Stochastic evolution equations in Banach spaces and applications to the Heath-Jarrow-Morton-Musiela equations : HJMM equations in weighted $L^p$ spaces

applications to the Heath-Jarrow-Morton-Musiela equations : HJMM equations in weighted

$L^p$ spaces

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Brzezniak, Zdzislaw orcid.org/0000-0001-8731-6523 and Kok, Tayfun (2018) Stochastic

evolution equations in Banach spaces and applications to the

Heath-Jarrow-Morton-Musiela equations : HJMM equations in weighted $L^p$ spaces.

Finance and Stochastics. pp. 959-1006. ISSN 1432-1122

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Stochastic evolution equations in Banach spaces

and applications to the Heath–Jarrow–Morton–Musiela

equations

Zdzisław Brze´zniak1·Tayfun Kok1

Received: 21 August 2016 / Accepted: 3 July 2018 / Published online: 28 September 2018 © The Author(s) 2018

Abstract The aim of this paper is threefold. Firstly, we study stochastic evolution equations (with the linear part of the drift being a generator of aC0-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 solu-tions for these stochastic evolution equasolu-tions in a class of Banach spaces satisfying the so-calledH-condition. Moreover, we analyse the Markov property of the solu-tions.

Secondly, we apply the abstract results obtained in the first part to prove the ex-istence 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.

Keywords Stochastic evolution equations_·Heath–Jarrow–Morton–Musiela equations·Markov semigroup·Invariant measures·Martingale-type 2 Banach spaces

Mathematics Subject Classification (2010) 60H15·91G80·37L40·37L55·

35R10·47D06·91B24·91B25·91B55·91B70

JEL Classification E43_·C60

B

[email protected]

T. Kok

[email protected]

1 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 ad-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 coef-ficients. For example, we assume that the linear part of the drift is the infinitesimal generator of a non-analyticC0-semigroup. The lack of analyticity of the semigroup leads to some difficulties, and in this article, we show how to overcome this prob-lem. For instance, the time-continuity of the solution cannot be a consequence of the Da Prato–Kwapie´n–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 evolu-tion equaevolu-tions in Hilbert spaces. Recently, the first author, Long and Simão [10] paid attention to the theory of invariant measures for stochastic evolution equations in Ba-nach 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 appli-cations in finance. One of these appliappli-cations 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 au-thors proved the existence and uniqueness of solutions to the HJMM model in some appropriate Hilbert spaces such as the weighted Lebesgue spaceL2and the weighted Sobolev spaceW1,2. Also some of these authors analysed ergodic properties of the solutions.

weightedLpandW1,pspaces, withp >2, are ‘locally’ smaller than the correspond-ing weightedL2 andW1,2 spaces. Thus, proving the existence of solutions taking values in these spaces provides additional information about the properties of the so-lutions. For instance, a solution with values in the weightedW1,2 space is locally Hölder-continuous with order strictly less than 1−12, while a solution with values

in the weightedW1,pspace is locally Hölder-continuous with order strictly less than 1_−p1. 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 De-partment 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

pshould make it possible to extend the deep results about finite-dimensional reali-sations, which are usually formulated in terms of the very small setD(A∞)to the spacesW1,porWθ,pwith largepin 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

L2or weighted SobolevW1,2spaces), 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 aC0-semigroup and the coefficients

satis-fying Lipschitz conditions) in the class of Banach spaces satissatis-fying theH-condition. Firstly, we prove the existence and uniqueness of solutions for corresponding equa-tions with globally Lipschitz coefficients. Next, using the previous existence result and approximation, we prove the existence and uniqueness of solutions for corre-sponding equations with locally Lipschitz coefficients. We also analyse the Markov property of the solutions. Finally, we introduce a theorem proposed by Brze´zniak et al. [10] about the existence and uniqueness of invariant measures for correspond-ing 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.3to the HJMM equations. In particular, we prove the exis-tence and uniqueness of solutions to the HJMM equations in the weighted Lebesgue and Sobolev spaces, respectively. We also find a sufficient condition for the exis-tence and uniqueness of invariant measures for the Markov semigroup associated to the HJMM equations in the weighted Lebesgue spaces. We mentioned some impor-tant features of our results above. Another imporimpor-tant 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-calledγ-radonifying operators from a Hilbert space to anLpspace and a Sobolev spaceW1,pfound recently by the first author and Peszat in [14,13].

2 Stochastic preliminaries

Definition 2.1 A Banach space(X, _· X)is calledmartingale-type 2(orM-type 2)

if there exists a constantC >0 depending only on X such that for anyX-valued martingale(Mn)n∈N,

sup

n∈N

E_[Mn2X] ≤C

n

E_[Mn−Mn₋12X].

For eachp_≥2, the Lebesgue spaceLpand the Sobolev spaceW1,pare examples of martingale-type 2 Banach spaces.

Remark 2.2 Another type of Banach space that has been frequently used in the re-cent years is the UMD, i.e., having the unconditional martingale difference property, Banach space; see [5,16,46]. Let us recall that a Banach spaceXis a UMD space if for anyp_∈(1,_∞), there exists a constantβp(X) >0 such that for anyX-valued

martingale(Mn)n∈NwithM0=0, for anyε:N→ {−1,1}and for anyn∈N,

E

n

j=1

εj(Mj−Mj)

p

X

≤βp(X)E[Mnp_X].

It is known, see e.g. [51, Proposition 2.4], that if a Banach spaceXis both UMD and of type 2, thenXis 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 martingale-type 2 spaces are more suitable for studying general SPDEs; see e.g. [13,17]. For our purposes, we consider the latter spaces in which allL2-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 as-sumptions on their indices; see e.g. [5,6]. This has important consequences in nu-merical 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 struc-tures (and in rough path theory) in order to apply methods from stochastic integration (extending Hölder-type spaces).

Definition 2.3 Let(,F,F,P), whereF₌(F_t)t≥0, be a filtered probability space.

Assume thatH is a separable Hilbert space endowed with an inner product·,_· H.

A family(W (t ))t≥0of bounded linear operatorsW (t ):H→L2(,F,P)is called

anH-valuedF-cylindrical canonical Wiener processif

(i) for allt_≥0 andh1, h2∈H,E[W (t )h1W (t )h2] =th1, h2 H;

(ii) for everyh_∈H,(W (t )h)t≥0is a real-valuedF-adapted Wiener process.

Definition 2.4 LetXbe a separable Banach space andH a separable Hilbert space. A bounded linear operatorL_:H_→Xis calledγ-radonifyingif the imageL(γH)

byLof the canonical Gaussian measureγH onH extends to a Gaussian probability

The following result is originally due to Neidhardt [44, Sect. 4]; see also [11,38,39,45].

Theorem 2.5 LetXbe a separable Banach space with a norm _· XandHa

sep-arable Hilbert space.Assume thatνLis the Gaussian probability measure onX

gen-erated byL_∈γ (H, X).Then the mapping _· γ (H,X)defined by

Lγ (H,X)= X

x2_XdνL(x)

1₂

, L_∈γ (H, X)

is a norm on the spaceγ (H, X)and γ (H, X)is a separable Banach space with respect to this norm.

Theorem 2.6 [2,Lemma8.4]Assume thatH1, H2are separable Hilbert spaces and

B1, B2 are separable Banach spaces.If h:H1→H2 and b:B2→B1 are linear

bounded operators andk_:H2→B2is aγ-radonifying operator,thenk◦handb◦k

are alsoγ-radonifying operators.

