Stochastic evolution equations in Banach spaces and applications to the Heath–Jarrow–Morton–Musiela equations

The aim of this paper is threefold. Firstly, we study stochastic evolution equations (with the linear part of the drift being a generator of a C0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{0}$\end{document}-semigroup) driven by an infinite-dimensional cylindrical Wiener process. In particular, we prove, under some sufficient conditions on the coefficients, the existence and uniqueness of solutions for these stochastic evolution equations in a class of Banach spaces satisfying the so-called H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H$\end{document}-condition. Moreover, we analyse the Markov property of the solutions. Secondly, we apply the abstract results obtained in the first part to prove the existence and uniqueness of solutions to the Heath–Jarrow–Morton–Musiela (HJMM) equations in weighted Lebesgue and Sobolev spaces. Finally, we study the ergodic properties of the solutions to the HJMM equations. In particular, we find a sufficient condition for the existence and uniqueness of invariant measures for the Markov semigroup associated to the HJMM equations (when the coefficients are time-independent) in the weighted Lebesgue spaces. Our paper is a modest contribution to the theory of financial models in which the short rate can be undefined.


Introduction
The theory of stochastic integration in the class of so-called M-type 2 Banach spaces was initiated independently by Neidhardt [44] in 1978 and Dettweiler [23] in 1983. Using this stochastic integration theory, the first of the present authors developed the theory of stochastic evolution equations in M-type 2 Banach spaces; see [5,6]. In [5], the first author studied linear stochastic evolution equations (with the drift being the infinitesimal generator of an analytic semigroup and the coefficients of the stochastic part being linear operators) driven by a d-dimensional Wiener process and proved the existence and uniqueness of solutions for such equations in some real interpolation spaces. In [6], he generalized the results from [5] by proving the existence and uniqueness of solutions for stochastic evolution equations (with the linear part of the drift being the infinitesimal generator of an analytic semigroup and the coefficients satisfying Lipschitz conditions) driven by an infinite-dimensional Wiener process in M-type 2 Banach spaces.
One of the aims of this article is to study stochastic evolution equations driven by an infinite-dimensional cylindrical Wiener process. We continue the line of research originated in [5] and [6]. However, we consider different assumptions on the coefficients. For example, we assume that the linear part of the drift is the infinitesimal generator of a non-analytic C 0 -semigroup. The lack of analyticity of the semigroup leads to some difficulties, and in this article, we show how to overcome this problem. For instance, the time-continuity of the solution cannot be a consequence of the Da Prato-Kwapień-Zabczyk factorisation method, but one needs to use a more sophisticated approach based on the maximal inequality from [14].
The notion of invariant measures is an important topic in the theory of stochastic dynamical systems. Many authors (see, for instance, [24,25,22]) have studied the question of the existence and uniqueness of invariant measures for stochastic evolution equations in Hilbert spaces. Recently, the first author, Long and Simão [10] paid attention to the theory of invariant measures for stochastic evolution equations in Banach spaces. They found a sufficient condition about the existence and uniqueness of invariant measures for stochastic evolution equations in Banach spaces, which we introduce and use in this article.
It is now a widely accepted fact that mathematics has a lot of interesting applications in finance. One of these applications appears to be the theory of stochastic evolution equations. The so-called HJMM model proposed by Heath-Jarrow-Morton (HJM) [34] is an example of stochastic evolution equations in finance. This model contains the dynamics of the forward rates. The HJMM model was studied in Hilbert spaces by many authors; see, for instance, [3,26,27,28,30,31,43,53,54]. These authors proved the existence and uniqueness of solutions to the HJMM model in some appropriate Hilbert spaces such as the weighted Lebesgue space L 2 and the weighted Sobolev space W 1,2 . Also some of these authors analysed ergodic properties of the solutions.
Another aim of this article is to apply the abstract results from Sect. 3 to the HJMM model. Thus, we focus on the question of the existence and uniqueness of solutions to the HJMM model in the weighted Lebesgue spaces (L p , p ≥ 2) and the weighted Sobolev spaces (W 1,p , p ≥ 2). Also, we aim to study the existence and uniqueness of invariant measures for the HJMM model in the weighted Lebesgue spaces. The weighted L p and W 1,p spaces, with p > 2, are 'locally' smaller than the corresponding weighted L 2 and W 1,2 spaces. Thus, proving the existence of solutions taking values in these spaces provides additional information about the properties of the solutions. For instance, a solution with values in the weighted W 1,2 space is locally Hölder-continuous with order strictly less than 1 − 1 2 , while a solution with values in the weighted W 1,p space is locally Hölder-continuous with order strictly less than 1 − 1 p . This could have profound consequences in the numerical studies of the HJMM model (to be investigated, for instance, in joint works with Boda Kang from the Department of Mathematics at the University of York). Similar comments apply to the invariant measures. Another motivation is that the use of Sobolev spaces with big p should make it possible to extend the deep results about finite-dimensional realisations, which are usually formulated in terms of the very small set D(A ∞ ) to the spaces W 1,p or W θ,p with large p in a similar way to what has been done in [8]. We should point out that as it was proved in [15], the HJMM equations are not well-posed in the space of continuous functions. Hence, one seeks for other type of spaces. So far, the literature has exclusively concentrated on the use of Hilbert spaces (weighted L 2 or weighted Sobolev W 1,2 spaces), and our paper is the first attempt to study the HJMM equations in the framework of a specific class of Banach spaces.
Let us now describe briefly the contents of this paper. In Sect. 2, we provide all the necessary preliminaries about the theory of stochastic integration in Banach spaces. In Sect. 3, we study stochastic evolution equations (with the linear part of the drift being only the infinitesimal generator of a C 0 -semigroup and the coefficients satisfying Lipschitz conditions) in the class of Banach spaces satisfying the H -condition. Firstly, we prove the existence and uniqueness of solutions for corresponding equations with globally Lipschitz coefficients. Next, using the previous existence result and approximation, we prove the existence and uniqueness of solutions for corresponding equations with locally Lipschitz coefficients. We also analyse the Markov property of the solutions. Finally, we introduce a theorem proposed by Brzeźniak et al. [10] about the existence and uniqueness of invariant measures for corresponding equations with time-independent coefficients. In Sect. 4, we introduce the Heath-Jarrow-Morton-Musiela (HJMM) equations, but not in detail. Then we apply the abstract results from Sect. 3 to the HJMM equations. In particular, we prove the existence and uniqueness of solutions to the HJMM equations in the weighted Lebesgue and Sobolev spaces, respectively. We also find a sufficient condition for the existence and uniqueness of invariant measures for the Markov semigroup associated to the HJMM equations in the weighted Lebesgue spaces. We mentioned some important features of our results above. Another important feature is that we are able to consider the HJMM equations driven by a cylindrical Wiener process in a (possibly infinite-dimensional) Hilbert space. For this purpose, we use the characterizations of the so-called γ -radonifying operators from a Hilbert space to an L p space and a Sobolev space W 1,p found recently by the first author and Peszat in [14,13].

