Stochastic Evolution Equations in Banach Spaces and Applications to Heath-Jarrow-Morton-Musiela Equation

In this paper we study the stochastic evolution equation (1.1) in martingale-type 2 Banach spaces (with the linear part of the drift being only a generator of a C0-semigroup). We prove the existence and the uniqueness of solutions to this equation. We apply the abstract results to the Heath-Jarrow-Morton-Musiela (HJMM) equation (6.3). In particular, we prove the existence and the uniqueness of solutions to the latter equation in the weighted Lebesgue and Sobolev spaces respectively. We also find a sufficient condition for the existence and the uniqueness of an invariant measure for the Markov semigroup associated to equation (6.3) in the weighted Lebesgue spaces.


Introduction
Let (Ω, F , F, P), where F = {F t } t≥0 , be a filtered probability space, and (H, ·, · H ) be a separable Hilbert space. Assume that W is an H-valued, F-cylindrical canonical Wiener process, i.e. W is a family {W(t)} t≥0 of bounded linear operators from H into L 2 (Ω, F , P) such that (i) for all t ≥ 0 and h 1 , h 2 ∈ H, EW(t)h 1 W(t)h 2 = t h 1 , h 2 H , (ii) for any h ∈ H, {W(t)h} t≥0 is real-valued, F-adapted Wiener process. The aim of this paper is twofold. The first one is to prove, under the local Lipschitz conditions on the coefficients, the existence and the uniqueness of solutions to the following stochastic evolution equation in a Banach space X satisfying the H p condition, which is stated below.
Here γ(H, X) denotes the Banach space of γ-radonifying operators from H into X, see [23] for details. The norm in the space γ(H, X) will be denoted by · γ(H,X) . We say that a Banach space X with the norm · X satisfies H p condition, see [32] for details, if for some p ≥ 2, the function ψ : x X → ψ(x) = x p X ∈ R is of C 2 class on X (in the Fréchet derivative sense) and there exist constants K 1 (p), K 2 (p) > 0 depending on p such that for every x ∈ X, ψ (x) ≤ K 1 (p) x p−1 X and ψ (x) ≤ K 2 (p) x p−2 X .
The second one is to apply the previous abstract results to the Heath-Jarrow-Morton-Musiela (HJMM) equation (6.3) introduced in section 6. The existence and the uniqueness of solutions to equation (6.3) in Hilbert spaces, and ergodic properties of the solutions have been studied, in particular, in [2], [12], [13], [14], [16], [18], [22], [25], [27], [28], [29], [31]. In this paper we prove the existence and the uniqueness of solutions to equation (6.3) in the weighted Banach spaces L p ν and W 1,p ν defined in section 6. We also find a sufficient condition from Theorem 3.7 in [10] for the existence and the uniqueness of invariant measures for the solution to equation (6.3), when the coefficients are time independent, in the spaces L p ν . An important feature of our results is that we are able to consider HJMM Equations driven by a cylindrical Wiener process on a (possibly infinite dimensional) Hilbert spaces. For this purpose we use a characterization of γ-radonifying operators from a Hilbert space to an L p space found recently by the first named author and Peszat in [4].
Definition 2.1. A Banach space X with the norm · X is called a martingale-type 2 Banach space if there exists a constant C > 0 depending only on X such that for any X-valued martingale {M n } n∈N , the following inequality holds The Lebesgue function spaces L p , p ≥ 2, are examples of martingale-type 2 Banach spaces.
Proposition 2.2. [32] If X is a Banach space satisfying the H p condition, then X is an martingale-type 2 Banach space. Definition 2.3. Let X be a Banach space and L(X) be the space of all bounded linear operators from X to X. A C 0 -semigroup S = {S (t)} t≥0 on X is called contraction type iff there exists a constant β ∈ R such that S (t) L(X) ≤ e βt , t ≥ 0.
(2.1) Definition 2.4. Let X be a martingale-type 2 Banach space and S be a contraction type C 0 -semigroup on X with the infinitesimal generator A. A process u is called an X-valued mild solution to SEE (1.1) if for each t ≥ 0, where u is such that each term on the right hand side is well-defined.