The next result is originally due to [44, Lemma 28]; see also [6, Theorem 2.3].

Theorem 2.7 Let (,F,F,P),whereF₌(F_t)t≥0,be a filtered probability space.

Assume that (X, _· X)is a martingale-type2 Banach space and H is a

separa-ble Hilbert space.Assume that (W (t ))t≥0 is an H-valued F-cylindrical canonical

Wiener process. If ξ is a γ (H, X)-valued F-progressively measurable process on

[0, T_],T >0,such that

E

T

0

ξ(t )2_{γ (H,X)}dt

<_∞, (2.1)

then the stochastic integral ₀Tξ(t ) dW (t )is well defined and there exists a constant

C >0such that

E

T

0

ξ(t ) dW (t )

2 X

≤C

T

0

E_[ξ(t )2_{γ (H,X)]}dt,

i.e., ₀Tξ(t ) dW (t )belongs to the spaceL2(,F,P_;X).

Theorem 2.8 [6,Theorem2.4]Suppose that all the assumptions of Theorem2.7are satisfied.Letξ be aγ (H, X)-valuedF-progressively measurable process on_[0, T_],

T >0,such that(2.1)holds.Then the process(φ (t ))t_∈[0,T_]defined by

φ (t )_:=

t

0

ξ(s) dW (s), t_{∈ [}0, T_],

is a martingale and has a continuous modification;in particular,it isF-progressively measurable.Moreover,there exists a constantC >0 (independent ofξ)such that

E

sup

t∈[0,T] φ (t )2_X

≤C

t

0

Definition 2.9 A Banach space(X, _· X)is said to satisfy theH-condition if for

someq_≥2, the functionψ_:X_→Rdefined by

ψ (x)₌ xq_X, x_∈X,

is of class C2 on X (in the Fréchet derivative sense) and if there exist constants

K1, K2>0 depending onq such that for everyx_∈X,

|ψ′(x)_{| ≤}K1xqX−1 and |ψ′′(x)| ≤K2xqX−2,

whereψ′(x)andψ′′(x)are the first and second Fréchet derivatives ofψatx_∈X.

For eachq_≥p_≥2, the Lebesque spaceLp and the Sobolev spaceW1,p satisfy theH-condition.

Proposition 2.10 [57]If a Banach space(X, _· X)satisfies theH-condition,then

it is also a martingale-type2Banach space.

Theorem 2.11 [14]Let (,F,F,P),whereF₌(F_t)t≥0,be a filtered probability

space.Assume that(X, _· X)is a Banach space satisfying theH-condition.Assume

that H is a separable Hilbert space and (W (t ))t≥0 is anH-valued F-cylindrical

canonical Wiener process. Let (S(t ))t≥0 be a contraction C0-semigroup on X. If ξ is aγ (H, X)-valuedF-progressively measurable process on_[0, T_],T >0, such that(2.1)holds,then there exists a constantK >0depending onH,XandK1, K2

appearing in theH-condition such that

E

sup

t_∈[0,T_]

t

0

S(t₋r)ξ(r) dW (r)

2 X

≤K

T

0

E_[ξ(t )2_{γ (H,X)]}dt. (2.2)

Definition 2.12 Let(X, _· X)be a Banach space. AC0-semigroup(S(t ))t≥0onX

is calledcontraction-typeif there exists a constantβ_∈Rsuch that

S(t ) _≤eβt, t_≥0.

Remark 2.13 If(S(t ))t_≥0is a contraction-typeC0-semigroup on a Banach spaceX,

then the family (T (t ))t≥0 of operators defined by T (t )=e−βtS(t ) is a

contrac-tion C0-semigroup on X. Therefore, the following corollary follows from

Theo-rem2.11.

Corollary 2.14 Let(,F,F,P),whereF₌(F_t)t_≥0,be a filtered probability space.

Assume that(X, _· X)is a Banach space satisfying theH-condition.Assume thatH

is a separable Hilbert space and(W (t ))t≥0is anH-valuedF-cylindrical canonical

holds,then there exists a constantKT >0depending onT >0andKappearing in

(2.2)such that

E

sup

t∈[0,T]

t

0

S(t₋r)ξ(r) dW (r)

2 X

≤KT

T

0

E_[ξ(t )2_{γ (H,X)]}dt.

3 Stochastic evolution equations in Banach spaces

3.1 Assumptions and definitions

Throughout this paper,we assume that

– (,F_{,F,P), whereF=(}F_{t)t≥0, is a filtered probability space.}

– (X, _· X)is a separable Banach space satisfying theH-condition.

– (S(t ))t≥0is a contraction-typeC0-semigroup onXwith infinitesimal generatorA.

– (H,_·,_· H)is a separable Hilbert space.

– (W (t ))t≥0is anH-valuedF-cylindrical canonical Wiener process.

– F andGare maps fromR_+×X intoX andγ (H, X), respectively (we impose some sufficient conditions on them later).

– u0∈L2(,F0,P;X).

We consider the stochastic evolution equation inXgiven by

du(t )₌Au(t )₊F

t, u(t )

dt₊G

t, u(t )

dW (t ), t >0,

u(0)₌u0.

(3.1)

Definition 3.1 AnX-valuedF-progressively measurable process(u(t ))t≥0is called

amild solutionto (3.1) if for everyt_≥0,

t

0

E_[u(s)2_X]ds <_∞

andP-a.s.

u(t )₌S(t )u0+

t

0

S(t₋r)F

r, u(r)

dr

+

t

0

S(t₋r)Gr, u(r)dW (r), t_≥0. (3.2)

3.2 Existence and uniqueness of solutions to Equation (3.1) with globally Lipschitz coefficients

Assume that for each T >0, ZT denotes the space of all X-valued, continuous, F-adapted processesuon[0, T_]such that

uT :=

E

sup

t∈[0,T] u(t )2_X

1₂

<_∞.

Lemma 3.2 Assume thatG_:R_+×X_→γ (H, X)is a globally Lipschitz map onX, i.e.,for eachT >0,there exists a constantLG>0such that for allt∈ [0, T],

G(t, x1)−G(t, x2)γ (H,X)≤LGx1−x2X, x1, x2∈X. (3.3)

Moreover,we assume that for everyx_∈X,the mapG(_·, x)_{: [}0,_∞)_→γ (H, X)is Borel-measurable and for everyT >0,

sup

t_∈[0,T_]

G(t,0)γ (H,X)<∞. (3.4)

Then the mappingIG:ZT →ZT defined by

IG(u)(t )=

t

0

S(t₋r)Gr, u(r)dW (r), u_∈ZT, t∈ [0, T],

is well defined.Moreover,it is of linear growth and globally Lipschitz onZT.

Proof Define

MT := sup t∈[0,T]

S(t ). (3.5)

Fixu_∈ZT. We first show that for eacht≥0, the integral 0tS(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_], isγ (H, X)-valued. SinceG(_·, x)_{: [}0,_∞)_→γ (H, X)is a Borel-measur-able function, the processS(t₋r)G(r, u(r)),r_{∈ [}0, t_], isF-progressively measur-able. It follows from (3.3) and (3.4) that the mapGis of linear growth, i.e., for each

T >0, there exists a constantL˜G>0 such that for allt∈ [0, T],

G(t, x)2_{γ (H,X)≤ ˜}L2_G(1+ x2_X), x_∈X. (3.6)

Therefore, by (3.5) and (3.6), we obtain

E

t

0

S(t₋r)G(r, u(r))2_{γ (H,X)}dr

≤M_t2L˜2_Gt₊M_t2L˜2_GtE

sup

r∈[0,t] u(r)2_X

.