Stochastic preliminaries
In this section, we provide all the necessary preliminaries about the theory of stochastic integration in Banach spaces.
if there exists a constant C > 0 depending only on X such that for any X-valued For each p ≥ 2, the Lebesgue space L p and the Sobolev space W 1,p are examples of martingale-type 2 Banach spaces.

Remark 2.2
Another type of Banach space that has been frequently used in the recent years is the UMD, i.e., having the unconditional martingale difference property, Banach space; see [5,16,46]. Let us recall that a Banach space X is a UMD space if for any p ∈ (1, ∞), there exists a constant β p (X) > 0 such that for any X-valued martingale (M n ) n∈N with M 0 = 0, for any ε : N → {−1, 1} and for any n ∈ N, It is known, see e.g. [51,Proposition 2.4], that if a Banach space X is both UMD and of type 2, then X is of martingale-type 2. From the point of view of applications, in UMD spaces, one can precisely characterise the processes which are stochastically integrable with respect to a cylindrical Wiener process, see [46], while the martingaletype 2 spaces are more suitable for studying general SPDEs; see e.g. [13,17]. For our purposes, we consider the latter spaces in which all L 2 -integrable processes turn out to be stochastically integrable (see [46,Theorem 3.6]).
It should be pointed out that many important spaces from functional analysis are martingale-type 2 spaces, e.g. classical Sobolev or Besov spaces under certain assumptions on their indices; see e.g. [5,6]. This has important consequences in numerical analysis and regularity theory. Recently, Hairer and Labbé [32] as well as Liu et al. [40] introduced martingale-type 2 spaces in the theory of regularity structures (and in rough path theory) in order to apply methods from stochastic integration (extending Hölder-type spaces).
The following result is originally due to Neidhardt [44,Sect. 4]; see also [11,38,39,45]. Theorem 2.5 Let X be a separable Banach space with a norm · X and H a separable Hilbert space. Assume that ν L is the Gaussian probability measure on X generated by L ∈ γ (H, X). Then the mapping · γ (H,X) defined by is a norm on the space γ (H, X) and γ (H, X) is a separable Banach space with respect to this norm.

Definition 2.9
A Banach space (X, · X ) is said to satisfy the H -condition if for some q ≥ 2, the function ψ : X → R defined by is of class C 2 on X (in the Fréchet derivative sense) and if there exist constants K 1 , K 2 > 0 depending on q such that for every x ∈ X, where ψ (x) and ψ (x) are the first and second Fréchet derivatives of ψ at x ∈ X.
For each q ≥ p ≥ 2, the Lebesque space L p and the Sobolev space W 1,p satisfy the H -condition.