The integrals in equation (2.2) are not always well-defined for any process u, and functions F and G. Therefore, we also introduce some notations which make the integrals in equation (2.2) well-defined in martingale type 2 Banach spaces. Let X be a martingale type 2 Banach space endowed with the norm · X and S be a contraction type C 0 -semigroup on X. It follows from [6] that if ξ is a γ(H, X)-valued, F-progressively measurable process such that for all t ≥ 0, then for each t ≥ 0, the stochastic integral t 0 ξ(s)dW(s) is well-defined. Moreover, the stochastic integral process t 0 ξ(s)dW(s), t ≥ 0, is an X-valued, F-progressively measurable process, and there exists a constant C > 0 (independent of ξ) such that In particular, for each t ≥ 0, the stochastic convolution integral t 0 S (t − r)ξ(r)dW(r) is well-defined provided that ξ is a γ(H, X)-valued, F-progressively measurable process such that for each t ≥ 0, Moreover, the stochastic convolution process t 0 S (t − r)ξ(r)dW(r), t ≥ 0, is an X-valued, F progressively measurable process. If for each t ≥ 0, the function G(t, ·) : X → γ(H, X) is bounded, and for each T > 0 and x ∈ X, the function G(·, x) : [0, T ] → γ(H, X) is Borel measurable, then the process G(t, u(t)), t ≥ 0, is γ(H, X)-valued, F-progressively measurable such that for each t ≥ 0, provided that u is an X-valued, F-progressively measurable process such that for each t ≥ 0, Hence, for each t ≥ 0, the stochastic convolution integral t 0 S (t − r)G(r, u(r))dW(r) is well-defined, and the stochastic convolution process t 0 S (t − r)G(r, u(r))dW(r), t ≥ 0, is an X-valued, F progressively measurable process.
It follows from [1] that if u is an X-valued, F-progressively measurable process such that for each t ≥ 0, t 0 u(r) X dr < ∞, P − a.s. then for each t ≥ 0, the deterministic convolution integral t 0 S (t − r)u(r)dr is well defined P-a.s, and the deterministic convolution process t 0 S (t−r)u(r)dr, t ≥ 0, is X-valued, F-progressively measurable. If for each t ≥ 0, the function F(t, ·) : X → X is bounded, and for each T > 0 and x ∈ X, the function F(·, x) : [0, T ] → X is Borel measurable, then the process F(t, u(t)), t ≥ 0, is an X-valued F-progressively measurable process such that for each t ≥ 0, t 0 F(r, u(r)) X dr < ∞ P − a.s.
Thus, for each t ≥ 0, the deterministic convolution integral t 0 S (t − r)F(r, u(r))dr is well-defined, and the deterministic convolution process t 0 S (t − r)F(r, u(r))dr, t ≥ 0, is X-valued F-progressively measurable.
It follows from [5] that if X is a Banach space satisfying the H p condition with the norm · X , {S (t)} t≥0 is a contraction C 0 -semigroup on X and ξ is a γ(H, X)-valued, F-progressively measurable process such that for some T > 0, then there exists a constant K > 0 depending on H, X and K 1 (p), K 2 (p) appearing in the H p condition such that Note that A semigroupS defined byS (t) = e −βt S (t), t ≥ 0, is a contraction C 0 -semigroup on X. Therefore, we can prove that The main result of this section is the following theorem.
Theorem 3.1. Let X be a Banach space satisfying the H p condition endowed with the norm · X . Assume that {S (t)} t≥0 is a contraction type C 0 -semigroup on X with the infinitesimal generator A. Assume that for each t ≥ 0, F(t, ·) : X → X and G(t, ·) : X → γ(H, X) are globally lipschitz maps on X, i.e. for each T > 0 there exist constants L F > 0 and L G > 0 such that for all t ∈ [0, T ], (3.2) Moreover, we assume that for every x ∈ X the functions F(·, x) : [0, ∞) t → F(t, x) ∈ X and G(·, x) : [0, ∞) t → G(t, x) ∈ γ(H, X) are Borel measurable, and for all T > 0 and x ∈ X, Then for each u 0 ∈ L 2 (Ω, F 0 , P; X), there exists a unique X-valued mild solution to SEE (1.1).