Thus, in view of Theorem2.7, the integral ₀tS(t₋r)G(r, u(r)) dW (r)is well de-fined. Now we show thatIG is well defined. By Theorem2.8, the processIG(u)is

continuous andF-adapted. Furthermore, by Corollary2.14and (3.6), we get

E

sup

t∈[0,T]

IG(u)(t )2X

≤T KTL˜2G+T KTL˜2GE

sup

t∈[0,T] u(t )2_X

.

HenceIG(u)∈ZT and thusIGis well defined. It follows from the last inequality that

which implies thatIGis of linear growth. By Corollary2.14and (3.3), we have

IG(u1)−IG(u2)2T ≤KTE

T

0

G(t, u1(r))−G(t, u2(r))2γ (H,X)dr

≤KTT L2GE

sup

r_∈[0,T_]

u1(r)−u2(r)2X

, u1, u2∈ZT.

Therefore, we infer that

IG(u1)−IG(u2)T ≤LG

KTTu1−u2T, u1, u2∈ZT, (3.7)

which implies thatIGis globally Lipschitz onZT.

Lemma 3.3 Assume thatF _:R_+×X_→Xis a globally Lipschitz map onX,i.e.,for eachT >0,there exists a constantLF >0such that for allt∈ [0, T],

F (t, x1)−F (t, x2)X≤LFx1−x2X, x1, x2∈X. (3.8)

Moreover, we assume that for every x _∈X, the function F (_·, x)_{: [}0,_∞)_→X is Borel-measurable and for everyT >0,

sup

t_∈[0,T_]

F (t,0)X<∞. (3.9)

Then the mapIF :ZT →ZT defined by

IF(u)(t )=

t

0

S(t₋r)Fr, u(r)dr, u_∈ZT, t∈ [0, T],

is well defined.Moreover,it is of linear growth and globally Lipschitz onZT.

Proof Fixu_∈ZT. By (3.8) and (3.9),F is of linear growth, i.e., for eachT >0,

there exists a constantL˜F such that for allt∈ [0, T],

F (t, x)2_{X≤ ˜}L2_F(1+ x2_X), x_∈X. (3.10) It follows from (3.5) and (3.10) that the integral ₀tS(t₋r)F (r, u(r)) dr is well defined P-a.s. for each t _≥ 0. Let us now show that IF is well defined. By

[37, Proposition 2.22], the processIF(u)is continuous and by [37, Corollary 2.7], it

has anF-progressively measurable modification. Moreover, by the Cauchy–Schwarz inequality and (3.5) and (3.10), we obtain

E

sup

t∈[0,T]

IF(u)(t )2X

≤T2M_T2L˜2_{F +}T2M_T2L˜2_FE

sup

t∈[0,T] u(r)2_X

.

ThereforeIF(u)∈ZT and thusIF is well defined. Moreover, it follows from the last

inequality that

IF(u)2T ≤T 2_M2

TL˜ 2

which implies that IF is of linear growth. Finally, we show that IF is globally

Lipschitz onZT. By the Cauchy–Schwarz inequality and (3.5) and (3.8), we infer

that

IF(u1)−IF(u2)2T

≤TE

sup

t_∈[0,T_]

t

0

S(t−r)

F

r, u1(r)−Fr, u2(r)

2 Xdr

≤T M_T2L2_FE

sup

t_∈[0,T_]

t

0

u1(r)−u2(r)2Xdr

≤T2M_T2L2_FE

sup

r_∈[0,T_]

u1(r)−u2(r)2X

, u1, u2∈ZT,

which implies that

IF(u1)−IF(u2)T ≤MTLFTu1−u2T, u1, u2∈ZT. (3.11)

This completes the proof.

Theorem 3.4 Assume that F _: R_{+ ×}X _→ X and G _: R_{+ ×}X _→ γ (H, X)

are globally Lipschitz maps on X. Moreover, we assume that for every x _∈X,

F (_·, x)_{: [}0,_∞)_→X and G(_·, x)_{: [}0,_∞)_→γ (H, X)are Borel-measurable maps and for everyT >0,

sup

t∈[0,T]

F (t,0)X+ G(t,0)γ (H,X)<∞.

Then there exists a uniqueX-valued continuous mild solution to Eq. (3.1).

Proof By the definition of a mild solution, it is sufficient to show that the inte-gral equation (3.2) has a unique X-valued continuous solution. Define the map

_:ZT→ZT by

(u)(t )₌S(t )u0₊

t

0

S(t₋r)Fr, u(r)dr

+

t

0

S(t₋r)Gr, u(r)dW (r), u_∈ZT.

It is obvious that the processS(_·)u0 isF-adapted. Moreover, for every ω∈, the

mapS(_·)u0(ω)_{: [}0,_∞)_→X is continuous andE_[u02_X]<_∞. Thus the process

S(_·)u0belongs toZT. Therefore, by Lemmas3.2and3.3,is well defined and of

linear growth. Moreover, by (3.7) and (3.11), we infer that

(u1)₋(u2)T ≤C(T )u1−u2T, u1, u2∈ZT,

whereC(T )₌MTLFT +LG√KTT. Hence,is globally Lipschitz onZT. If we

theorem, there exists a unique processu1_∈ZT0 such that(u

1_)=u1_{. Therefore, the}

integral equation (3.2) has a uniqueX-valued continuous solutionu1on[0, T0].

DefineZ(k−1)T0,kT0,k=1,2,3, . . ., to be the space of allX-valued, continuous,

F-adapted stochastic processesuon[(k₋1)T0, kT0_]such that

E

sup

t∈[(k−1)T0,kT0]

u(t )2_X

<_∞.

It is obvious that for eachk,Z(k−1)T0,kT0 is a Banach space endowed with the norm

u(k₋1)T0,kT0=

E

sup

t∈[(k−1)T0,kT0]

u(t )2_X

1₂

.

As above, it can be easily shown that the equation

u(t )₌St₋(k₋1)T0u(k₋1)T0₊

t

(k−1)T0

S(t₋r)Fr, u(r)dr

+

t

(k−1)T0

S(t₋r)Gr, u(r)dW (r) (3.12)

has a uniqueX-valued, continuous solutionukon_[(k₋1)T0, kT0_]such that

uk(kT0)=uk+1(kT0).

Consequently, we have a sequence(uk)k_∈N of solutions. Define a process u from

these solutions by

u(t )₌ ∞

k₌1

uk(t )1_[(k₋1)T0,kT0](t ), t∈ [0,∞).

We claim that this process is a uniqueX-valued continuous solution to the integral equation (3.2). It is obvious that the processuis continuous andF-adapted. There-fore,uisF-progressively measurable. Furthermore, for eacht_≥0,

t

0

E_[u(r)2_X]dr <_∞.

