## Abstract

We propose an alternative approach on the existence of affine realizations for Heath, Jarrow and Morton interest rate models. It is applicable to a wide class of models, and simultaneously it is conceptually rather comprehensible. We also supplement some known existence results for particular volatility structures and provide further insights into the geometry of term structure models.

## 1. Introduction

A zero coupon bond with maturity *T* is a financial asset, which pays the holder one unit of cash at *T*. Its price at *t*≤*T* can be written as the continuous discounting of one unit of cash
where *f*(*t*,*T*) is the rate prevailing at time *t* for instantaneous borrowing at time *T*, also called the forward rate for date *T*. The classical continuous framework for the evolution of the forward rates goes back to Heath, Jarrow and Morton (HJM; Heath *et al.* 1992). They assume that, for every date *T*, the forward rates *f*(*t*,*T*) follow an Itô process of the form
where *W* is a Wiener process. Note that such an HJM interest rate model is an infinite-dimensional object, because for every date of maturity *T*≥0, we have an Itô process.

There are several reasons why, in practice, we are interested in the existence of a finite-dimensional realization, that is, the forward rate evolution can be described by a finite-dimensional state process. Such a finite-dimensional realization ensures larger analytical tractability of the model, for example, in view of option pricing. Moreover, as argued in Baudoin & Teichmann (2005), HJM models without a finite-dimensional realization do not seem reasonable, because then the support of the forward rate curves *f*(*t*,*t*+⋅), *t*>0 becomes too large, and hence any ‘shape’ of forward rate curves, which we assume from the beginning to model the market phenomena, is destroyed with positive probability.

The problem concerning the existence and construction of finite-dimensional realizations for HJM interest rate models has been studied, for various special cases, in Jeffrey (1995), Ritchken & Sankarasubramanian (1995), Duffie & Kan (1996), Bhar & Chiarella (1997), Inui & Kijima (1998), Björk & Christensen (1999), Björk & Gombani (1999), Chiarella & Kwon (2001, 2003), and has finally completely been solved in (Björk & Svensson 2001; Björk & Landén 2002; Filipović & Teichmann 2003); see also Björk (2003) for a survey.

The main idea is to switch to the Musiela parametrization of forward curves *r*_{t}(*x*)=*f*(*t*,*t*+*x*) (Musiela 1993), and to consider the forward rates as the solution of a stochastic partial differential equation (SPDE), the so-called HJMM (Heath–Jarrow–Morton–Musiela) equation
1.1
on a suitable Hilbert space *H* of forward curves, where (*d*/*dx*) denotes the differential operator, which is generated by the strongly continuous semigroup (*S*_{t})_{t≥0} of shifts.

The bank account *B* is the riskless asset, which starts with one unit of cash and grows continuously at time *t* with a short rate *r*_{t}(0), i.e.
According to Delbaen & Schachermayer (1994), the implied bond market, which we can now express as
1.2
is free of arbitrage if there exists an equivalent (local) martingale measure , such that the discounted bond prices
are local -martingales for all maturities *T*. If we formulate the HJMM equation (1.1) with respect to such an equivalent martingale measure , then the drift is determined by the volatility, i.e. in equation (1.1) is given by the HJM drift condition (Heath *et al.* 1992)
1.3
Now, we can consider the problem from a geometric point of view, and the existence of a finite-dimensional realization just means the existence of an invariant manifold, i.e. a finite-dimensional submanifold, which the forward rate process never leaves. Applying the Frobenius Theorem, we obtain the following necessary and sufficient condition for the existence of an invariant manifold, namely
i.e. the so-called Lie algebra generated by the vector fields
and *h*↦*σ*(*h*) must be locally of finite dimension. These are the essential ideas of the mentioned articles (Björk & Svensson 2001; Björk & Landén 2002; Filipović & Teichmann 2003).

The technical problem with this approach is that the differential operator (*d*/*dx*) is, in general, an unbounded and, therefore, non-smooth operator. Björk and co-workers (Björk & Svensson 2001; Björk & Landén 2002) choose the state space *H* to be very small such that (*d*/*dx*) becomes bounded. As a consequence, not all forward curves of basic HJM models belong to this space, as for example the forward curves implied by a Cox–Ingersoll–Ross model, see (Filipović & Teichmann 2003, §1).

Filipović & Teichmann (2003) solved this problem by using convenient analysis on Fréchet spaces, developed in Kriegl & Michor (1997), which, however, is far from being trivial to carry out. In their paper, they in particular show that any HJM model with a finite-dimensional realization necessarily has an affine term structure.