Proposition 2.10 [57]
If a Banach space (X, · X ) satisfies the H -condition, then it is also a martingale-type 2 Banach space.
Theorem 2.11 [14] Let ( , F, F, P), where F = (F t ) t≥0 , be a filtered probability space. Assume that (X, · X ) is a Banach space satisfying the H -condition. Assume that H is a separable Hilbert space and (W (t)) t≥0 is an H -valued F-cylindrical canonical Wiener process. Let (S(t)) t≥0 be a contraction C 0 -semigroup on X. If ξ is a γ (H, X)-valued F-progressively measurable process on [0, T ], T > 0, such that (2.1) holds, then there exists a constant K > 0 depending on H , X and K 1 , K 2 appearing in the H -condition such that

Definition 2.12
Let (X, · X ) be a Banach space. A C 0 -semigroup (S(t)) t≥0 on X is called contraction-type if there exists a constant β ∈ R such that S(t) ≤ e βt , t≥ 0.

Remark 2.13
If (S(t)) t≥0 is a contraction-type C 0 -semigroup on a Banach space X, then the family (T (t)) t≥0 of operators defined by T (t) = e −βt S(t) is a contraction C 0 -semigroup on X. Therefore, the following corollary follows from Theorem 2.11.

Assumptions and definitions
Throughout this paper, we assume that t≥0 , is a filtered probability space.
-(X, · X ) is a separable Banach space satisfying the H -condition.
-(S(t)) t≥0 is a contraction-type C 0 -semigroup on X with infinitesimal generator A.
-F and G are maps from R + × X into X and γ (H, X), respectively (we impose some sufficient conditions on them later).
We consider the stochastic evolution equation in X given by (3.1) Definition 3.1 An X-valued F-progressively measurable process (u(t)) t≥0 is called a mild solution to (3.1) if for every t ≥ 0, and P-a.s.

Existence and uniqueness of solutions to Equation (3.1) with globally Lipschitz coefficients
Assume that for each T > 0, Z T denotes the space of all X-valued, continuous, F-adapted processes u on [0, T ] such that For each T > 0, the space Z T is a Banach space with respect to the norm · T .
Moreover, we assume that for every x ∈ X, the map G(·, x) : [0, ∞) → γ (H, X) is Borel-measurable and for every T > 0, Then the mapping I G : Z T → Z T defined by is well defined. Moreover, it is of linear growth and globally Lipschitz on Z T .

Proof
Define Fix u ∈ Z T . We first show that for each t ≥ 0, the integral t 0 S(t −r)G(r, u(r)) dW (r) is well defined. Fix t ≥ 0. Due to Theorem 2.6, the process S(t − r)G(r, u(r)), r ∈ [0, t], is γ (H, X)-valued. Since G(·, x) : [0, ∞) → γ (H, X) is a Borel-measurable function, the process S(t − r)G(r, u(r)), r ∈ [0, t], is F-progressively measurable. It follows from (3.3) and (3.4) that the map G is of linear growth, i.e., for each T > 0, there exists a constantL G > 0 such that for all t ∈ [0, T ], Therefore, by (3.5) and (3.6), we obtain Thus, in view of Theorem 2.7, the integral t 0 S(t − r)G(r, u(r)) dW (r) is well defined. Now we show that I G is well defined. By Theorem 2.8, the process I G (u) is continuous and F-adapted. Furthermore, by Corollary 2.14 and (3.6), we get Hence I G (u) ∈ Z T and thus I G is well defined. It follows from the last inequality that which implies that I G is of linear growth. By Corollary 2.14 and (3.3), we have Therefore, we infer that which implies that I G is globally Lipschitz on Z T .

Lemma 3.3
Assume that F : R + × X → X is a globally Lipschitz map on X, i.e., for each T > 0, there exists a constant L F > 0 such that for all t ∈ [0, T ], Moreover, we assume that for every x ∈ X, the function F (·, x) : [0, ∞) → X is Borel-measurable and for every T > 0, Then the map I F : Z T → Z T defined by is well defined. Moreover, it is of linear growth and globally Lipschitz on Z T .
Proof Fix u ∈ Z T . By (3.8) and (3.9), F is of linear growth, i.e., for each T > 0, there exists a constantL F such that for all t ∈ [0, T ], It follows from (3.5) and (3.10) that the integral t 0 S(t − r)F (r, u(r)) dr is well defined P-a.s. for each t ≥ 0. Let us now show that I F is well defined. By [ Therefore I F (u) ∈ Z T and thus I F is well defined. Moreover, it follows from the last inequality that which implies that I F is of linear growth. Finally, we show that I F is globally Lipschitz on Z T . By the Cauchy-Schwarz inequality and (3.5) and (3.8), we infer that which implies that This completes the proof.