Thus, the integrals in (3.2) are well defined for the processu. Let us now show that the processusolves the integral equation (3.2). We have already proved that fork₌1, the processuon[0, T0_]solves (3.2). By induction, we assume that the processuon

[0, kT0_]solves (3.2). Thus, we have

u(kT0)=uk(kT0)=S(kT0)u0+

kT0

0

S(kT0−r)F

r, u(r)dr

+

kT0

0

We need to show that the processuon [0, (k₊1)T0] solves (3.2). Sinceuk+1 on [kT0, (k+1)T0]solves (3.12), we get, for eacht∈ [kT0, (k+1)T0],

uk+1(t )₌S(t₋kT0)uk+1(kT0)₊

t

kT0

S(t₋r)Ft, uk+1(r)dr

+

t

kT0

S(t₋r)Gt, uk+1(r)dW (r).

Sinceuk(kT0)₌uk+1(kT0)and by (3.13), we obtain

u(t )₌S(t )u0+

kT0

0

S(t₋r)F

r, u(r)

dr₊

kT0

0

S(t₋r)G

r, u(r)

dW (r)

+

t

kT0

S(t₋r)Ft, uk+1(r)dr₊

t

kT0

S(t₋r)Gt, uk+1(r)dW (r)

=S(t )u0+

t

0

S(t₋r)Fr, u(r)dr

+

t

0

S(t₋r)G

r, u(r)

dW (r), t_{∈ [}0, (k₊1)T0].

Therefore, the processu on_[0, (k₊1)T0]solves (3.2). Hence, the processu 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. Letu1andu2be two solutions to (3.2). Define the processzby z(t )₌u1(t )−u2(t ), t≥0. Thus

z(t )₌

t

0

S(t₋r)F

r, u1(r)−Fr, u2(r)

dr

+

t

0

S(t₋r)G

r, u1(r)−Gr, u2(r)

dW (r), t_≥0.

By Theorem2.8and (3.3) and (3.5), we have

E t 0

S(t₋r)Gr, u1(r)₋Gr, u2(r)dW (r)

2 X

≤CL2_GM_t2

t

0

E_[u1(r)−u2(r)2X]dr.

Moreover, using the Cauchy–Schwarz inequality and (3.5) and (3.8), we have

E t 0

S(t₋r)F

r, u1(r)−Fr, u2(r)

dr 2 X

≤t L2_FM_t2

t

0

Taking into account the last two estimates, we infer that

E_[z(t )2_{X] ≤}(2t L2_FM_t2₊2CL2_GM_t2)

t

0

E_[z(r)2_X]dr, t_≥0.

Applying Gronwall’s lemma, we infer that for allt_≥0,E_[z(t )2_{X] =}0. Therefore, for allt _≥0, z(t )₌0 and sou1₌u2. Thus, the processu is a unique X-valued continuous solution to the integral equation (3.2). Therefore, the processuis a unique

X-valued continuous solution to (3.1).

Theorem 3.5 Assume that for eachζ _∈L2(,F₀,P_;X),u(_·, ζ )denotes the solution of(3.1)with the initial valueζ.Then for eachT >0,there exists a constantCT >0

such that for allt_{∈ [}0, T_],

E_[u(t, ζ )2_{X] ≤}CT(1+E[ζ2X]), ζ ∈L 2_(,F

0,P;X), (3.14)

and

E_[u(t, ζ )₋u(t, δ)2_{X] ≤}CTE[ζ−δ2X], ζ, δ∈L2(,F0,P;X). (3.15)

Proof FixT >0 andζ_∈L2(,F_s,P_;X). Then for allt_{∈ [}0, T_],

E_[u(t, ζ )2_{X] ≤}3E_[S(t )ζ2_{X] +}3E

t 0

S(t₋r)F

r, u(r, ζ )

dr 2 X

+3E

t 0

S(t₋r)G

r, u(r, ζ )

dW (r) 2 X

, t_{∈ [}0, T_].

Using Theorem2.8and (3.5) and (3.6), we get, for eacht_{∈ [}0, T_],

E t 0

S(t₋r)Gr, u(r, ζ )dW (r)

2 X

≤CTL˜2_GM_T2₊CL˜2_GM_T2

t

0

E_[u(r, ζ )2_X]dr. (3.16)

By the Cauchy–Schwarz inequality and (3.5) and (3.10), we obtain, for each

t_{∈ [}0, T_],

E t 0

S(t₋r)F

r, u(r, ζ )

dr 2 X

≤T2L˜2_FM_T2₊TL˜2_FM_T2

t

0

E_[u(r, ζ )2_X]dr. (3.17)

Moreover, by (3.5), we have

It follows from (3.16)–(3.18) that there exist constantsKT,K˜T >0 such that

E_[u(t, ζ )2_{X] ≤}KT+ ˜KT

t

0

E_[u(r, ζ )2_X]dr, t_{∈ [}0, T_].

Applying Gronwall’s lemma, we obtain the desired result (3.14). Similarly, one can

show (3.15).

3.3 Existence and uniqueness of solutions to Equation (3.1) with locally Lipschitz coefficients

Lemma 3.6 Assume thatF _:R_+×X_→Xis a locally Lipschitz map on balls,i.e., for eachT >0andR >0,there exists a constantLF(R) >0such that ift∈ [0, T]

andx1X,x2X≤R,then

F (t, x1)−F (t, x2)X≤LF(R)x1−x2X.

Moreover,assume thatF is of linear growth(uniformly int),i.e.,for allT >0,there exists a constantL¯F >0such that for allt∈ [0, T],

F (t, x)2_{X≤ ¯}L2_F(1+ x2_X), x_∈X. (3.19) For eachn_∈N,define the mapFn_:R_+×X_→Xby

Fn(t, x)₌

F (t, x), xX≤n, t≥0,

F (t, n_xx_X), xX> n, t≥0.

(3.20)

Then for eachn_∈N,Fnis globally Lipschitz onXwith constant3LF(n)(

indepen-dent oft).Moreover,Fnis of linear growth,i.e.,

Fn(t, x)2_{X≤ ¯}L2_F(1₊ x2_X), x_∈X. (3.21) Proof It follows from [7, Lemma 7] that for eachn_∈N,Fn is globally Lipschitz with constant 3LF(n)(independent oft). Moreover, it follows from (3.19) thatFnis

of linear growth.

Similarly, we have the following result.

Lemma 3.7 Assume that G_:R_+×X_→γ (H, X) is a locally Lipschitz map on balls,i.e.,for eachT >0andR >0,there exists a constantLF(R) >0such that if t_{∈ [}0, T_]andx1X,x2X≤R,then

G(t, x1)₋G(t, x2)γ (H,X)≤LG(R)x1−x2X.

Moreover,assume thatGis of linear growth(uniformly int),i.e.,for allT >0,there exists a constantL¯G>0such that for allt∈ [0, T],

G(t, x)2_{γ (H,X)≤ ¯}L2_G

1+ x2_X

For eachn_∈N,define the mapGn_:R_+×X_→γ (H, X)by

Gn(_·, x)₌

G(_·, x), xX≤n, t≥0, G(_·, n x

x_X), xX> n, t≥0.

(3.22)

Then for eachn_∈N,Gnis globally Lipschitz onXwith constant3LG(n)(

indepen-dent oft).Moreover,Gnis of linear growth,i.e.,

Gn(t, x)2_{γ (H,X)≤ ¯}L2_G(1+ x2_X), x_∈X. (3.23)

The previous lemmas yield the following natural conclusion of Theorem3.4.

