℎisabsolutelycontinuouswith‖ℎ‖_𝛾<∞, (3) for some constant 𝛾>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 all 𝛾>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 space 𝐻_𝛾, applying these operators its solutions can be approximated by a sequence of finite dimensional processes in the larger state space 𝐿²_𝛽⊕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 𝐿²_𝛽⊕ℝ and 𝐻_𝛾 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∈𝐻_𝛾, the limit ℎ(∞)∶=lim_𝑥→∞ℎ(𝑥) exists, and the subspace 𝐻⁰_𝛾∶=ℎ∈𝐻_𝛾∶ℎ(∞)=0 (6) is a closed subspace of 𝐻_𝛾; see [1]. For an open set Ω⊂ℝ, we denote by 𝑊¹(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 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)∶=lim_𝑥→0ℎ(𝑥) exists. Then, we define the reflection ℎ^∗∶ℝ→ℂ as ℎ^∗(𝑥)ℎ(𝑥)ℎ()∶=,if𝑥≥0,−𝑥,if𝑥<0. (9)

Lemma 1. The following statements are true.(1)For each ℎ∈𝑊¹((0,∞)), one has ℎ^∗∈𝑊¹(ℝ).(2)The mapping 𝑊¹((0,∞))→𝑊¹(ℝ), ℎ↦ℎ^∗ is a bounded linear operator.(3)For each ℎ∈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. Let 𝛾>𝛽>0 be arbitrary. Then, the following statements are true.(1)One has 𝐻⁰_𝛾⊂𝐻⁰_𝛽, and ‖ℎ‖_𝛽≤‖ℎ‖_𝛾∀ℎ∈𝐻⁰_𝛾. (11)(2)One has 𝐻⁰_𝛾⊂𝐿²_𝛽, and there is a constant 𝐶₁=𝐶₁(𝛽,𝛾)>0 such that ‖ℎ‖_𝐿²_𝛽≤𝐶₁‖ℎ‖_𝛾∀ℎ∈𝐻⁰_𝛾. (12)(3)For each ℎ∈𝐻⁰_𝛾, one has ℎ𝑒^(𝛽/2)•_(0,∞)∈𝑊¹((0,∞)),ℎ𝑒^(𝛽/2)•_(0,∞)^∗∈𝑊¹(ℝ), (13) and there is a constant 𝐶₂=𝐶₂(𝛽,𝛾)>0 such that ℎ𝑒^(𝛽/2)•_(0,∞)^∗_𝑊1(ℝ)≤𝐶₂‖ℎ‖_𝛾∀ℎ∈𝐻⁰_𝛾. (14)

Proof. The first statement is a direct consequence of the representation of the norm on 𝐻⁰_𝛾 given by (4). Let ℎ∈𝐻⁰_𝛾 be arbitrary. By the Cauchy-Schwarz inequality, we obtain ‖ℎ‖²_𝐿2𝛽=_ℝ+|ℎ(𝑥)|²𝑒^𝛽𝑥=𝑑𝑥_ℝ+∞𝑥ℎ^′𝜂𝑒^(𝛾/2)𝜂𝑒^{−(𝛾/2)𝜂}𝑑𝜂²𝑒^𝛽𝑥≤𝑑𝑥_ℝ+∞𝑥ℎ^′𝜂²𝑒^𝛾𝜂𝑑𝜂^∞_𝑥𝑒^−𝛾𝜂𝑑𝜂𝑒^𝛽𝑥≤𝑑𝑥_ℝ+ℝ+ℎ^′𝜂²𝑒^𝛾𝜂1𝑑𝜂𝛾𝑒^−𝛾𝑥𝑒^𝛽𝑥≤1𝑑𝑥𝛾_ℝ+𝑒^{−(𝛾−𝛽)𝑥}𝑑𝑥‖ℎ‖²_𝛾=1𝛾‖ℎ‖𝛾−𝛽^2𝛾, (15) proving the second statement. Furthermore, by (12) we have ℎ𝑒^(𝛽/2)•_(0,∞)²_{𝐿2((0,∞))}=_ℝ+ℎ(𝑥)𝑒^{(𝛽/2)𝑥2}𝑑𝑥=_ℝ+|ℎ(𝑥)|²𝑒^𝛽𝑥=‖ℎ‖𝑑𝑥_2𝐿2𝛽≤𝐶²₁‖ℎ‖²_𝛾, (16) and by estimates (11), (12), we obtain 𝑑𝑑𝑥ℎ𝑒^(𝛽/2)•_(0,∞)²_{𝐿2((0,∞))}=_ℝ+𝑑ℎ(𝑥)𝑒𝑑𝑥^{(𝛽/2)𝑥2}=𝑑𝑥_ℝ+ℎ^′(𝑥)𝑒^(𝛽/2)𝑥+𝛽2ℎ(𝑥)𝑒^{(𝛽/2)𝑥2}𝑑𝑥≤2_ℝ+ℎ^′(𝑥)²𝑒^𝛽𝑥𝛽𝑑𝑥+²4_ℝ+|ℎ(𝑥)|²𝑒^𝛽𝑥‖ℎ‖𝑑𝑥≤2²_𝛽+𝛽²2‖ℎ‖_𝐿²_𝛽≤𝛽2+²𝐶²₁2‖ℎ‖²_𝛾, (17) which, together with Lemma 1, concludes the proof.