Theorem 3.4 Assume that
are globally Lipschitz maps on X. Moreover, we assume that for every x ∈ X, Then there exists a unique X-valued continuous mild solution to Eq. (3.1).
Proof By the definition of a mild solution, it is sufficient to show that the integral equation (3.2) has a unique X-valued continuous solution. Define the map It is obvious that the process S(·)u 0 is F-adapted. Moreover, for every ω ∈ , the map S(·)u 0 (ω) Thus the process S(·)u 0 belongs to Z T . Therefore, by Lemmas 3.2 and 3.3, is well defined and of linear growth. Moreover, by (3.7) and (3.11), we infer that Hence, is globally Lipschitz on Z T . If we choose T small enough, say T 0 , such that C(T 0 ) ≤ 1 2 , then by the Banach fixed point theorem, there exists a unique process u 1 ∈ Z T 0 such that (u 1 ) = u 1 . Therefore, the integral equation (3.2) has a unique X-valued continuous solution It is obvious that for each k, Z (k−1)T 0 ,kT 0 is a Banach space endowed with the norm As above, it can be easily shown that the equation Consequently, we have a sequence (u k ) k∈N of solutions. Define a process u from these solutions by We claim that this process is a unique X-valued continuous solution to the integral equation (3.2). It is obvious that the process u is continuous and F-adapted. Therefore, u is F-progressively measurable. Furthermore, for each t ≥ 0, Thus, the integrals in (3.2) are well defined for the process u. Let us now show that the process u solves the integral equation (3.2). We have already proved that for k = 1, the process u on [0, T 0 ] solves (3.2). By induction, we assume that the process u on [0, kT 0 ] solves (3.2). Thus, we have We need to show that the process u on [0, Since u k (kT 0 ) = u k+1 (kT 0 ) and by (3.13), we obtain Therefore, the process u on [0, (k + 1)T 0 ] solves (3.2). Hence, the process u is an X-valued continuous solution to the integral equation (3.2).
Uniqueness. In principle, the uniqueness of solutions follows from our proof via the Banach fixed point theorem. However, for completeness, we now present our independent proof. Let u 1 and u 2 be two solutions to (3.2). Define the process z by

By Theorem 2.8 and (3.3) and (3.5), we have
Moreover, using the Cauchy-Schwarz inequality and (3.5) and (3.8), we have Taking into account the last two estimates, we infer that Applying Gronwall's lemma, we infer that for all t ≥ 0, E[ z(t) 2 X ] = 0. Therefore, for all t ≥ 0, z(t) = 0 and so u 1 = u 2 . Thus, the process u is a unique X-valued continuous solution to the integral equation (3.2). Therefore, the process u is a unique X-valued continuous solution to (3.1).

Existence and uniqueness of solutions to Equation (3.1) with locally Lipschitz coefficients
Lemma 3.6 Assume that F : R + × X → X is a locally Lipschitz map on balls, i.e., for each Then for each n ∈ N, F n is globally Lipschitz on X with constant 3L F (n) (independent of t). Moreover, F n is of linear growth, i.e., Proof It follows from [7, Lemma 7] that for each n ∈ N, F n is globally Lipschitz with constant 3L F (n) (independent of t). Moreover, it follows from (3.19) that F n is of linear growth.
Similarly, we have the following result.
Moreover, assume that G is of linear growth (uniformly in t), i.e., for all T > 0, there exists a constantL G > 0 such that for all t ∈ [0, T ], Then for each n ∈ N, G n is globally Lipschitz on X with constant 3L G (n) (independent of t). Moreover, G n is of linear growth, i.e., The previous lemmas yield the following natural conclusion of Theorem 3.4.
Corollary 3.8 Assume the maps F : R + × X → X and G : R + × X → γ (H, X) are of linear growth and locally Lipschitz on balls. Moreover, we assume that for each Then for each n ∈ N, the stochastic evolution equation where F n and G n are the mappings defined in (3.20) and (3.22), respectively, has a unique X-valued continuous mild solution.

Lemma 3.9
Assume that for each n ∈ N, u n is the unique solution of (3.24). Then for every T > 0, there exists a constant C(T ) > 0 (independent of n) such that for each n ∈ N, Proof Fix n ∈ N and T > 0. Since u n is the unique solution of (3.24), we have By Corollary 2.14 and (3.23), we have, for each s ∈ [0, T ], Using the Cauchy-Schwarz inequality, (3.5) and (3.21), we obtain Therefore, we deduce that In view of (3.26)-(3.28), there exist constants N T > 0 and V T > 0 such that Applying Gronwall's lemma, we get the desired result (3.25).
where for each n ∈ N, u n is the unique solution of (3.24), is a stopping time. Moreover, the sequence (τ n ) n∈N of these stopping times converges to infinity.
Proof It was proved in [35, Problem 1.2.7] that for each n ∈ N, τ n is a stopping time. Thus we only prove that the sequence (τ n ) n∈N converges to infinity. For this aim, we need to show that there exists¯ ∈ F with P[¯ ] = 1 such that for all ω ∈¯ , τ n (ω) → ∞, i.e., for every T > 0, there exists k ∈ N such that for all n ≥ k, τ n (ω) ≥ T . Therefore, it is sufficient to show that for each T > 0, Then by (3.25) and the Chebyshev inequality, we have, for each n ∈ N, Thus there is k ∈ N such that for all n ≥ k and t ∈ [0, T ], u n (t, ω) X < n, which implies that there is k ∈ N such that for all n ≥ k, τ n (ω) ≥ T . This gives the desired conclusion. Proof For each n ∈ N, let u n be the unique solution of (3.24). Consider the sequence (τ n ) n∈N in Lemma 3.10. Define the process u by In view of [6,Lemma 4.11], (τ n ) n∈N has the two properties that (i) τ n ≤ τ n+1 and (ii) u n (t, ω) = u n+1 (t, ω), t ≤ τ n (ω), P-a.s. Therefore, the process u is well defined. We claim that this process is a unique X-valued continuous mild solution to (3.1).
Let us prove this. By the definition of a mild solution, it is sufficient to show that the process u is a unique X-valued continuous solution to the integral equation (3.2). It is obvious that u is continuous and F-adapted. Hence, it is F-progressively measurable. Moreover, for each t ≥ 0, Thus, the integrals in (3.2) are well defined for the process u. Now we show that the process u solves (3.2). Since u n is the solution of (3.24), we have the integral equation However, we have a problem here. The stochastic integral part of the equation is not well defined because we integrate a process which is not adapted and hence not progressively measurable. To overcome this problem, let us define the process I by It is obvious that I is well defined. Consider the process It was shown in [11,Lemma A.1] that if the processes I and I τ n are continuous, then for all t ≥ 0, P-a.s.
Since r ≤ τ n , u n (r) X ≤ n. Therefore, from the definitions of F n and G n , we get, for every r ≤ τ n , F n r, u n (r) = F r, u n (r) and G n r, u n (r) = G r, u n (r) .
Also by the definition of u, u(r) = u n (r) if r ≤ τ n . Hence, we obtain We know that τ n → ∞ and so t ∧τ n → t. Thus u(t ∧τ n ) → u(t) and S(t ∧ τ n ) → S(t). Therefore, we deduce that Thus, the process u solves the integral equation (3.2). The uniqueness of the solution follows from Theorem 3.4. Thus, the process u is a unique X-valued continuous mild solution to (3.1).