The contribution of the present paper is to propose an alternative approach, which is characterized by the following two major features:

— We work on the Hilbert space

*H*from Filipović (2001, §5), which is large enough to capture any reasonable forward curve. As already mentioned, Björk and co-workers (Björk & Svensson 2001; Björk & Landén 2002) choose the space*H*such that the differential operator (*d*/*dx*) is bounded, hence it is rather small.— Simultaneously, this article does not require knowledge about convenient analysis on Fréchet spaces. This rather technical machinery is used in Filipović & Teichmann (2003). We avoid this framework by directly focusing on affine realizations, which, owing to the mentioned result from Filipović & Teichmann (2003), does not mean a restriction. This makes our approach rather comprehensible.

Summing up, we present an alternative approach to the existence of affine realizations for HJM models, which is applicable to a wide class of models, and which is conceptually accessible to a wide readership.

Our approach also allows us to supplement some existence results for particular volatility structures from Björk & Svensson (2001) (see our comments in remarks 6.6, 7.4) and to provide further insights into the geometry of term structure models (see our comments in remarks 6.3, 6.5, 8.2).

Before we finish this introduction with overviewing the rest of the paper, let us briefly mention another geometric approach for modelling zero coupon bonds, which is conceptually completely different from the present HJM framework, but also leads to an invariance problem. It was suggested by Brody & Hughston (2001) and is inspired by methods from information geometry. They define the bond prices as
where every *ρ*_{t} is a density on . In order to introduce the densities, the authors Brody & Hughston (2001) consider a process *ξ* on the state space , which—by construction—stays in the positive orthant of the unit sphere in the Hilbert space *H*. This implies that is indeed a density.

The rest of this paper is organized as follows. In §2, we provide results on invariant foliations and in §3 on affine realizations for general SPDEs. In §4, we introduce the space of forward curves. Working on this space, we present a result concerning invariant foliations for the HJMM equation (1.1) in §5. Using this result, we study the existence of affine realizations for the HJMM equation (1.1) with general volatility in §6, and for various particular volatility structures in §§7 to 9. Finally, §10 concludes the paper.

## 2. Invariant foliations for general SPDEs

In this section, we provide results on invariant foliations for general SPDEs, which we will apply to the HJMM equation (1.1) later on.

From now on, let be a filtered probability space satisfying the usual conditions and let *W* be a real-valued Wiener process.

Here, we shall deal with SPDEs of the type
2.1
on a separable Hilbert space (*H*,∥⋅∥,〈⋅,⋅〉). In equation (2.1), the operator is the infinitesimal generator of a *C*_{0}-semigroup (*S*_{t})_{t≥0} on *H* with adjoint operator . Recall that the domains and are dense in *H*, e.g. (Rudin 1991, theorems 13.35.c and 13.12).

Concerning the vector fields , we impose the following conditions.

### Assumption 2.1.

*We assume that α,σ∈C*^{1}*(H) and that there is a constant L>0 such that*
2.2
*and*
2.3
*for all h*_{1}*,h*_{2}*∈H.*

The Lipschitz assumptions (2.2) and (2.3) ensure that for each *h*_{0}∈*H* there exists a unique weak solution for equation (2.1) with *r*_{0}=*h*_{0}, (Da Prato & Zabczyk 1992, theorems 6.5 and 7.4).

### Definition 2.2.

A subset is called invariant for equation (2.1) if for every *h*∈*U* we have
where (*r*_{t})_{t≥0} denotes the weak solution for equation (2.1) with *r*_{0}=*h*.

In what follows, let be a finite-dimensional linear subspace and .

### Definition 2.3.

A family of affine subspaces , *t*≥0 is called a foliation generated by *V* if there exists such that
2.4
The map *ψ* is a parametrization of the foliation .

### Remark 2.4.

Note that the parametrization of a foliation generated by *V* is not unique. However, owing to condition (2.4), for two parametrizations *ψ*_{1}, *ψ*_{2} we have

In what follows, let be a foliation generated by *V* . For every *t*≥0, the set consists of exactly one point. Therefore, the map
is well-defined, and it is the unique parametrization of the foliation such that *ψ*(*t*)∈*V* ^{⊥} for all *t*≥0.

### Definition 2.5.

For each *t*≥0, we define the *tangent space*

By remark 2.4, the definition of the tangent is independent of the choice of the parametrization.

### Definition 2.6.

The foliation of submanifolds is invariant for equation (2.1) if for every and we have
2.5
where (*r*_{t})_{t≥0} denotes the weak solution for equation (2.1) with *r*_{0}=*h*.

As we shall see now, an invariant foliation generated by *V* , provided it exists, is unique.

### Lemma 2.7.

*Let* , * i=1,2 be two foliations generated by V with* ,

*which are invariant for equation (2.1). Then we have*

*for all*

*t*≥0.