For ℎ∈𝐿¹(ℝ), the Fourier transform ℱℎ∶ℝ→ℂ is defined as (𝜉)1ℱℎ)(∶=√2𝜋^ℝℎ(𝑥)𝑒^{−𝑖𝜉𝑥}𝑑𝑥,𝜉∈ℝ. (18) 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.(1)For each ℎ∈𝐻⁰_𝛾, one has (ℎ𝑒^(𝛽/2)•|_(0,∞))^∗∈𝐿¹(ℝ), and there is a constant 𝐶₃=𝐶₃(𝛽,𝛾)>0 such that ℎ𝑒^(𝛽/2)•_(0,∞)^∗_𝐿1(ℝ)≤𝐶₃‖ℎ‖_𝛾∀ℎ∈𝐻⁰_𝛾. (19)(2)For each 𝜉∈ℝ, the mapping 𝐻⁰_𝛾⟶ℝ,ℎ⟼ℱℎ𝑒^(𝛽/2)•_(0,∞)^∗(𝜉) (20) is a continuous linear functional.

Proof. We set 𝛿∶=(1/2)(𝛽+𝛾)∈(𝛽,𝛾). Let ℎ∈𝐻⁰_𝛾 be arbitrary. By the Cauchy-Schwarz inequality and Lemma 2, we have ℎ𝑒^(𝛽/2)•_(0,∞)^∗_𝐿1(ℝ)=2ℎ𝑒^(𝛽/2)•_𝐿1(ℝ+)=2_ℝ+ℎ(𝑥)𝑒^(𝛽/2)𝑥𝑑𝑥=2_ℝ+|ℎ(𝑥)|𝑒^(𝛿/2)𝑥𝑒^{−((𝛿−𝛽)/2)𝑥}𝑑𝑥≤2_ℝ+|ℎ(𝑥)|²𝑒^𝛿𝑥𝑑𝑥^1/2_ℝ+𝑒^{−(𝛿−𝛽)𝑥}𝑑𝑥^1/2=21‖ℎ‖𝛿−𝛽^𝐿2𝛿≤2𝐶₁𝛿,𝛾1‖ℎ‖𝛿−𝛽^𝛾, (21) showing the first statement. Moreover, we have 𝑒^{((𝛽/2)−𝛿)•2}_𝐿2𝛿=_ℝ+𝑒^{2((𝛽/2)−𝛿)𝑥}𝑒^𝛿𝑥=𝑑𝑥_ℝ+𝑒^{−(𝛿−𝛽)𝑥}1𝑑𝑥=,𝛿−𝛽 (22) showing that 𝑒^{((𝛽/2)−𝛿)•}∈𝐿²_𝛿. Let ℎ∈𝐻⁰_𝛾 and 𝜉∈ℝ be arbitrary. By Lemma 2, we have ℎ∈𝐿²_𝛿, and hence ℱℎ𝑒^(𝛽/2)•_(0,∞)^∗(𝜉)=1√2𝜋^∞0ℎ(𝑥)𝑒^(𝛽/2)𝑥𝑒^{−𝑖𝜉𝑥}𝑑𝑥+⁰_−∞ℎ(−𝑥)𝑒^{−(𝛽/2)𝑥}𝑒^{−𝑖𝜉𝑥}=1𝑑𝑥√2𝜋^∞0ℎ(𝑥)𝑒^(𝛽/2)𝑥𝑒^{−𝑖𝜉𝑥}𝑑𝑥+^∞₀ℎ(𝑥)𝑒^(𝛽/2)𝑥𝑒^𝑖𝜉𝑥=1𝑑𝑥√2𝜋ℎ,𝑒^{((𝛽/2)−𝛿)•}𝑒^{−𝑖𝜉•}+𝑒^𝑖𝜉•_𝐿2𝛿, (23) 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 ℱ𝑓,ℱ𝑔_𝐿2(ℝ)=𝑓,𝑔_𝐿2(ℝ)∀𝑓,𝑔∈𝐿²(ℝ). (24) Moreover, the two just reviewed definitions of the Fourier transform coincide on 𝐿¹(ℝ)∩𝐿²(ℝ). For each ℎ∈𝑊¹(ℝ), we have ℱℎ^′(𝜉)=𝑖𝜉(ℱℎ)(𝜉),𝜉∈ℝ. (25)

Lemma 5. For every ℎ∈𝑊¹(ℝ), one has ‖•ℱℎ‖_𝐿²_(ℝ)≤‖ℎ‖_𝑊¹_(ℝ). (26)

Proof. Let ℎ∈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 all 𝛾>𝛽>0, one has the compact embedding 𝐻_𝛾⊂⊂𝐿²_𝛽⊕ℝ. (28)