Markov property and invariant measures
In this section, we analyse the Markov property of the solution to Eq. (3.1). Because our main interests lies in the existence of an invariant measure, we assume in this and the following subsections that the coefficients in (3.1) are time-independent. It follows from the results proved in the previous sections that (3.1) has a unique X-valued continuous mild solution under the Lipschitz assumptions on the coefficients. Similarly, one can show that for every u 0 ∈ L 2 ( , F 0 , P; X), Eq. (3.1) with the initial value u(0) = u 0 has a unique X-valued continuous (mild) solution u on [0, ∞). We denote this solution by u(·, u 0 ). If u 0 = x ∈ X, we denote the solution by u(·, x). Assume that B b (X) denotes the space of all bounded measurable functions from X into R and B(X) the Borel σ -field of X.

Definition 3.12
The family (P t ) t≥0 of linear bounded operators on B b (X) defined by  [47] for the infinite-dimensional version of the Yamada-Watanabe theory, we infer that this is the case for our problem. Moreover, one can show, see e.g. Ondreját [47], that the family (P t ) t≥0 is indeed a Markov semigroup, i.e., we have P t+s = P t P s , t,s≥ 0.
Definition 3.14 A Borel probability measure μ on X is called an invariant probability measure for (3.30) if it is an invariant probability measure for the Markov semigroup, i.e., if for any time t ≥ 0, where C b (X) denotes the space of all bounded and continuous functions from X to R.
Remark 3.15 By Theorem 3.5, the transition semigroup (P t ) t≥0 is Feller, that is, for any t ≥ 0, P t : is the space of all continuous bounded functions from X into R, is well defined. It is known that the semigroup is in general not strongly continuous on the space C b (X). However, if μ is an invariant probability measure for the semigroup (P t ) t≥0 , then for every p ∈ [1, ∞), the semigroup (P t ) t≥0 has a unique extension to a C 0 -semigroup on the space L p (X, μ). Such statements can be found in many papers, see e.g. [18] and [42], but we could not find a proof. However, the proof of such a claim is standard since (i) the set of bounded functions in L p (X, μ) is dense in L p (X, μ); (ii) for every bounded function f ∈ L p (X, μ) and for every ε > 0, there exists a continuous bounded function f ε : Brzeźniak et al. [10] found some sufficient condition for the existence and uniqueness of an invariant measure for (3.1) with time-independent coefficients. Before introducing this, we present the following natural consequence of Theorem 3.11.

Corollary 3.16 Assume that the maps F : X → X and G : X → γ (H, X) are of linear growth and locally Lipschitz on balls. Then there exists a unique X-valued continuous mild solution to the equation
where A n is the Yosida approximation of A, K 2 is a constant appearing in the H -condition and [·, ·] X is the semi-inner product on X (see Definition 3.18 below), then there exists a unique invariant probability measure for (3.30).

Definition 3.18 A semi-inner product on a complex or real vector space
x, y ∈ V . Such a vector space V with the semi-inner product [·, ·] V is called a semi-inner product space.

Lemma 3.19 [10]
The mapping on X × X defined by where y * ∈ X * (X * is the dual space of X) is such that we have y * = y X and y, y * = y X , is a semi-inner product. Such a y * ∈ X * exists by the Hahn-Banach theorem.
Remark 3.20 Invariant measures are a subject related to semigroups, and an SDE generates a semigroup only when the coefficients are time-independent. However, there are some papers considering a generalization of an invariant measure for timedependent equations; see e.g. [48,20,21]. We plan to investigate this concept in relation to the HJMM equation in the future.