Proof. Existence : The proof for the case β < 0 is included in the proof for the case β ≥ 0, since for β < 0, e βt ≤ 1. Therefore, we assume that {S (t)} t≥0 is a contraction type C 0 -semigroup on X, i.e. there exists a constant β ≥ 0 such that S (t) L(X) ≤ e βt , t ≥ 0. For each T > 0, let Z T be the space of all X-valued, F-progressively measurable processes u on [0, T ] such that the trajectories of u are P-a.s continuous and Obviously Z T is a Banach space endowed with the norm Fix u 0 ∈ L 2 (Ω, F 0 , P; X) and T > 0. Define a map Φ T,u 0 by It follows from [1] and [6] that the deterministic convolution process t 0 S (t − r)F(r, u(r))dr, t ≥ 0, and the stochastic convolution process t 0 S (t −r)G(r, u(r))dW(r), t ≥ 0, are well-defined and belong to Z T . Moreover, the process S (·)u 0 belongs to Z T since E u 0 2 X < ∞. Thus, Φ T,u 0 is a well-defined map from Z T into Z T . Furthermore, we shall prove that the map Φ T,u 0 is globally Lipschitz on Z T . For this aim let us fix u 1 , u 2 ∈ Z T . Then, using the Cauchy-Schwarz inequality, (3.1) and (3.4) Therefore, we infer that and thus E sup Moreover, using (2.4) and (3.2) we have Taking into account inequalities (3.6) and (3.7) we infer that Thus, Φ T,u 0 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 Φ T,u 0 (u 1 ) = u 1 . Therefore, equation (2.2) has a unique solution u 1 on [0, T 0 ]. Let Z (k−1)T 0 ,kT 0 , k = 1, 2, 3..., be the space of all X-valued, F-progressively measurable stochastic processes u on [(k − 1)T 0 , kT 0 ] such that trajectories of u are P-a.s. continuous and For each k, Z (k−1)T 0 ,kT 0 is a Banach space endowed with the norm As shown above, it can be easily shown that the following equation has a unique solution u k in the space Z (k−1)T 0 ,kT 0 such that u k (kT 0 ) = u k+1 (kT 0 ). Consequently we have a sequence (u k ) k∈N of solutions.
Define a process u by We claim that this process u is a unique solution to equation (2.2).
Uniqueness : In principle, the uniqueness of solutions follows from our proof via the Banach Fixed Point Theorem. However, for completeness and educational purposes, we will present now our independent proof.
Let u 1 and u 2 be two solutions to equation (2.2). Define a process z by It follows from inequalities (3.5) and (3.11) that . Applying Gronwall's inequality, see [30], to function E z(t) 2 X , t ≥ 0, we obtain that for all t ≥ 0, E z(t) 2 X = 0, which completes the proof of the claim. Therefore, u defined in (3.10) is a unique mild solution to SEE (1.1). Hence the proof of Theorem 3.1 is complete. 4. The existence and the uniqueness of solutions to see (1.1) in Banach spaces with locally Lipschitz coefficients The main result of this section is the following theorem.
Theorem 4.1. Let X be a Banach space satisfying the H p condition endowed with the norm · X . Assume that {S (t)} t≥0 is a contraction type C 0 -semigroup on X with the infinitesimal generator A. Assume that for each t ≥ 0, F(t, ·) : X → X and G(t, ·) : X → γ(H, X) are locally Lipschitz maps, i.e. for all T > 0 and R > 0 and Then for each u 0 ∈ L 2 (Ω, F 0 , P; X), there exists a unique X-valued mild solution u to SEE (1.1).
Proof. We assume β ≥ 0 as in the proof of Theorem 3.1. For each n ∈ N and t ≥ 0, define mappings F n (t, ·) : X → X and G n (t, ·) : X → γ(H, X) by respectively, and For each n ∈ N and t ≥ 0, F n (t, ·) and G n (t, ·) are globally Lipschitz on X with constants 3L F (n) and 3L G (n) (independent of t), see [8]. Moreover, for each n ∈ N and t ≥ 0, F n (t, ·) and G n (t, ·) satisfy the linear growth conditions (4.3) and (4.4) with the same constantsL F andL G . We will prove linear growth of F n (t, ·).