Proof. Noting that 𝐻_𝛾≅𝐻⁰_𝛾⊕ℝ, it suffices to prove the compact embedding 𝐻⁰_𝛾⊂⊂𝐿²_𝛽. Let (ℎ_𝑗)_𝑗∈ℕ⊂𝐻⁰_𝛾 be a bounded sequence. Then, there exists a subsequence which converges weakly in 𝐻⁰_𝛾. Without loss of generality, we may assume that the original sequence (ℎ_𝑗)_𝑗∈ℕ converges weakly in 𝐻⁰_𝛾. We will prove that (ℎ_𝑗)_𝑗∈ℕ is a Cauchy sequence in 𝐿²_𝛽. According to Lemma 2, the sequence (𝑔_𝑗)_𝑗∈ℕ given 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 all 𝑗,𝑘∈ℕ, we get ℎ_𝑘−ℎ_𝑗²_𝐿2𝛽=ℎ_𝑘𝑒^(𝛽/2)•−ℎ_𝑗𝑒^(𝛽/2)•2_𝐿2(ℝ+)≤𝑔_𝑘−𝑔_𝑗²_𝐿2(ℝ)=ℱ𝑔_𝑘−ℱ𝑔_𝑗²_𝐿2(ℝ)=_ℝℱ𝑔_𝑘(𝑥)−ℱ𝑔_𝑗(𝑥)²𝑑𝑥. (30) Thus, for every 𝑅>0 we obtain the estimate ℎ_𝑘−ℎ_𝑗²_𝐿2𝛽≤_{{|𝑥|≤𝑅}}ℱ𝑔_𝑘(𝑥)−ℱ𝑔_𝑗(𝑥)²+𝑑𝑥_{{|𝑥|>𝑅}}ℱ𝑔_𝑘(𝑥)−ℱ𝑔_𝑗(𝑥)²𝑑𝑥. (31) By Lemma 5, the sequence (•ℱ𝑔_𝑗)_𝑗∈ℕ is bounded in 𝐿²(ℝ). Therefore, for an arbitrary 𝜖>0 there exists a real number 𝑅>0 such that _{{|𝑥|>𝑅}}ℱ𝑔_𝑘(𝑥)−ℱ𝑔_𝑗(𝑥)²≤1𝑑𝑥𝑅²_{{|𝑥|>𝑅}}|𝑥|²ℱ𝑔_𝑘(𝑥)−ℱ𝑔_𝑗(𝑥)²𝑑𝑥<𝜖∀𝑗,𝑘∈ℕ. (32) By Lemma 4, for each 𝜉∈ℝ the mapping 𝐻⁰_𝛾⟶ℝ,ℎ⟼ℱℎ𝑒^(𝛽/2)•_(0,∞)^∗(𝜉) (33) is a continuous linear functional. Consequently, since (ℎ_𝑗)_𝑗∈ℕ converges weakly in 𝐻⁰_𝛾, for each 𝜉∈ℝ, the real-valued sequence ((ℱ𝑔_𝑗)(𝜉))_𝑗∈ℕ is convergent. Moreover, by Lemmas 3 and 4, for all ℎ∈𝐻⁰_𝛾, we have the estimate ℱℎ𝑒^(𝛽/2)•_(0,∞)^∗_𝐶0(ℝ)≤1√2𝜋ℎ𝑒^(𝛽/2)•_(0,∞)^∗_𝐿1(ℝ)≤𝐶₃√‖ℎ‖2𝜋^𝛾. (34) Therefore, the sequence (ℱ𝑔_𝑗)_𝑗∈ℕ is bounded in 𝐶₀(ℝ). Using Lebesgue’s dominated convergence theorem, we deduce that _{{|𝑥|≤𝑅}}ℱ𝑔_𝑘(𝑥)−ℱ𝑔_𝑗(𝑥)²𝑑𝑥⟶0for𝑗,𝑘⟶∞. (35) Combining (31) together with (32) and (35) shows that (ℎ_𝑗)_𝑗∈ℕ is a