### Proof.

Choose and let (*r*_{t})_{t≥0} be the weak solution for equation (2.1) with *r*_{0}=*h*_{0}. Then we have
which completes the proof. ■

### Proposition 2.8.

*Suppose the foliation of submanifolds is invariant for equation (2.1) and let be a continuous linear operator with . Then, for every and we have almost surely*
2.6
*where ( r_{t})_{t≥0} denotes the weak solution for equation (2.1) with r_{0}=h, and equation (2.6) is the decomposition of (r_{t})_{t≥0} according to V ^{⊥}⊕V* .

### Proof.

By condition (2.5) we obtain almost surely 2.7 Therefore, we obtain almost surely 2.8 Inserting equation (2.8) into equation (2.7), we arrive at equation (2.6). ■

### Remark 2.9.

If the foliation is invariant for equation (2.1), then for every continuous linear operator , with the decomposition (2.6) provides a realization of the solution (*r*_{t})_{t≥0} by means of the finite-dimensional process, ℓ(*r*).

We shall now approach our main result of this section, theorem 2.11 below, which provides consistency conditions for invariance of the foliation .

### Lemma 2.10.

*There exist and an isomorphism such that*
2.9
*where we use the notation* .

### Proof.

By the Gram–Schmidt method, there exists an orthonormal basis {*e*_{1},…,*e*_{d}} of *V* . Since is dense in *H*, there exist with ∥*ζ*_{i}−*e*_{i}∥<2^{−d} for *i*=1,…,*d*. Hence, we obtain
for all *i*,*j*=1,…,*d* with *i*≠*j* and
for all *i*=1,…,*d*. Thus, we have
for all *i*=1,…,*d*, and hence, owing to the theorem of Gerschgorin, the (*d*×*d*)-matrix
is invertible. Let be the isomorphism
Then, the isomorphism has the representation
Defining the isomorphism completes the proof. ■

Now, let be a parameterization of and let be an isomorphism as in lemma 2.10. We define as
and
By assumption 2.1, we have and there exists a constant *K*>0 such that
and
for all and all . Thus, for each and each there exists a unique strong solution for
2.10
We define the vector field as

Here is our main result concerning invariance of the foliation for SPDE (2.1).

### Theorem 2.11.

*The foliation is an invariant foliation for equation (2.1) if and only if for all t≥0 we have*
2.11
2.12

*and*2.13

*If the previous conditions are satisfied, the map*2.14

*is continuous, and for every and the weak solution for (2.1) with*.

*r*_{0}=*h*is also a strong solution

### Proof.

‘’: Let and *h*_{0}∈*V* be arbitrary. Then we have . Let (*r*_{t})_{t≥0} be the weak solution for equation (2.1) with *r*_{0}=*h* and set *z*_{0}:=〈*ζ*,*h*_{0}〉. Since and is an invariant foliation for equation (2.1), we obtain, by using equation (2.9),
This identity shows that almost surely
where (*Z*_{t})_{t≥0} denotes the strong solution for equation (2.10) with *Z*_{0}=*z*_{0}. By equation (2.9), we have almost surely
Let be arbitrary. Using Itô’s formula and by applying the linear functional 〈*ξ*,⋅〉 afterwards, we obtain
2.15
Since (*r*_{t})_{t≥0} is a weak solution for equation (2.1) with *r*_{0}=*h*, we have
2.16
Combining equations (2.15) and (2.16), we get
2.17
Therefore, all integrands in equation (2.17) vanish and, since was arbitrary, setting *s*=0 yields , proving (2.11) and the identities
2.18
and
2.19
which show equations (2.12) and (2.13). Furthermore, identity (2.18) proves the continuity of the map defined in equation (2.14).

‘’: Let and be arbitrary. There exists a unique such that *h*=*ψ*(*t*_{0})+*ϕ*(*z*_{0}). Let (*Z*_{t})_{t≥0} be the strong solution for equation (2.10) with *Z*_{0}=*z*_{0}. By using equations (2.11)–(2.13) and (2.9), Itô’s formula yields,
By the uniqueness of solutions for equation (2.1), we obtain almost surely
where (*r*_{t})_{t≥0} denotes the weak solution for equation (2.1) with *r*_{0}=*h*, whence is an invariant foliation, and we get that (*r*_{t})_{t≥0} is also a strong solution. ■

### Remark 2.12.

Note that equations (2.11)–(2.13) are consistency conditions on the tangent spaces (for related results e.g. Filipović 2000). Since the foliation consists of affine manifolds, we do not need a Stratonovich correction term for the drift.

Now, we express the consistency conditions from theorem 2.11 by means of a coordinate system. Let be a parametrization of and let {*λ*_{1},…,*λ*_{d}} be a basis of *V* .