Corollary 3.8 Assume the mapsF _:R_+×X_→XandG_:R_+×X_→γ (H, X)are of linear growth and locally Lipschitz on balls.Moreover,we assume that for each

x_∈X,F (_·, x)_{: [}0,_∞)_→XandG(_·, x)_{: [}0,_∞)_→γ (H, X)are Borel-measurable functions.Then for eachn_∈N,the stochastic evolution equation

dun(t )₌

Aun(t )₊Fnt, un(t )

dt₊Gnt, un(t )dW (t ), t_≥0,

un(0)₌u0,

(3.24)

whereFn andGn are the mappings defined in(3.20)and(3.22),respectively,has a uniqueX-valued continuous mild solution.

Lemma 3.9 Assume that for eachn_∈N,un is the unique solution of (3.24).Then for everyT >0,there exists a constantC(T ) >0 (independent ofn)such that for eachn_∈N,

E

sup

t∈[0,T] un(t )2_X

≤C(T ). (3.25)

Proof Fixn_∈NandT >0. Sinceunis the unique solution of (3.24), we have

un(t )₌S(t )u0+

t

0

S(t₋r)Fn

r, un(r)

dr

+

t

0

S(t₋r)Gnr, un(r)dW (r) t_{∈ [}0, T_].

Thus, for eachs_{∈ [}0, T_], we obtain

E

sup

t∈[0,s] un(t )2_X

≤3E

sup

t∈[0,s]

S(t )u02_X

+3E

sup

t∈[0,s]

t 0

S(t₋r)Fnr, un(r)dr

2 X

+3E

sup

t∈[0,s]

t 0

S(t₋r)Gnr, un(r)dW (r)

By Corollary2.14and (3.23), we have, for eachs_{∈ [}0, T_],

E

sup

t∈[0,s]

t 0

S(t₋r)Gnr, un(r)dW (r)

2 X

≤KTE s

0

Gn

r, un(r)

2 γ (H,X)dr

≤KTL¯2GE s

0

(1₊ un(r)2_X) dr

≤KTTL¯2G+KTL¯2G

s

0 E

sup

t∈[0,r] un(t )2_X

dr. (3.26)

Using the Cauchy–Schwarz inequality, (3.5) and (3.21), we obtain

t 0

S(t₋r)Fnr, un(r)dr

2 X ≤t t 0

S(t−r)Fn

r, un(r)

2 Xdr

≤t M_t2L¯2_F

t

0

1+ un(r)2_X

dr

≤t2M_t2L¯_F2 ₊t M_t2L¯2_F

t

0

sup

t∈[0,r]

un(t )2_Xdr.

Therefore, we deduce that

E

sup

t∈[0,s]

t 0

S(t₋r)Fnr, un(r)dr

2 X

≤T2M_T2L¯_F2 ₊T M_T2L¯2_F

s

0 E

sup

t_∈[0,r_] un(t )2_X

dr. (3.27)

Moreover, by (3.5), we get

E

sup

t_∈[0,s_]

S(t )u02X

≤MTu0X2, s∈ [0, T]. (3.28)

In view of (3.26)–(3.28), there exist constantsNT >0 andVT >0 such that

E

sup

t∈[0,s] un(t )2_X

≤NT +VT

s

0 E

sup

t∈[0,r] un(t )2_X

dr, s_{∈ [}0, T_].

Applying Gronwall’s lemma, we get the desired result (3.25).

Lemma 3.10 For eachn_∈N,the random variableτn:→ [0,∞]defined by

τn(ω)=inf{t∈ [0,∞): un(t, ω)X≥n}, ω∈,

Proof It was proved in [35, Problem 1.2.7] that for each n_∈N, τn is a stopping

time. Thus we only prove that the sequence(τn)n∈N converges to infinity. For this

aim, we need to show that there exists ¯ _∈F _with P_{[ ¯] =}1 such that for all

ω_{∈ ¯},τn(ω)→ ∞, i.e., for everyT >0, there existsk∈Nsuch that for alln≥k, τn(ω)≥T. Therefore, it is sufficient to show that for eachT >0,

P_[{ω_{∈: ∃}k_∈Nwithτn(ω)≥T ,∀n≥k}] =1.

FixT >0. For eachn_∈N, set

An:=

ω_∈: sup

t_∈[0,T_]

un(t )X≥n

.

Then by (3.25) and the Chebyshev inequality, we have, for eachn_∈N,

P_[An] ≤C(T )

1

n2.

Therefore, since∞

n₌1_n12 <∞, we have ∞

n₌1P[An]<∞. Thus, by the Borel–

Cantelli lemma, we infer thatP_[∞

k=1

∞

n=kAn] =0 and so

P

∞

k=1 ∞

n=k

(_\An)

=1.

Choose ¯ ₌∞

k=1

∞

n=k(\An) and fix ω∈ ¯. Then there is k∈N such that ω_∈∞

n=k(\An), i.e.,ω∈\An,∀n≥k. Therefore, for alln≥k,

sup

t_∈[0,T_]

un(t, ω)X< n.

Thus there isk_∈Nsuch that for alln_≥k andt_{∈ [}0, T_],un(t, ω)X< n, which

implies that there isk_∈Nsuch that for alln_≥k,τn(ω)≥T. This gives the desired

conclusion.

Theorem 3.11 Assume the mapsF_:R_+×X_→XandG_:R_+×X_→γ (H, X)are of linear growth and locally Lipschitz on balls.Moreover,we assume that for each

x_∈X,F (_·, x)_{: [}0,_∞)_→XandG(_·, x)_{: [}0,_∞)_→γ (H, X)are Borel-measurable functions.Then there exists a uniqueX-valued continuous mild solution to(3.1).

Proof For eachn_∈N, letunbe the unique solution of (3.24). Consider the sequence

(τn)n∈Nin Lemma3.10. Define the processuby

u(t )₌un(t ), ift_≤τn.

In view of [6, Lemma 4.11], (τn)n∈N has the two properties that (i)τn≤τn+1and

(ii)un(t, ω)₌un+1(t, ω), t_≤τn(ω),P-a.s. Therefore, the processuis well defined.

Let us prove this. By the definition of a mild solution, it is sufficient to show that the processuis a uniqueX-valued continuous solution to the integral equation (3.2). It is obvious thatuis continuous andF-adapted. Hence, it isF- progressively measurable. Moreover, for eacht_≥0,

t

0

E_[u(r)2_X]dr <_∞.

Thus, the integrals in (3.2) are well defined for the processu. Now we show that the processu solves (3.2). Sinceun is the solution of (3.24), we have the integral equation

u(t_∧τn)=un(t∧τn)

=S(t_∧τn)u0+

t∧τn 0

S(t_∧τn−r)Fnr, un(r)dr

+

t∧τn 0

S(t_∧τn−r)Gnr, un(r)dW (r), t≥0. (3.29)

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

I (t )_:=

t

0

S(t₋r)Gnr, un(r)dW (r), t_≥0.

It is obvious thatI is well defined. Consider the process

Iτn(t )=

t

0

S(t₋r)1_[0,τn)(r)G n

r_∧τn, un(r∧τn)dW (r), t≥0.

It was shown in [11, Lemma A.1] that if the processes I and Iτn are

continu-ous, then

S(t₋t₋τn)I (t∧τn)=Iτn(t ) for allt≥0,P-a.s.

In particular,

I (t_∧τn)=Iτn(t∧τn) for allt≥0,P-a.s.