Cauchy sequence in 𝐿²_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 embedding 𝐻¹₀(Ω)⊂⊂𝐿²(Ω) for an open, bounded subset Ω⊂ℝ^𝑛. Here, 𝐻¹₀(Ω) denotes the Sobolev space 𝐻¹₀(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.(i)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)=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; see Lemma 4.(ii)𝐻_𝛾 and 𝐻¹₀(Ω) are different kinds of spaces. While the norm on 𝐻¹₀(Ω) given by (8) involves the 𝐿²-norms of a function ℎ and its derivative ℎ^′, the norm (4) on 𝐻_𝛾 only involves the 𝐿²-norm of the derivative ℎ^′ and a point evaluation. Therefore, the embedding 𝐻¹₀(Ω)⊂𝐿²(Ω) follows right away, whereas we require the assumption 𝛽<𝛾 for the embedding 𝐻⁰_𝛾⊂𝐿²_𝛽; see Lemma 2.(iii)The classical Rellich embedding theorem does not need to be true with 𝐻¹₀(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(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, for example, [9] for further details. In our situation, we have to ensure that every function ℎ∈𝐻⁰_𝛾 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 𝑑𝑟_𝑡=𝐴𝑟_𝑡+𝛼𝑡,𝑟_𝑡𝑑𝑡+𝜎𝑡,𝑟_𝑡𝑑𝑊_𝑡+_𝐸𝛾𝑡,𝑟_𝑡−(𝔭()())𝑟,𝜉𝑑𝑡,𝑑𝜉−𝜈𝑑𝜉𝑑𝑡₀=ℎ₀, (36) 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 𝜁,𝑟_𝑡=𝜁,ℎ_0𝐻1+^𝑡₀𝐴^∗𝜁,𝑟_𝑠𝐻1+𝜁,𝛼𝑠,𝑟_𝑠𝐻1+𝑑𝑠_𝑡0𝜁,𝜎𝑠,𝑟_𝑠𝐻1𝑑𝑊_𝑠+^𝑡_0𝐸𝜁,𝛾𝑠,𝑟_𝑠−,𝜉_𝐻1(𝔭(𝑑𝑠,𝑑𝜉)−𝜈(𝑑𝜉)𝑑𝑠)∀𝑡≥0; (37) 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=𝐻_𝛾 and 𝐻₂=𝐿²_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_𝑡ℎ=ℎ(𝑡+•), and 𝛼=𝛼_HJM is given by the so-called HJM drift condition 