### Corollary 2.13.

*The following statements are equivalent*:

—

*is an invariant foliation for equation (2.1)*.—

*We have*2.20*and*2.21*and there exist**such that*2.22*and*2.23

*If the previous conditions are satisfied, μ and γ are uniquely determined, we have , there exists a constant K>0 such that*
2.24

*and*2.25

*for all and , and for every and the weak solution for equation (2.1) with*.

*r*_{0}=*h*is also a strong solution

### Proof.

The asserted equivalence follows from theorem 2.11. By the linear independence of *λ*_{1},…,*λ*_{d}, the mappings *μ* and *γ* are uniquely determined. Denoting by , the isomorphism , we can express them as
and

Since the map defined in equation (2.14) is continuous by theorem 2.11, we have and equations (2.24) and (2.25) by virtue of assumption 2.1. ■

Suppose the foliation is invariant for equation (2.1). We shall now identify the underlying coordinate process *Y* . Let and be arbitrary. There exists a unique such that . Taking into account equations (2.24) and (2.25), we let (*Y*_{t})_{t≥0} be the strong solution for
2.26
where are given by equations (2.22) and (2.23). By Itô’s formula, the process
2.27
is the strong solution for equation (2.1) with *r*_{0}=*h*.

### Remark 2.14.

If we think of interest rate models, the state process *Y* has no direct economic interpretation. Proposition 2.8 shows that for any continuous linear operator with , we can choose ℓ(*r*) as a state process. We may think of (benchmark yields) or ℓ_{i}(*h*)=*h*(*x*_{i}) (benchmark forward rates). We refer to Björk & Landén (2002, §7), Björk & Gombani (1999, proposition 5.1), Björk & Svensson (2001, theorem 3.3), Chiarella & Kwon (2003, proposition 2) and Duffie & Kan (1996, §5) for related results.

## 3. Affine realizations for general SPDEs

The results of the previous section lead to the following definition of an affine realization.

### Definition 3.1.

Let be a finite-dimensional linear subspace. The SPDE (2.1) has an *affine realization generated by V* if for each there exists a foliation generated by

*V*with , which is invariant for equation (2.1).

We call the *dimension* of the affine realization.

### Lemma 3.2.

*Let and λ_{1},…,λ_{d}∈H be linearly independent. Suppose the SPDE (2.1) has a d-dimensional affine realization generated by V =〈λ_{1},…,λ_{d}〉. Then, there exist such that*
3.1

### Proof.

Relation (2.13) from theorem 2.11 yields *σ*(*h*)∈*V* for all . Since is dense in *H* and *V* is closed, we obtain *σ*(*h*)∈*V* for all *h*∈*H*. Hence, there exist such that equation (3.1) is satisfied. Since *σ*∈*C*^{1}(*H*), we have . ■

Suppose the SPDE (2.1) has an affine realization generated by a finite-dimensional subspace . Then, for each , the foliation is uniquely determined by lemma 2.7. We define the *singular set* *Σ* as
3.2
From a geometric point of view, the singular set consists of all starting points , for which the corresponding foliation only consists of a single leaf, that is, the solution process even stays on the *d*-dimensional affine space *h*_{0}+*V* . For *h*_{0}∈*Σ*, the mappings in equation (2.26) do not depend on the time *t*, whence the coordinate process *Y* is time-homogeneous, and the parametrization *ψ* in the affine realization (2.27) may be chosen as *ψ*≡*h*_{0}.

A consequence of the definition of the singular set in equation (3.2) is the identity
3.3
In particular, *Σ* is an invariant set for equation (2.1).

### Proposition 3.3.

*Suppose the SPDE (2.1) has an affine realization generated by V , then, the singular set Σ is given by*
3.4

*for each and , we have either or , and for every with , we have for all*.

*t*≥*t*_{0}

### Proof.

Let be arbitrary. By condition (2.12) of theorem 2.11 we have *ν*(*h*_{0})∈*V* if and only if *ν*(*h*)∈*V* for all *h*∈*h*_{0}+*V* , which means that *h*_{0}+*V* is an invariant manifold, proving equation (3.4). Taking into account equation (3.3), we obtain the remaining statements. ■

### Remark 3.4.

Suppose the SPDE (2.1) has an affine realization generated by *V* . For any , we define the deterministic stopping time
which, by remark 2.4 and equation (3.3), does not depend on the choice of the parametrization *ψ* of . By proposition 3.3, the strong solution (*r*_{t})_{t≥0} for equation (2.1) with *r*_{0}=*h*_{0} has the dichotomic behaviour
3.5
and
3.6
i.e. up to time *t*_{0}, the solution proceeds outside the singular set *Σ*, afterwards it stays in *Σ*, and therefore even on an affine manifold. In particular, if *t*_{0}=0 we have for all *t*≥0, and if , we have for all *t*≥0.