Therefore, (3.29) can be rewritten as

u(t_∧τn)=S(t∧τn)u0

+

t∧τn 0

Sincer_≤τn,un(r)X≤n. Therefore, from the definitions ofFn andGn, we get,

for everyr_≤τn,

Fn

r, un(r)

=F

r, un(r)

and Gn

r, un(r)

=G

r, un(r)

.

Also by the definition ofu,u(r)₌un(r)ifr_≤τn. Hence, we obtain

u(t_∧τn)=S(t∧τn)u0+

t∧τn 0

S(t_∧τn−r)Fr, u(r)dr

+

t

0

S(t₋r)1_[0,τn)(r)G

r_∧τn, u(r∧τn)dW (r).

We know thatτn→ ∞and sot∧τn→t. Thusu(t∧τn)→u(t )andS(t∧τn)→S(t ).

Therefore, we deduce that

u(t )₌S(t )u0+

t

0

S(t₋r)F

r, u(r)

dr

+

t

0

S(t₋r)G

r, u(r)

dW (r), t_≥0.

Thus, the processusolves the integral equation (3.2). The uniqueness of the solution follows from Theorem3.4. Thus, the processuis a uniqueX-valued continuous mild

solution to (3.1).

3.4 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 fol-lows from the results proved in the previous sections that (3.1) has a uniqueX-valued continuous mild solution under the Lipschitz assumptions on the coefficients. Sim-ilarly, one can show that for everyu0∈L2(,F0,P;X), Eq. (3.1) with the initial

valueu(0)₌u0has a uniqueX-valued continuous (mild) solutionuon[0,_∞). We denote this solution byu(_·, u0). If u0₌x_∈X, we denote the solution byu(_·, x). Assume thatBb(X)denotes the space of all bounded measurable functions fromX

intoRandB(X)the Borelσ-field ofX.

Definition 3.12 The family(Pt)t≥0of linear bounded operators onBb(X)defined

by

(Ptϕ)(x):=Eϕu(t, x), t≥0, ϕ∈Bb(X), x∈X,

see [50, Definition 1.7], is called theMarkov semigroupcorresponding to Eq. (3.1).

Remark 3.13 As discussed in [9, Sect. 2.4], for everyφ_∈Bb(X), the bounded

func-tion Ptφ is also measurable when it is well posed, i.e., when weak existence and

latter generalizes to the infinite-dimensional setting the finite-dimensional result of Stroock and Varadhan [52, Exercise 6.7.4]. Since our results imply weak existence and pathwise uniqueness and since pathwise uniqueness implies uniqueness in law, see [47] for the infinite-dimensional version of the Yamada–Watanabe theory, we in-fer that this is the case for our problem. Moreover, one can show, see e.g. Ondreját [47], that the family(Pt)t≥0is indeed a Markov semigroup, i.e., we have

Pt+s=PtPs, t, s≥0.

Definition 3.14 A Borel probability measureμ onX is called aninvariant prob-ability measurefor (3.30) if it is an invariant probability measure for the Markov semigroup, i.e., if for any timet_≥0,

X

Ptφ dμ=

X

φ dμ, _∀φ_∈Cb(X),

whereCb(X)denotes the space of all bounded and continuous functions fromXtoR.

Remark 3.15 By Theorem3.5, the transition semigroup(Pt)t_≥0is Feller, that is, for

anyt_≥0,Pt:Cb(X)→Cb(X), whereCb(X)is the space of all continuous bounded

functions fromXintoR, is well defined. It is known that the semigroup is in general not strongly continuous on the spaceCb(X). However, ifμis an invariant probability

measure for the semigroup(Pt)t_≥0, then for everyp∈ [1,∞), the semigroup(Pt)t_≥0

has a unique extension to aC0-semigroup on the space Lp(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 inLp(X, μ)is dense inLp(X, μ); (ii) for every bounded functionf _∈Lp(X, μ)and for everyε >0, there exists a continuous bounded function fε:X→Rsuch that μ(_{x_∈X_:f (x)₌fε(x)}) < ε, see [4, Theorem 7.1.13]; (iii) the trajectories of the

processu(t ),t_≥0, are continuous. Indeed, properties (i) and (ii) imply that the set

Cb(X)is dense in Lp(X, μ). See also Da Prato’s survey article [19] and Yosida’s

monograph [56, Theorem XIII.1].

Brze´zniak et al. [10] found some sufficient condition for the existence and unique-ness of an invariant measure for (3.1) with time-independent coefficients. Before in-troducing this, we present the following natural consequence of Theorem3.11.

Corollary 3.16 Assume that the mapsF _:X_→X and G_:X_→γ (H, X)are of linear growth and locally Lipschitz on balls.Then there exists a uniqueX-valued continuous mild solution to the equation

du(t )₌Au(t )₊F

u(t )

dt₊G

u(t )

dW (t ), t_≥0,

u(0)₌u0.

(3.30)

x1, x2∈X,

[An(x1−x2)+F (x1)−F (x2), x1−x2]X+ K2(q)

q G(x1)−G(x2) 2 γ (H,X)

≤ −ωx1−x22X,

where An is the Yosida approximation of A, K2 is a constant appearing in the H-condition and_[·,_·]Xis the semi-inner product onX (see Definition3.18below),

then there exists a unique invariant probability measure for(3.30).

Definition 3.18 Asemi-inner producton a complex or real vector spaceV is a map-ping_[·,_·]V :V×V →C(orR) such that

(i)[x₊y, z_]V = [x, z]V + [y, z]V, x, y, z∈V;

(ii)_[λx, y_]V =λ[x, y]V, x, y∈V,λ∈C(orR);

(iii)[x, x_]V >0 forx=0;

(iv)_|[x, y_]V|2≤ [x, x]V[y, y]V, x, y∈V.

Such a vector spaceV with the semi-inner product[·,_·]V is called asemi-inner

prod-uct space.

Lemma 3.19 [10]The mapping onX_×Xdefined by

[x, y_]X= x, y∗ , x, y∈X,

wherey∗_∈X∗(X∗is the dual space ofX)is such that we havey∗ ₌ yXand y, y∗ ₌ yX,is a semi-inner product.Such ay∗∈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 time-dependent equations; see e.g. [48, 20,21]. We plan to investigate this concept in relation to the HJMM equation in the future.

4 Application to the Heath–Jarrow–Morton–Musiela (HJMM)

equations

4.1 The HJMM equations

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 eachT _≥0 andt _{∈ [}0, T_],P (t, T ) is an

R-valued random variable defined on a probability space(,F_{,P). Therefore, for}

eachT >0, (P (t, T ))t_∈[0,T_]is anR-valued stochastic process. Under the

assump-tion that for eacht_{∈ [}0, T_],[0,_∞)_∋T _→P (t, T )is a differentiable function, the functionf defined by

f (t, T )_{= −} ∂

∂T logP (t, T ), T >0, t∈ [0, T],

is called the forward rate function. It contains all the original bond price information. For eachT >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 additionP (T , T )₌1, then one can write the equality

P (t, T )₌e− tTf (t,s) ds_{, T >0, t}∈ [_{0, T}]_.

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 fort∈ [0, T]the stochastic

differential equation

df (t, T )₌

d

i

σi(t, T )

T

t

σi(t, u) du

dt₊ d

i

σi(t, T ) dWi(t ), (4.1)

whereW (t )₌(W1(t ), W2(t ), . . . , Wd(t )),t≥0, is a standardd-dimensional

Brown-ian motion and for eachT >0,(σi(t, T ))t∈[0,T]is a real-valued stochastic process.