𝛼_HJM(𝑡,ℎ)=_𝑗𝜎^𝑗(𝑡,ℎ)^•₀𝜎^𝑗(𝜂−𝑡,ℎ)𝑑𝜂_𝐸𝛾(𝑡,ℎ,𝜉)exp−^•₀𝛾(𝑡,ℎ,𝜉)𝜂𝑑𝜂−1𝜈(𝑑𝜉), (38) 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 (𝑒_𝑘)_𝑘∈ℕ of 𝐻₁ and (𝑓_𝑘)_𝑘∈ℕ of 𝐻₂, and a decreasing sequence (𝑠_𝑘)_𝑘∈ℕ⊂ℝ₊ with 𝑠_𝑘→0 such that ℎ=^∞_𝑘=1𝑠_𝑘ℎ,𝑒_𝑘𝐻1𝑓_𝑘∀ℎ∈𝐻₁; (39) see, for example, [7, Theorem VI.3.6]. The numbers 𝑠_𝑘 are the singular numbers of the identity operator Id∶𝐻₁→𝐻₂. Defining the sequence (𝑇_𝑛)_𝑛∈ℕ of finite-rank operators 𝑇_𝑛∶𝐻₁⟶𝐹_𝑛,𝑇_𝑛ℎ∶=^𝑛_𝑘=1𝑠_𝑘ℎ,𝑒_𝑘𝐻1𝑓_𝑘, (40) where 𝐹_𝑛∶=⟨𝑓₁,…,𝑓_𝑛⟩, we even have 𝑇_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 by 𝑟 the weak solution to the SPDE (36) for some initial condition ℎ₀∈𝐻₁, the sequence (𝑇_𝑛(𝑟))_𝑛∈ℕ is a sequence of 𝐹_𝑛-valued stochastic processes, and we have almost surely 𝑇_𝑛𝑟_𝑡−𝑟_𝑡𝐻2≤𝑇_𝑛−Id𝑟_𝑡𝐻1⟶0∀𝑡≥0, (42) showing that the weak solution 𝑟—when considered on the larger state space 𝐻₂—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 (42). 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 8. Let (𝜖_𝑛)_𝑛∈ℕ⊂(0,∞) be an arbitrary decreasing sequence with 𝜖_𝑛→0. Then, for every initial condition ℎ₀∈𝐻₁, there exists a sequence (𝑟^(𝑛))_𝑛∈ℕ of 𝐹_𝑛-valued Itô processes such that almost surely 𝑟_𝑡^(𝑛)−𝑟_𝑡𝐻2≤𝑇_𝑛−Id+𝜖_𝑛𝑟_𝑡𝐻1⟶0∀𝑡≥0, (43) 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),…,𝜁_𝑛^(𝑛)∈𝒟(𝐴^∗) such that 𝜁_𝑘^(𝑛)−𝑒_𝑘𝐻1<𝜖_𝑛2^𝑘⋅𝑠_𝑘∀𝑘=1,…,𝑛, (44) where we use the convention 𝑥/0∶=∞ for 𝑥>0. We define the sequence (𝑆_𝑛)_𝑛∈ℕ of finite-rank operators as 𝑆_𝑛∶𝐻₁⟶𝐹_𝑛,𝑆_𝑛ℎ∶=^𝑛_𝑘=1𝑠_𝑘ℎ,𝜁_𝑘^(𝑛)_𝐻1𝑓_𝑘. (45) By the geometric series, for all 𝑛∈ℕ, we have 