For our later investigations on the existence of affine realizations, quasi-exponential functions (cf. Björk & Svensson 2001, §5), which we shall now introduce in this general context, will play an important role. Inductively, we define the domains as well as the intersection

### Definition 3.5.

An element is called *quasi-exponential* if
3.7

### Lemma 3.6.

*Let h∈H be arbitrary. The following statements are equivalent*.

—

*h is quasi-exponential*.—

*There exists**such that**and**A*^{d}*h*∈〈*h*,*Ah*,…,*A*^{d−1}*h*〉.—

*There exists a finite-dimensional subspace with*3.8*h*∈*V*such that

### Proof.

(1) (2): This is clear for *h*=0, and for *h*≠0 there exists, by equation (3.7), a minimal integer such that *h*,*Ah*,…,*A*^{d−1}*h* are linearly independent. Consequently, we have *A*^{d}*h*∈〈*h*,*Ah*,…,*A*^{d−1}*h*〉.

(2) (3): The finite-dimensional subspace *V* =〈*h*,*Ah*,…,*A*^{d−1}*h*〉 has the desired properties.

(3) (1): Using equation (3.8), by induction, for each we have and *A*^{n}*h*∈*V* , which yields and equation (3.7), whence *h* is quasi-exponential. ■

In the subsequent sections, *H* will be a function space and *A*=(*d*/*dx*) the differential operator. Then, the domain consists of all -functions such that any derivative belongs to the function space *H*. As lemma 3.6 shows, a function is quasi-exponential if it satisfies a linear ordinary differential equation of *d*th order
3.9
for some . In particular, any exponential function has this property, which explains the term *quasi-exponential*. Note that equation (3.9) implies that the finite-dimensional subspace is invariant under the operator *A*, i.e. we have .

Quasi-exponential functions will play a decisive role for the characterization of term structure models with an affine realization, see the subsequent §§6–8.

## 4. The space of forward curves

In this section, we define the space of forward curves, on which we will study the HJMM equation (1.1) in the forthcoming sections. These spaces have been introduced in Filipović (2001, §5).

We fix an arbitrary constant *β*>0. Let *H*_{β} be the space of all absolutely continuous functions such that
4.1
Let (*S*_{t})_{t≥0} be the shift semigroup on *H*_{β} defined by *S*_{t}*h*:=*h*(*t*+⋅) for .

Since forward curves should flatten for large time to maturity *x*, the choice of *H*_{β} is reasonable from an economic point of view.

### Theorem 4.1.

*Let β>0 be arbitrary*.

—

*The space (*.*H*_{β},∥⋅∥_{β}), is a separable Hilbert space—

*For each , the point evaluation is a continuous linear functional*.— (

*S*_{t})_{t≥0}*is a**C*_{0}-semigroup on*H*_{β}with infinitesimal generator , (*d*/*dx*)*h*=*h*′, and domain—

*Each*.*h*∈*H*_{β}is continuous, bounded and the limit exists—

*is a closed subspace of**H*_{β}.—

*There exists a universal constant*4.2*C*>0, only depending on*β*, such that for all*h*∈*H*_{β}we have the estimate—

*For each*4.3*β*′>*β*, we have and the relation

### Proof.

Note that *H*_{β} is the space *H*_{w} from (Filipović 2001, §5.1) with weight function *w*(*x*)=*e*^{βx}, . Hence, the first six statements follow from (Filipović 2001, theorem 5.1.1, corollary 5.1.1). For each *β*′>*β*, the observation
shows and equation (4.3). ■

### Lemma 4.2.

*The following statements are valid.*

*For all*.*h*,*g*∈*H*_{β}, we have*hg*∈*H*_{β}and the multiplication map defined as*m*(*h*,*g*):=*hg*is a continuous, bilinear operator*For all , we have*.

### Proof.

The function *hg* is absolutely continuous, because *h* and *g* are absolutely continuous and bounded, see theorem 4.1. By estimate equation (4.2), we obtain
Hence, we have *hg*∈*H*_{β} and the estimate
proving that *m* is a continuous, bilinear operator.

If , we have with (*hg*)′=*h*′*g*+*hg*′, whence by the first statement. ■

For *λ*∈*H*_{β} we define , which belongs to , the space of all continuous functions from to .

### Lemma 4.3.

*Let 0< β<β′ be arbitrary real numbers. For each we have Λ∈H_{β} and the map is a continuous linear operator*.

### Proof.

Let be arbitrary. Then, is absolutely continuous. Since , using the Cauchy–Schwarz inequality, we obtain proving the assertion. ■

## 5. Invariant foliations for the HJMM equation

We shall now investigate invariant foliations for the HJMM equation (1.1) by working on the space of forward curves from the previous section.

Let 0<*β*<*β*′ be arbitrary real numbers and let be given.