The HJMM equations
The value of one dollar at time t ∈ [0, T ] with maturity T ≥ 0 is called the zerocoupon bond, and is denoted by P (t, T ). This is a contract that guarantees the holder one dollar to be paid at the maturity date T . Thus, this is the most basic interest rate contract. Because of some additional factors like changes of the economy in time, the value of one dollar today could be better than the value of one dollar tomorrow and even the value of one dollar next year. Therefore, the bond prices are unknown in advance. Thus, it is assumed that for each T ≥ 0 and t ∈ [0, T ], P (t, T ) is an R-valued random variable defined on a probability space ( , F, P). Therefore, for each T > 0, (P (t, T )) t∈[0,T ] is an R-valued stochastic process. Under the assumption that for each t ∈ [0, T ], [0, ∞) T → P (t, T ) is a differentiable function, the function f defined by is called the forward rate function. It contains all the original bond price information.
For each T > 0, the family (f (t, T )) t∈[0,T ] is a stochastic process, and called forward rate process. For each t ∈ [0, T ], the function [t, ∞) T → f (t, T ) is called the forward curve. It is always assumed that the forward curves are locally integrable with respect to Lebesgue measure. If in addition P (T , T ) = 1, then one can write the equality In the framework of Heath-Jarrow-Morton [34], it was assumed that for each T > 0, the forward rate process (f (t, T )) t∈[0,T ] satisfies for t ∈ [0, T ] the stochastic differential equation  where W (t) = (W 1 (t), W 2 (t), . . . , W d (t)), t ≥ 0, is a standard d-dimensional Brownian motion and for each T > 0, (σ i (t, T )) t∈[0,T ] is a real-valued stochastic process. Filipović [26] extended the framework of Heath-Jarrow-Morton by considering a Wiener process in a (possibly infinite-dimensional) Hilbert space instead of a standard finite-dimensional Brownian motion. Thus, he assumed that for an arbitrary but fixed T > 0, the forward rate process (f (t, T )) t∈[0,T ] satisfies the stochastic differential equation

df (t, T ) =
where (W (t)) t≥0 is a Wiener process in an infinite-dimensional Hilbert space H endowed with an inner product ·, · H and for each T > 0, (σ (t, T )) t∈[0,T ] is an H -valued stochastic process.
Using the Musiela parametrization [43], an important connection between the HJMM model and stochastic partial differential equations can be provided as follows. Define where x is called time to maturity, and for each t ≥ 0, the function r(t) is a random variable on ( , F, P) taking values in the space of real-valued functions on [0, ∞) and called forward curve. Therefore, the family (r(t)) t≥0 of forward curves is a stochastic process taking values in the space of real-valued functions on [0, ∞) and called forward curve process. By the framework of Heath-Jarrow-Morton [34], the forward curve process (r(t)) t≥0 satisfies the stochastic partial differential equation where α is the function defined by Let g : [0, ∞) × [0, ∞) × R → H be a given function which is locally integrable with respect to the second variable. Assume that the volatility α is defined by i.e., the volatility α depends on the forward curve process (r(t)) t≥0 . Then (4.2) becomes

Existence and uniqueness of solutions to the HJMM equations in weighted Lebesgue spaces
In this section, we study the existence and uniqueness of solutions to the HJMM equations in the weighted L p spaces with p ≥ 2. The motivation for this is at least threefold. Firstly, it is of purely theoretical curiosity to see to what extent the results obtained in the L 2 -framework remain valid in the more general one. The second one, to which our paper is the very first step, is to consider the HJMM equations in the weighted fractional Sobolev spaces W θ,p with θ > 1 p (allowing thus to have θ < 1). Taking p large and θ just a bit bigger than 1 p will allow us to use a space which in some sense is much closer to the space C of continuous functions. Also using either weighted L p or fractional Sobolev spaces W θ,p allows one to consider the HJMM equations with coefficients satisfying less stringent regularity assumptions than in the classical approach. Thirdly, if the functions σ i (t, ·) are not regular enough, then the natural state spaces for the solutions of the HJMM equations are the weighted L p spaces; see Example 4.10. Let us point out that in their monograph [50], Peszat and Zabczyk used the weighted L 2 spaces. For each ν ∈ R and p ≥ 1, define L p ν to be the space of all (equivalence classes of) Lebesgue-measurable functions f : [0, ∞) → R such that For each ν ∈ R and p ≥ 1, L p ν is a separable Banach space with the norm · ν,p .