Fix n ∈ N and t ≥ 0. Then Case 2 : Consider x ∈ X such that x X > n. Then, F n (t, x) = F t, n x x X . Therefore, from (4.3) we get Hence, we infer that F n (t, x) 2 Similarly, we can prove that For each n ∈ N, consider the following stochastic evolution equation By Theorem 3.1, SEE (4.9) has a unique X-valued mild solution u n . Moreover, we shall prove that for each Proof of (4.10): Fix n ∈ N and T > 0. Let u n be the unique solution of equation (4.9). Then Thus, we have, for each s ∈ [0, T ], By the Cauchy-Schwarz inequality, (3.4) and (4.7), we obtain, for each t ≥ 0, u n (t)) 2 X dr. For each n ∈ N, τ n is a stopping time, see [20] for the proof. Moreover, we shall prove that the sequence (τ n ) n∈N converges to ∞. For this aim it is sufficient to show that for each T > 0, (4.13) Fix T > 0. Then using inequality (4.10) and the Chebyshev Inequality we have
Since r ≤ τ n and by the definition of τ n , u n (r) X ≤ n. Therefore, from the definition of F n (t, ·) and G n (t, ·) we get F n (r, u n (r)) = F(r, u n (r)) and G n (r, u n (r)) = G(r, u n (r)).

Existence and uniqueness of an invariant measure to SEE (1.1). In this section we consider the following SEE
where P * t is the dual semigroup defined on the space M(X) of all bounded measures on (X, B(X)), and is defined by ϕ, P * t µ = P t ϕ, µ , ϕ ∈ C b (X), µ ∈ M(X).
The following theorem obtained in [10] gives a sufficient condition for the existence and the uniqueness of invariant measures for the solution to problem (5.1), see Theorem 3.7 and Theorem 4.6 therein for the proof.
Theorem 5.6. Assume that the functions F : X → X and G : X → γ(H, X), satisfy all the assumptions of Theorem 3.1 or Theorem 4.1. If there exist a constant ω > 0 and a natural number n 0 ∈ N such that for all x 1 , x 2 ∈ X and n ≥ n 0 , where A n is the Yosida approximation of A, and [·, ·] is the semi-inner product on X × X, see Definition 5.7 below, then there exists a unique invariant measure for the solution to SEE (5.1).

Definition 5.7. A semi-inner product on a complex or real vector space V is a mapping
Such a vector space V is called a semi-inner product space.
Remark 5.8. A semi-inner product on a Banach space X is given by where y * ∈ X * such that y * = y X and y, y * = y X . Such a y * ∈ X * exists by the Hahn-Banach theorem.
6. Application to Heath-Jarrow-Morton-Musiela (HJMM) equation 6.1. The HJMM equation. The value of one dollar at time t ∈ [0, T ] with maturity T ≥ 0 is called the zero-coupon 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 . It is assumed that for each T ≥ 0 and t ∈ [0, T ], P(t, T ) denotes an R-valued random variable defined on a probability space (Ω, F , P). Therefore, for each T > 0, {P(t, T )} t∈[0,T ] denotes an R-valued stochastic process. Under some assumptions, P(t, T ) can be written as follows, see [15], where for each 0 ≤ t ≤ T , f (t, T ) is an R-valued random variable on (Ω, F , P), and called forward rate. Therefore, for each T ≥ 0, the family { f (t, T )} t∈[0,T ] of forward rates is a stochastic process, and called forward rate process. In the framework of Heath-Jarrow-Morton [19], it was assumed that for each T ≥ 0, forward rate process f (t, T ), t ∈ [0, T ], satisfies the following stochastic differential equation, where W is a standard d-dimensional Brownian motion and ·, · denotes the usual inner product in R d .
Using the Musiela parametrization [22] an important connection between HJM model and stochastic partial differential equations can be provided. Define where x is called time to maturity, and for each t ≥ 0, the function r(t) is a random variable on (Ω, F , P) taking values in the space of functions of x, and called forward curve. Therefore, the family {r(t)} t≥0 of forward curves is a stochastic process taking values in the space of functions of x, and called forward curve process. By the framework of Heath-Jarrow-Morton [19], the forward curve process satisfies the following stochastic partial differential equation, where α(t)(x) = σ(t, t + x), t, x ≥ 0, (H, ·, · H ) is a (possibly infinite dimensional) Hilbert space and W is an F-cylindrical canonical Wiener process on H, see [21] for the derivation of equation (6.2).