### Assumption 5.1.

*We assume that σ∈C*^{1}*(H*_{β}*) with* *and that there exist L,M>0 such that*

Using the notation of the previous section, the HJM drift term (1.3) is given by Recall that this choice of the drift ensures that the implied bond market (1.2) will be free of arbitrage opportunities.

According to (Filipović 2001, corollary 5.1.2), we have and there exists a constant *K*>0 such that
Hence, for each *h*_{0}∈*H*_{β} there exists a unique weak solution for equation (1.1) with *r*_{0}=*h*_{0}, see (Da Prato & Zabczyk 1992, theorems 6.5 and 7.4). Note that equation (1.1) is a particular example of the SPDE (2.1) on the state space *H*=*H*_{β} with generator *A*=(*d*/*dx*) and drift *α*=*α*_{HJM}. Moreover, lemmas 4.2, and 4.3 yield *α*_{HJM}∈*C*^{1}(*H*_{β}), whence all required conditions from assumption 2.1 are fulfilled.

Now let be a foliation generated by a finite-dimensional subspace . We set . In order to investigate invariance of for the HJMM equation (1.1), we directly switch to a coordinate system. Let be a parametrization of and let {*λ*_{1},…,*λ*_{d}} be a basis of *V* . Then, the set {*Λ*_{1},…,*Λ*_{d}} is linearly independent in .

### Remark 5.2.

Let be an index set. We set
Then, there are subsets and such that
5.1
is a basis of the vector space
5.2
For each *m*∈*E*_{1}∖*D*_{1} there exist unique , and such that
5.3
and for each (*m*,*n*)∈*E*_{2}∖*D*_{2} there exist unique , and such that
5.4

### Theorem 5.3.

*The foliation is invariant for the HJMM equation (1.1) if and only if we have equations (2.20) and (2.21), there exist such that*
5.5
*and equation (2.23), there are , and such that for all we have*
5.6
5.7*and*
5.8
*where is chosen such that and the further quantities are chosen as in remark 5.2, and we have the Riccati equations*
5.9

### Proof.

‘’ Suppose is an invariant foliation for equation (1.1). According to corollary 2.13 we have equations (2.20)–(2.23). Relation (5.5) follows by setting *y*=0 in equation (2.22). Inserting equation (2.23) into (2.22) we get, by taking into account the HJM drift condition (1.3),
for all . Differentiating with respect to *y*_{k} we obtain
for all . Integrating yields
for all . Noting that , we can express this equation as
for all . Introducing the functions , *i*=1,…,*d* and , *i*∈*D*_{1} as well as , (*i*,*j*)∈*D*_{2} by
we obtain, by taking into account equations (5.3) and (5.4),
for all . Since *B* defined in equation (5.1) is a basis of the vector space *W* in equation (5.2), we deduce equations (5.6)–(5.8) and the Riccati equations (5.9).

‘’: Relations (1.3), (2.23), (5.3) and (5.4) yield
5.10
for all . In particular, by setting *y*=0, we have
5.11
for all . Relations (5.10), (5.6), (5.7), (5.8), (5.11) and the Riccati equations (5.9) give us
for all . We conclude, by furthermore incorporating equation (5.5),
for all , showing equation (2.22). According to corollary 2.13, the foliation is an invariant for equation (1.1). ■

Note that in particular, system (5.9) of Riccati equations is useful in order to gain knowledge about the existence of an affine realization. We will exemplify theorem 5.3 in the subsequent sections in order to characterize volatility structures for which the HJMM equation (1.1) admits an affine realization. We will start with general volatilities in §6, and will obtain results for particular volatility structures as corollaries in §§7–9.

## 6. Affine realizations for the HJMM equation with general volatility

In this section, we assume that the volatility *σ* in the HJMM equation (1.1) is of the form
6.1
where denotes a positive integer, are functionals and are linearly independent. We assume that for *i*=1,…,*p* and that there exist *L*,*M*>0 such that for all *i*=1,…,*p*, we have
and
Then, assumption 5.1 is fulfilled.

Note that, in view of lemma 3.2, this is the most general volatility, which we can have for the HJMM equation (1.1) with an affine realization. The corresponding HJM drift term (1.3) is given by 6.2

### Proposition 6.1.

*Suppose there exist h_{1},…,h_{p}∈H_{β} such that σ(h_{1}),…,σ(h_{p}) are linearly independent, and such that one of the following conditions is satisfied*:

*We have*6.3*There exist**k*,*l*∈{1,…,*p*}*such that the functions*6.4*are linearly independent for*1≤*i*≤*j*≤*p*.

*If the HJMM equation (1.1) has an affine realization, then λ_{1},…,λ_{p} are quasi-exponential*.