Filipovi´c [26] extended the framework of Heath–Jarrow–Morton by considering a Wiener process in a (possibly infinite-dimensional) Hilbert space instead of a stan-dard finite-dimensional Brownian motion. Thus, he assumed that for an arbitrary but fixedT >0, the forward rate process(f (t, T ))t∈[0,T]satisfies the stochastic

differ-ential equation

df (t, T )₌

σ (t, T ),

T

t

σ (t, u) du

H

dt₊ σ (t, T ), dW (t ) H, t∈ [0, 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 fol-lows. Define

wherex is called time to maturity, and for eacht _≥0, the function r(t )is a ran-dom 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≥0of 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≥0satisfies the stochastic partial differential

equa-tion

dr(t )(x)₌

_∂

∂xr(t )(x)+

α(t )(x),

x

0

α(t )(y) dy

H

dt

+ α(t )(x), dW (t )H, t≥0, (4.2)

whereαis the function defined by

α(t )(x)₌σ (t, t₊x), t_≥0, x_{∈ [}0,_∞).

Letg_{: [}0,_∞)_{× [}0,_∞)_×R_→H be a given function which is locally integrable with respect to the second variable. Assume that the volatilityαis defined by

α(t )(x)₌gt, x, r(t )(x), t, x_≥0,

i.e., the volatilityαdepends on the forward curve process(r(t ))t≥0. Then (4.2)

be-comes

dr(t )(x)₌

_∂

∂xr(t )(x)+

gt, x, r(t )(x),

x

0

gt, y, r(t )(y)dy

H

dt

+

gt, x, r(t )(x), dW (t )_H. (4.3)

This equation is known as the Heath–Jarrow–Morton–Musiela (HJMM) equation. In this paper, we analyse, under some sufficient conditions on the functiong, the exis-tence and uniqueness of solutions to (4.3) in certain Banach spaces. Moreover, we analyse the existence and uniqueness of invariant measures for (4.3).

4.2 Existence and uniqueness of solutions to the HJMM equations in weighted Lebesgue spaces

some sense is much closer to the spaceCof continuous functions. Also using either weightedLp or fractional Sobolev spacesWθ,p allows one to consider the HJMM equations with coefficients satisfying less stringent regularity assumptions than in the classical approach. Thirdly, if the functionsσi(t,·)are not regular enough, then the

natural state spaces for the solutions of the HJMM equations are the weightedLp

spaces; see Example4.10. Let us point out that in their monograph [50], Peszat and Zabczyk used the weightedL2spaces.

For eachν_∈Randp_≥1, defineLpν to be the space of all (equivalence classes

of) Lebesgue-measurable functionsf _{: [}0,_∞)_→Rsuch that

fν,p:=

∞

0 |

f (x)_|peνxdx

_p1

<_∞.

For eachν_∈Randp_≥1,Lpν is a separable Banach space with the norm · ν,p.

Lemma 4.1 For eachν >0andp_≥1,the spaceLpν is continuously embedded into

the spaceL1.In particular,

f1≤

_p

νq

_q1

fν,p, f ∈Lpν,

whereq_{∈ [}1,_∞)is such that_p1₊1_q=1.

The proof follows from the Hölder inequality.

Theorem 4.2 Letν >0andp_≥2.Assume thatg_{: [}0,_∞)_{× [}0,_∞)_×R_→H is a measurable function with respect to the second variable such that there exist functions

¯

g_∈Lpν andgˆ∈Lνp∩L∞such that for allt∈ [0,∞),

g(t, x, u)H≤ | ¯g(x)|, u∈R, x∈ [0,∞), (4.4)

and

g(t, x, u)₋g(t, x, v)H≤ | ˆg(x)||u−v|, u, v∈R, x∈ [0,∞). (4.5)

Then for each r0 ∈ L2(,F0,P;Lpν), (4.3) with the function g has a unique Lpν-valued continuous mild solution with the initial valuer(0)=r0.

The proof is given below.

Lemma 4.3 For eachν_∈Randp_≥2,the spaceLpν satisfies theH-condition.In

particular,ifψis the function defined by

then

ψ′(f ) _≤pfν,pp−1 and ψ′′(f ) ≤p(p−1)fν,pp−2, f ∈Lpν,

whereψ′(f )andψ′′(f )are the first and second Fréchet derivatives ofψatf _∈Lpν,

respectively.

Proof Fixν_∈Randp_≥2. Define the linear operatorT _:Lpν →Lpby

Tf (x)₌f (x)epνx_{, f ∈L}p

ν, x∈ [0,∞).

It is obvious that the mapT is well defined, bijective and

Tfp= fν,p, f ∈Lpν. (4.6)

It is clear that any linear operator, soT, is twice Fréchet differentiable and for each

f _∈Lpν,T′(f )=T andT′′(f )=0. By [13, Proposition 2.1], for everyp≥2, the

spaceLpsatisfies theH-condition for anyq_≥p, i.e., for someq_≥p(in particular,

q₌p), the mapφ_:Lp_∋f_→φ (f )₌ fpp∈Ris of classC2and

φ′(f ) _≤pfpp−1 and φ′′(f ) ≤p(p−1)fpp−2, f ∈Lp. (4.7)

We can writeψ₌φ_◦T. SinceT andφare of classC2,ψis of classC2. Moreover, it follows from (4.6) and (4.7) that

ψ′(f ) _≤pfν,pp−1, f ∈Lpν.

Similarly,

ψ′′(f ) _≤p(p₋1)fpν,p−2, f ∈Lpν,

which finishes the proof.

Lemma 4.4 For each ν >0and p_≥1,the shift semigroup(S(t ))t≥0on the space Lpν is a contraction-typeC0-semigroup,in particular,

S(t ) _≤e−pνt_{, t≥0.}

Moreover,its infinitesimal generatorAis characterized by

D_{(A)= {f ∈L}p

ν :Df∈Lpν}

and

Af ₌Df, f_∈D(A),

This result can be proved along the lines of [56, Examples IX.2.1 and IX.5.1]. See also [37, Lemma 5.3] for the first and [37, Lemma 5.8] for the second part.

For eachν_∈Randp_≥1, defineLpν(H )to be the space of all (equivalence classes

of) Borel-measurable functionsf _{: [}0,_∞)_→H such that

f_Lp ν(H ):=

_∞

0

f (x)p_Heνxdx

_p1

<_∞.

For eachν_∈Randp_≥1,Lpν(H )is a separable Banach space endowed with the

norm · Lpν(H ). The following lemma gives a sufficient condition under which an Lpν-valued operator defined onHisγ-radonifying.

Lemma 4.5 For each ν_∈Randp_≥2,the bounded linear operatorK_:H_→Lpν

defined by

K_[h_](x)₌ κ(x), h H, h∈H, x∈ [0,∞),

whereκ_∈Lpν(H ),is aγ-radonifying operator,i.e.,K∈γ (H, Lpν).Moreover,there

exists a constantN >0independent ofκ such that

K_{γ (H,L}p

ν)≤NκLpν(H ).

Proof Define the linear operatorV_:Lpν(H )→Lp(H )by

Vf ₌f eνp_{, f ∈L}p ν(H ).