If the volatility α depends on the forward curve r, i.e. α(t)(x) = ζ(t, r(t))(x), then equation (6.2) becomes x 0 ζ(t, r(t))(y) dy H dt + ζ(t, r(t))(x), dW(t) H . (6.3) In this paper we will analyse, for a certain function ζ, the existence and the uniqueness of solutions to equation (6.3) in some certain spaces of functions, and the existence and the uniqueness of invariant measures for the solution to equation (6.3).

Existence and uniqueness of solutions to HJMM equation with globally Lipschitz coefficients.
For each ν ∈ R and p ≥ 1, let L p ν be the space of all (equivalence classes of) Lebesgue measurable functions It is well know that for each ν ∈ R and p ≥ 1, L p ν is a Banach space endowed with the norm If p ≥ 2, then L p ν satisfies the H p condition [5]. Moreover, see [7], if ψ : Proposition 6.1. Let ν > 0 and p ≥ 1. Then the space L p ν is continuously embedded into the space L 1 ([0, ∞)), and for all f ∈ L p ν , Proof. Fix ν > 0, p ≥ 1 and f ∈ L p ν . Then by Hölder inequality, we get This gives the desired conclusion.
Lemma 6.2. Let S = {S (t)} t≥0 be a family of operators on L p ν defined by Then S is a contraction type C 0 -semigroup on L p ν such that Moreover, the infinitesimal generator A of S on L p ν is given by where D f is the first weak derivative of f . Remark 6.3. The semigroup S defined in (6.5) is called the shift semigroup.
For the suitability of our aim, throughout this section we assume that p ≥ 2 and ν > 0.
In view of Theorem 3.1, the proof of the existence and the uniqueness of solutions follows from Lemma 6.2 and the following lemma. The proof of the Markov property follows from Theorem 5.3. Lemma 6.5. Under the assumptions of Theorem 6.4, for each t ≥ 0, F(t, ·) : L p ν → L p ν and G(t, ·) : L p ν → γ(H, L p ν ) are well-defined. Moreover, for all t ≥ 0, (ii) F(t, ·) and G(t, ·) are globally Lipschitz on L p ν with Lipschitz constants independent of t ∈ [0, ∞]. Proof. Our proof is based mainly on the proposition taken from [4] Proposition 6.6. Let H be a separable Hilbert space and O ⊂ R d . Let (O, F , µ) be a measurable space and L p = L p (O, F , µ; R), p ∈ [2, ∞). Then K : H → L p defined by where κ ∈ L p (O, F , µ; H), is a γ-radonifying operator from H into L p ν and for some N > 0, Proof of (i) : Fix t ≥ 0 and f ∈ L p ν . Then using (6.9) we get Therefore, by Proposition 6.6, G(t, f ) is a γ-radonifying operator from H into L p ν and for some N > 0, Thus, G(t, ·) is well defined. Using the Cauchy-Schwarz inequality we get y, f (y)) H dy.
It follows from Proposition 6.1 and (6.9) that Therefore, we obtain Hence, F(t, ·) is well-defined, and by inequalities (6.12) and (6.13), we infer that Proof of (ii) : Fix t ≥ 0 and f 1 , Therefore, by Proposition 6.6, there exist a constant N > 0 such that Hence, G(t, ·) is globally Lipschitz on L p ν .
6.3. Existence and uniqueness of solutions to HJMM equation with locally Lipschitz coefficients. For each ν ∈ R and p ≥ 1, let W 1,p ν be the weighted Sobolev space defined by where D f is the first weak derivative of f . It is well known that W 1,p ν is a Banach space endowed with the norm Moreover, for p ≥ 2, W 1,p ν satisfies the H p condition, see [21] for the proof. Proposition 6.7. Let ν ∈ R and p ≥ 1. Then there exists a constant C(ν, p) > 0 such that Proof. Take fixed ν ∈ R, p ≥ 1 and f ∈ W 1,p ν . Fix ε > 0. Since Therefore, we get Lemma 6.9. Let S = {S (t)} t≥0 be the family of operators defined in (6.5) for the space W 1,p ν . Then S is a contraction type C 0 -semigroup on W 1,p ν and its infinitesimal generator A satisfies where D f and D 2 f denote the first and second weak derivative of f respectively.
For the suitability of our aim, in this section we again assume that p ≥ 2 and ν > 0.