### Proof.

Let be a finite-dimensional subspace generating the affine realization and set . Lemma 3.2 yields that *σ*(*h*)∈*V* for all *h*∈*H*_{β}. Since *σ*(*h*_{1}),…,*σ*(*h*_{p}) are linearly independent, we obtain *λ*_{1},…,*λ*_{p}∈*V* , because relation (6.1) yields that
Choose *λ*_{p+1},…,*λ*_{d}∈*V* such that {*λ*_{1},…,*λ*_{d}} is a basis of *V* . Let be such that one of the conditions above is satisfied. Now we apply theorem 5.3 to the invariant foliation . In view of equations (2.23) and (6.1), the function is given by
and
In particular, we have and we can choose *E*_{1}={1,…,*p*}.

If equation (6.3) is satisfied, then equations (5.7) and (5.8) give us for all *i*∈*D*_{1}, *k*=1,…,*p* and for all (*i*,*j*)∈*D*_{2}, *k*=1,…,*p*. Consequently, the Riccati equations (5.9) show that *λ*_{1},…,*λ*_{p} are quasi-exponential.

If there exist *k*,*l*∈{1,…,*p*} such that the functions (6.4) are linearly independent for 1≤*i*≤*j*≤*p*, then we claim that *D*_{1}=*D*_{2}=∅, which, in view of the Riccati equations (5.9), implies that *λ*_{1},…,*λ*_{p} are quasi-exponential. Suppose, on the contrary, that *D*_{1}≠∅ or *D*_{2}≠∅.

If *D*_{1}≠∅, choose *i*∈*D*_{1} and differentiate equation (5.7) with respect to *y*_{l}, which yields
for all *h*∈*h*_{0}+〈*λ*_{1},…,*λ*_{p}〉. This contradicts the linear independence of equation (6.4) for 1≤*i*≤*j*≤*p*.

Analogously, if *D*_{2}≠∅, choosing (*i*,*j*)∈*D*_{2} and differentiating equation (5.8) with respect to *y*_{l} yields a contradiction to the linear independence of equation (6.4) for 1≤*i*≤*j*≤*p*. ■

### Proposition 6.2.

*If λ_{1},…,λ_{p} are quasi-exponential, then the HJMM equation (1.1) has an affine realization*.

### Proof.

Since *λ*_{1},…,*λ*_{p} are quasi-exponential, the linear space
is finite-dimensional and we have
6.5
Since for all *t*≥0, we have . Set . There exist *λ*_{p+1},…,*λ*_{q}∈*W*, such that {*λ*_{1},…,*λ*_{q}} is a basis of *W*. We define the subspace
Note that by lemmas 4.2 and 4.3. Set and choose *λ*_{q+1},…,*λ*_{d}∈*V* such that {*λ*_{1},…,*λ*_{d}} is a basis of *V* . Relation (6.5) implies
6.6
By equations (6.6) and (6.5), we have
whence we have
6.7
Let be arbitrary. We define the map ,
6.8
the map ,
6.9
and as
6.10
where, due to (6.7), the are chosen such that
Then, conditions (2.20)–(2.23) are fulfilled, and therefore, by corollary 2.13, the foliation generated by *V* =〈*λ*_{1},…,*λ*_{d}〉 with parametrization *ψ* is invariant for the HJMM equation (1.1). ■

### Remark 6.3.

Note that the proof of proposition 6.2 simultaneously provides the construction of the affine realization. For the invariant foliation is generated by 〈*λ*_{1},…,*λ*_{d}〉 and has the parametrization *ψ* defined in equation (6.8). For with some the strong solution (*r*_{t})_{t≥0} for equation (1.1) with *r*_{0}=*h* is given by equation (2.27), where the maps for the state process (2.26) are defined in equations (6.9) and (6.10). We refer to (Björk & Landén 2002, propositions 5.1 and 6.1) for similar results.

### Remark 6.4.

According to remark 2.14, for any continuous linear operator with , we can choose ℓ(*r*) as a state process. For example, the components ℓ_{i} could be evaluations of benchmark yields or benchmark forward rates. This provides an economic interpretation of the affine realization.

### Remark 6.5.

Combining proposition 3.3 and relations (6.2) and (6.8), the singular set *Σ* is given by the (*d*+1)-dimensional linear space
and, by remark 3.4, for each , we have equations (3.5) and (3.6), where *t*_{0} denotes the deterministic stopping time
and where (*r*_{t})_{t≥0} denotes the strong solution for equation (1.1) with *r*_{0}=*h*_{0}.