Lemma 4.1
For each ν > 0 and p ≥ 1, the space L p ν is continuously embedded into the space L 1 . In particular, The proof follows from the Hölder inequality.
and The proof is given below.
where ψ (f ) and ψ (f ) are the first and second Fréchet derivatives of ψ at f ∈ L p ν , respectively.
Proof Fix ν ∈ R and p ≥ 2. Define the linear operator T : L p ν → L p by It is obvious that the map T is well defined, bijective and It is clear that any linear operator, so T , is twice Fréchet differentiable and for each f ∈ L p ν , T (f ) = T and T (f ) = 0. By [13, Proposition 2.1], for every p ≥ 2, the space L p satisfies the H -condition for any q ≥ p, i.e., for some q ≥ p (in particular, q = p), the map φ : We can write ψ = φ • T . Since T and φ are of class C 2 , ψ is of class C 2 . Moreover, it follows from (4.6) and (4.7) that Similarly, which finishes the proof.

Lemma 4.4
For each ν > 0 and p ≥ 1, the shift semigroup (S(t)) t≥0 on the space L p ν is a contraction-type C 0 -semigroup, in particular,

Moreover, its infinitesimal generator A is characterized by
where D is the weak derivative (of order one).
where φ ∈ L p (H ), is a γ -radonifying operator and, for a constant N > 0 independent of φ, Therefore, the operatorK : H → L p defined bȳ is a γ -radonifying operator, and for a constant N > 0, It follows from (4.8) that For the linear operator T in the proof of Lemma 4.3, we can write K = T −1 •K. Thus by Theorem 2.6, the map K is a γ -radonifying operator. Moreover, since T −1 ≤ 1 and by (4.9), we infer that which completes the proof.
is well defined. Moreover, we have: This function is continuous. Indeed, by (4.4), we have for every sequence (x n ) n∈N in [0, ∞) converging to x ∈ [0, ∞) that as n → ∞, Thus h is continuous and so it is measurable. Therefore, F (t, f ) is measurable. Moreover, by the Cauchy-Schwarz inequality, Lemma 4.1 and (4.4), we obtain Therefore F (t, f ) ∈ L p ν and thus F is well defined. Moreover, (4.11) gives the desired conclusion (4.10). Finally, we prove that F is globally Lipschitz on L p ν . Fix t ≥ 0 and f 1 , f 2 ∈ L p ν . Then by the Cauchy-Schwarz inequality, we have It follows from Lemma 4.1 and (4.4) and (4.5) that for t ≥ 0, x ∈ [0, ∞), Taking into account the last inequality, we infer that concluding that F is globally Lipschitz on L p ν with a Lipschitz constant independent of time t.
is well defined. Moreover, we have: Then G(t, f ) can be written as It follows from (4.4) that κ ∈ L p ν (H ). Therefore, by Lemma 4.5, G(t, f ) is a γ -radonifying operator from H to L p ν and so G is well defined. Moreover, again by Lemma 4.5 and (4.4), we have for a constant N > 0 independent of κ that G(t, f ) γ (H,L p ν ) ≤ N ḡ ν,p , which gives the desired result (4.13). Finally, using Lemma 4.5 and (4.5), we have for each t ≥ 0 and f 1 , where σ 0 , λ are constants and λ > 0. This corresponds to the volatility σ in (4.1) having the form With this function g, (4.3) (driven by a standard one-dimensional Wiener process W = (W (t)), t ≥ 0) transforms to As g satisfies all assumptions of Theorem 4.2, we infer that (4.16) has a unique L p ν -valued continuous mild solution with initial value r 0 ∈ L 2 ( , F 0 , P; L p ν ). Moreover, by the definition of a mild solution, this unique solution is of the exact form We can also consider the functions g 1 and/or g 2 defined by where λ > 0 and β ∈ R. As these functions obviously satisfy all the assumptions of Theorem 4.2, we infer that for each r 0 ∈ L 2 ( , F 0 , P; L p ν ), (4.3) with the function g i , i = 1, 2, has a unique L p ν -valued continuous mild solution with initial value r 0 . Example 4.9 Let us next consider the function g defined by where λ > 0. With this function g, (4.3) changes to x 0 e −λy sin r(t)(y) dy dt where (W (t)) t≥0 is a standard one-dimensional Wiener process. Since the function sin is bounded and globally Lipschitz on R, the function g satisfies all the assumptions of Theorem 4.2. Thus, (4.9) has a unique L p ν -valued continuous mild solution with the initial value r 0 ∈ L 2 ( , F 0 , P; L p ν ). Example 4.10 Suppose that W (t), t ≥ 0, is a standard 1-dimensional Brownian motion and the function σ in (4.1) is of the form we infer that for all t ≥ 0, Hence, for every t ≥ 0, α(t) ∈ L p ν . However, unless σ 0 ∈ W 1,2 ν , α(t) does not belong to the space W 1,2 ν . In particular, one can expect that the problem