It is obvious thatV is well defined, bijective and

VfLp_{(H )}= f

Lpν(H ), f ∈L p

ν(H ). (4.8)

Fixκ_∈Lpν(H )so thatV κ ∈Lp(H ). By [12, Proposition 2.1], the bounded linear

operatorM_:H_→Lpdefined by

M_[h_](x)₌ φ (x), hH, h∈H, x∈ [0,∞),

whereφ_∈Lp_{(H ), is aγ-radonifying operator and, for a constantN >0 independent}

ofφ,

Mγ (H,Lp₎≤Nφ_Lp_{(H )}.

Therefore, the operatorK¯ _:H_→Lpdefined by

¯

K_[h_](x)₌ V κ(x), hH, h∈H, x∈ [0,∞),

is aγ-radonifying operator, and for a constantN >0,

It follows from (4.8) that

¯Kγ (H,Lp₎≤Nκ

Lpν(H ). (4.9)

For the linear operatorT 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

K_{γ (H,L}p ν)≤ T

−1

¯Kγ (H,Lp₎≤Nκ Lpν(H ),

which completes the proof.

Lemma 4.6 Letν >0andp_≥2.Assumeg_{: [}0,_∞)_×[0,_∞)_×R_→His a function satisfying the assumptions of Theorem4.2.ThenF _{: [}0,_∞)_×Lpν →Lpν defined by

F (t, f )(x)₌

g

t, x, f (x)

,

x

0 g

t, y, f (y)

dy

H

, f _∈Lp_ν, x, t_{∈ [}0,_∞),

is well defined.Moreover,we have: (i)For allt_{∈ [}0,_∞)andf _∈Lpν,

F (t, f )ν,p≤

_p

νq

1_q

¯g2_ν,p. (4.10)

(ii)F is globally Lipschitz onLpν with a Lipschitz constant independent of timet.

Proof Fix t_≥0 and f _∈Lpν. It follows from (4.4) that for each x ∈ [0,∞), the

integral ₀xg(t, y, f (y)) dyexists. Define the functionhby

h(x)₌

x

0 g

t, y, f (y)

dy, x_{∈ [}0,_∞).

This function is continuous. Indeed, by (4.4), we have for every sequence(xn)n_∈Nin

[0,_∞)converging tox_{∈ [}0,_∞)that asn_{→ ∞},

|h(xn)−h(x)| ≤

xn

x

g

t, y, f (y)dy≤ ∞

0

1_[x,x_n](y)| ¯g(y)|dy−→0.

Thushis continuous and so it is measurable. Therefore,F (t, f )is measurable. More-over, by the Cauchy–Schwarz inequality, Lemma4.1and (4.4), we obtain

|F (t, f )(x)_{| ≤} g(t, x, f (x)H

x

0

g(t, y, f (y)) dy

H

≤ | ¯g(x)_|

∞

0 | ¯

g(y)_|dy_≤

_p

νq

1_q

Taking into account the last inequality, we deduce that

∞

0 |

F (t, f )(x)_|peνxdx_≤

_p

νq

p_q

¯g2ν,pp. (4.11)

ThereforeF (t, f )_∈Lpν and thusFis well defined. Moreover, (4.11) gives the desired

conclusion (4.10). Finally, we prove thatF is globally Lipschitz onLpν. Fixt≥0 and f1, f2∈Lpν. Then by the Cauchy–Schwarz inequality, we have

|F (t, f1)(x)−F (t, f2)(x)|

≤

gt, x, f1(x)

,

x

0

gt, y, f1(y)

−gt, y, f2(y)

dy H +

gt, x, f1(x)₋gt, x, f2(x),

x

0

gt, y, f2(y)dy

H ≤ g

t, x, f1(x)_H ∞

0

g

t, x, f1(x)−gt, x, f2(x)_Hdx

+g

t, x, f1(x)₋gt, x, f2(x)

H ∞ 0 g

t, x, f2(x)

Hdx, x∈ [0,∞).

It follows from Lemma4.1and (4.4) and (4.5) that fort_≥0, x_{∈ [}0,_∞),

|F (t, f1)(x)−F (t, f2)(x)| ≤

_p

νq

_q1

ˆg_∞f1−f2ν,p| ¯g(x)|

+

_p

νq

_q1

¯gν,p| ˆg(x)| |f1(x)−f2(x)|. (4.12)

Taking into account the last inequality, we infer that

F (t, f1)−F (t, f2)ν,p≤2

_p

νq

_q1

ˆg_{∞ ¯}gν,pf1−f2ν,p,

concluding thatF is globally Lipschitz onLpν with a Lipschitz constant independent

of timet.

Lemma 4.7 Letν >0andp_≥2.Assumeg_{: [}0,_∞)_{× [}0,_∞)_×R_→H is a func-tion satisfying the assumpfunc-tions of Theorem4.2.ThenG_{: [}0,_∞)_×Lpν →γ (H, Lpν)

defined by

G(t, f )_[h_](x)₌gt, x, f (x), h_H, f _∈Lp_ν, h_∈H, x, t_{∈ [}0,_∞),

(i)For allt_{∈ [}0,_∞)andf _∈Lpν,there exists a constantN >0such that

G(t, f )_{γ (H,L}p

ν)≤ N ¯gν,p. (4.13)

(ii)Gis globally Lipschitz onLpν with a Lipschitz constant independent of timet.

Proof Fixt_≥0 andf _∈Lpν. Define the functionκ: [0,∞)→H by

κ(x)₌g

t, x, f (x)

, x_{∈ [}0,_∞).

ThenG(t, f )can be written as

G(t, f )_[h_](x)₌ κ(x), h H, h∈H, x∈ [0,∞).

It follows from (4.4) that κ _∈Lpν(H ). Therefore, by Lemma 4.5, G(t, f ) is a γ-radonifying operator fromH toLpν and so G is well defined. Moreover, again

by Lemma4.5and (4.4), we have for a constantN >0 independent ofκthat

G(t, f )_{γ (H,L}p

ν)≤N ¯gν,p,

which gives the desired result (4.13). Finally, using Lemma4.5and (4.5), we have for eacht_≥0 andf1, f2_∈Lpν that

G(t, f1)−G(t, f2)γ (H,Lpν)≤N ˆg∞f1−f2ν,p, (4.14)

which implies thatG(t,_·)is globally Lipschitz onLpν with a Lipschitz constant

inde-pendent of timet.

Proof of Theorem4.2Fixν >0 andp_≥2. The abstract form of (4.3) in the space

Lpν is

dr(t )₌Ar(t )₊Ft, r(t )dt₊Gt, r(t )dW (t ), t_≥0, (4.15)

whereAis the infinitesimal generator of the shift semigroup onLpν (see Lemma4.4),

and F and G are the functions defined in Lemmas 4.6 and 4.7, respectively. Now (4.3) is of the same form as (3.1) and it follows from the previous lem-mas that all the assumptions of Theorem 3.4 hold. Therefore, in view of Theo-rem3.4, (4.15) has a uniqueLpν-valued continuous mild solution with the initial value r0∈L2(,F0,P;Lpν).

Example 4.8 Let us consider the functiongdefined by

α(t, x)₌g(x)₌σ0e−λx, t_≥0, x_{∈ [}0,_∞),

whereσ0, λ are constants and λ >0. This corresponds to the volatility σ in (4.1)

having the form