𝑆_𝑛−Id≤𝑆_𝑛−𝑇_𝑛+𝑇_𝑛≤−Id_𝑛𝑘=1𝑠_𝑘•,𝜁_𝑘^(𝑛)−𝑒_𝑘𝐻1+𝑇_𝑛−Id≤𝜖_𝑛^𝑛_𝑘=112^𝑘+𝑇_𝑛−Id≤𝜖_𝑛+𝑇_𝑛−Id. (46) For each 𝑛∈ℕ, let 𝑟^(𝑛) be the 𝐹_𝑛-valued Itô process 𝑟_𝑡^(𝑛)=ℎ₀^(𝑛)+^𝑡₀𝛼_𝑠^(𝑛)𝑑𝑠+^𝑡₀𝜎_𝑠^(𝑛)𝑑𝑊_𝑠+^𝑡_0𝐸𝛿_𝑠^(𝑛)(𝜉)(𝔭(𝑑𝑠,𝑑𝜉)−𝜈(𝑑𝜉,𝑑𝑠)), (47) with parameters given by ℎ₀^(𝑛)=^𝑛_𝑘=1𝑠_𝑘𝜁_𝑘^(𝑛),ℎ_0𝐻1𝑓_𝑘,𝛼_𝑡^(𝑛)=^𝑛_𝑘=1𝑠_𝑘𝐴^∗𝜁_𝑘^(𝑛),𝑟_𝑡𝐻1+𝜁_𝑘^(𝑛),𝛼𝑡,𝑟_𝑡𝐻1𝑓_𝑘,𝜎_𝑡^(𝑛)=^𝑛_𝑘=1𝑠_𝑘𝜁_𝑘^(𝑛),𝜎𝑡,𝑟_𝑡𝐻1𝑓_𝑘,𝛿_𝑡^(𝑛)(𝜉)=^𝑛_𝑘=1𝑠_𝑘𝜁_𝑘^(𝑛),𝛿𝑡,𝑟_𝑡−,𝜉_𝐻1𝑓_𝑘. (48) Since 𝑟 is a weak solution to (36), we obtain almost surely 𝑆_𝑛𝑟_𝑡=^𝑛_𝑘=1𝑠_𝑘𝜁_𝑘^(𝑛),𝑟_𝑡𝐻1𝑓_𝑘=^𝑛_𝑘=1𝑠_𝑘𝜁_𝑘^(𝑛),ℎ_0𝐻1+^𝑡₀𝐴^∗𝜁_𝑘^(𝑛),𝑟_𝑠𝐻1+𝜁_𝑘^(𝑛),𝛼𝑠,𝑟_𝑠𝐻1+𝑑𝑠_𝑡0𝜁_𝑘^(𝑛),𝜎𝑠,𝑟_𝑠𝐻1𝑑𝑊_𝑠+^𝑡_0𝐸𝜁_𝑘^(𝑛),𝛿𝑠,𝑟_𝑠−,𝜉_𝐻1×(𝔭(𝑑𝑠,𝑑𝜉)−𝜈(𝑑𝜉,𝑑𝑠))𝑓_𝑘=ℎ₀^(𝑛)+^𝑡₀𝛼_𝑠^(𝑛)𝑑𝑠+^𝑡₀𝜎_𝑠^(𝑛)𝑑𝑊_𝑠+^𝑡_0𝐸𝛿_𝑠^(𝑛)(𝜉)(𝔭(𝑑𝑠,𝑑𝜉)−𝜈(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 by 𝑟 the weak solution to (36). Furthermore, let 𝑇>0 be a finite time horizon. Since 𝔼sup_{[]𝑡∈0,𝑇}𝑟_𝑡²_𝐻1<∞, (50) see, for example, [10, Corollary  10.3], by (43) there exists a constant 𝐾>0 such that 𝔼sup_{[]𝑡∈0,𝑇}𝑟_𝑡^(𝑛)−𝑟_𝑡²_𝐻2^1/2≤𝐾𝑇_𝑛−Id+𝜖_𝑛⟶0, (51) 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+𝛼𝑡,𝑟_𝑡𝑑𝑡+𝜎𝑡,𝑟_𝑡𝑑𝑊_𝑡𝑟₀=ℎ₀, (52) the sample paths of 𝑟 are continuous; for every constant 𝐾>‖ℎ₀‖_𝐻1 the stopping time 𝜏∶=inf𝑡≥0∶𝑟_𝑡≥𝐾 (53) is strictly positive, and by (43) for the stopped processes we obtain almost surely sup_𝑡∈ℝ+𝑟^(𝑛)_𝑡∧𝜏−𝑟_{𝑡∧𝜏𝐻2}≤𝐾𝑇_𝑛−Id+𝜖_𝑛⟶0; (54) that is, locally the solution 𝑟 stays in a bounded subset of 𝐻_𝛾 and we obtain the uniform convergence (54).

Acknowledgment

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