Existence and uniqueness of solutions to the HJMM equations in weighted Sobolev spaces
For each ν ∈ R and p ≥ 1, define W 1,p ν to be the space of all functions f ∈ L p ν such that the weak derivative Df belongs to L p ν , i.e., For each ν ∈ R and p ≥ 1, the space W Using the Hölder and Young inequalities, we get Taking into account the last inequality, we obtain Similarly, we can prove the above inequality for x ∈ [0, x 0 ).  (iv) For all t ∈ [0, ∞), In view of Theorem 3.11, the proof follows from the following lemmas. Similarly to the argument for Lemma 4.3, the proof follows from the fact that for each p ≥ 2, the space W 1,p satisfies the H -condition; see [13]. Lemma 4.14 For each ν ∈ R and p ≥ 1, the shift semigroup on W 1,p ν is a contraction-type C 0 -semigroup. Moreover, its infinitesimal generator is characterized by The proof of the first part of Lemma 4.14 can be found in [37,Lemma 5.13] and the proof of the second part in [37,Lemma 5.16].
For each ν ∈ R and p ≥ 1, define W is well defined. Moreover, we have: Since the functions g(t, x, f (x)), Dg(t, x, f (x)) and h(x) := x 0 g(t, y, f (y)) are measurable, DF (t, f ) is measurable. Moreover, by the Cauchy-Schwarz inequality, we obtain for each x ∈ [0, ∞) that Using (4.17) and Lemma 4.1, we obtain

Note that
Therefore, by (4.19) and (4.21), we get for each x ∈ [0, ∞) that Taking into account the last inequality, we obtain Thus, by the last inequality and Proposition 4.11, we infer that Therefore DF (t, f ) ∈ L P ν and hence F is well defined. Moreover, it follows from (4.10) and the last inequality that which gives the desired conclusion (4.23).
Let us now prove that F is locally Lipschitz on balls. Fix t ≥ 0 and f 1 , By the Cauchy-Schwarz inequality, we have Using (4.17) and (4.18), we obtain for each x ∈ [0, ∞) that It follows from Lemma 4.1 and Proposition 4.11 that Taking into account the last inequality and Proposition 4.11, we infer that By the chain rule, we get for x ∈ [0, ∞) that Note that Using the Cauchy-Schwarz inequality and (4.17) and (4.18), we obtain By Lemma 4.1 and Proposition 4.11, we get for x ∈ [0, ∞) that Note that Using the Cauchy-Schwarz inequality, we obtain It follows from (4.20)-(4.22) that for x ∈ [0, ∞), Taking into account (4.24) and the last inequality, we obtain for each x ∈ [0, ∞) that Using the last inequality and Proposition 4.11, we infer that is well defined. Moreover, we have:  Step 3. Taking into account (4.14), (4.33) and (4.34), we obtain where C n = 2( p νq ) 1 q ĝ ∞ ḡ ν,p + (p − 1)N 2 ĝ 2 ∞ + −νn np+ν . Since (4.31) gives there exists n 0 ∈ N such that for all ω ∈ (0, −C), C n ≤ −ω, n ≥ n 0 .

Remark 4.23
There are many papers on the existence of invariant measures for the HJMM equations. In addition to the already mentioned papers [53,54], one should also list [29,41].

Remark 4.24
It would be interesting to investigate the uniqueness of an invariant measure for the HJMM equation in the weighted Sobolev spaces. A use of Malliavin calculus could prove to be essential here. In this context, one should mention two papers. The first one by Baudoin and Teichmann [1] is about hypoellipticity for finite-dimensional realisations of the HJMM equations, whose generalisation could provide a first step in proving the uniqueness. The second one by Hairer and Mattingly [33] shows how to use the hypoellipticity for (infinite-dimensional) parabolic SPDEs in conjunction with an asymptotic strong Feller property in order to prove the uniqueness of an invariant measure.

An extension of the HJMM model
Klein et al. [36] extended the HJMM model by an additional constituent which is not absolutely continuous in terms of the maturity T . Let us fix a finite time horizon T * and a probability measure Q which is equivalent to the measure P| F T * . Assume that W and V are d-and 1-dimensional Q-independent Q-Brownian motions. Consider the filtration (F t ) t≥0 defined by (4.39) By the above calculations, the coefficient α 2 is equal to We see that for a fixed t ≥ 0, the function c(t, ·) is at most Hölder-continuous of order strictly less than 1 2 and therefore, the HJMM equations (4.38) and (4.39) need to be solved in larger spaces than the weighted W 1,2 spaces, for instance, weighted L p spaces or fractional Sobolev spaces.

The HJMM equations with real-world dynamics driven by Wiener processes
Tappe [55] presented new results concerning the HJMM equations with real-world dynamics. He assumed that under the real-world probability measure P, for every T , the forward rate process (f (t, T )) t∈[0,T ] satisfies the term structure equation df (t, T ) = α(t, T ) dt + σ (t, T ) dW (t) + jumps, t∈ [0, T ], (4.40) where W is a (possibly infinite-dimensional) Wiener process. We did not write the jumps in (4.40) in detail since we assume that the jump terms in our case are simply equal to zero. If the jumps are zero, then Tappe's model includes the original HJMM framework. As argued in [37], switching to the Musiela parametrization [43], i.e., leads to the partial differential equation  Tappe proved that in an arbitrage-free bond market, the drift term in (4.41) is given by where θ denotes the reference market price of risk with respect to the Wiener process W . He also proved, under some suitable conditions, the existence and uniqueness of mild solutions to the HJMM (4.41) with real-world dynamics in a separable Hilbert space.