### Remark 6.6.

Note that the conditions in proposition 6.1 are singular events, because the respective conditions only have to be satisfied for one single point *h*_{0}. Hence, propositions 6.1 and 6.2 yield that, apart from degenerate examples like the CIR model, the existence of an affine realization is essentially equivalent to the condition that *λ*_{1},…,*λ*_{p} are quasi-exponential (which means that all quadratic terms in the system (5.9) of Riccati equations disappear). This also supplements (Björk & Svensson 2001, proposition 6.4), which provides the sufficient implication.

## 7. Affine realizations for the HJMM equation with constant direction volatility

In this section, we study the existence of affine realizations for the HJMM equation (1.1) with constant direction volatility, that is, we assume that the volatility *σ* in the HJMM equation (1.1) is of the form
7.1
where is a functional and with *λ*≠0. We assume that and that there exist *L*,*M*>0 such that
Then, assumption 5.1 is fulfilled.

### Corollary 7.1.

*Suppose Φ≢0. If the HJMM equation (1.1) has an affine realization, then λ is quasi-exponential or we have*
7.2

*and*7.3

### Proof.

This is an immediate consequence of proposition 6.1. ■

### Remark 7.2.

Conditions (7.2) and (7.3) mean that at each forward curve the functional *Φ*^{2} is affine, but not constant, in direction *λ*, which is the typical feature for CIR type models.

### Corollary 7.3.

*If λ is quasi-exponential, then the HJMM equation (1.1) has an affine realization*.

### Proof.

This is a direct consequence of proposition 6.2. ■

### Remark 7.4.

Suppose we have *Φ*≢0 and there exists , such that
Then, by corollaries 7.1 and 7.3, the HJMM equation (1.1) has an affine realization if and only if *λ* is quasi-exponential. Hence, we have relaxed the assumptions from Björk & Svensson (2001, proposition 6.1), where it is assumed that *Φ*(*h*)≠0 for all *h*∈*H*_{β} and *D*^{2}*Φ*^{2}(*h*)(*λ*,*λ*)≠0 for all *h*∈*H*_{β}.

## 8. Affine realizations for the HJMM equation with constant volatility

In this section, we study the existence of affine realizations for the HJMM equation (1.1) with constant volatility, i.e. we have
with , *λ*≠0. Then, assumption 5.1 is fulfilled.

### Corollary 8.1.

*The HJMM equation (1.1) has an affine realization if and only if λ is quasi-exponential*.

### Proof.

The assertion is a direct consequence of corollaries 7.1 and 7.3, because *σ* is of the form (7.1) with *Φ*≡1. ■

### Remark 8.2.

If *λ* is quasi-exponential, we even obtain a *d*-dimensional affine realization, where . For , the invariant foliation is generated by 〈*λ*_{1},…,*λ*_{d}〉 with
and has the parametrization
which can be shown using corollary 2.13 (cf. Björk & Landén 2002; proposition 4.1). Using proposition 3.3, the singular set *Σ* is given by the (*d*+1)-dimensional affine space
and, by remark 3.4, for each we have equations (3.5) and (3.6), where *t*_{0} denotes the deterministic stopping time
and where (*r*_{t})_{t≥0} denotes the strong solution for equation (1.1) with *r*_{0}=*h*_{0}.

### Remark 8.3.

Not surprisingly, the statement of corollary 8.1 coincides with that of Björk & Svensson (2001, proposition 5.1).

## 9. Short rate realizations for the HJMM equation

In this last section, we deal with affine realizations of dimension *d*=1. As explained in remark 6.4, we can give an economic interpretation to the affine realization and choose (subject to slight regularity conditions) the short rate *r*(0) as the state process. In this case, we also speak about a *short rate realization*.

Let us assume that the volatility *σ* is of the form as in the following:
9.1
where denotes the evaluation of the short rate and where is an arbitrary map. Then, the short rate process will be the solution of a one-dimensional stochastic differential equation.

Using our previous results with *d*=1 and taking into account the particular structure (9.1) of the volatility, we see that *σ* is of one of the following three types:

We can have with a constant . This is the Ho–Lee model.

We can have with appropriate constants . This is the Hull–White extension of the Vasicek model.

We can have with appropriate constants , where satisfies a Riccati equation of the kind This is the Hull–White extension of the Cox–Ingersoll–Ross (CIR) model.

Thus, we have recognized the three well-known short rate models, which is completely in line with the existing literature (e.g. Jeffrey 1995; Björk & Svensson 2001; Filipović & Teichmann 2004).

## 10. Conclusion

We have presented an alternative approach on the existence of affine realizations for HJM interest rate models, which has the feature to be applicable to be a wide class of models and being conceptually rather comprehensible.

Applying this approach, we have been able to provide further insights into the structure of affine realizations. In particular, we have seen that essentially all volatility structures with an affine realization are of the form (6.1) with *λ*_{1},…,*λ*_{p} being quasi-exponential. All remaining volatilities with an affine realization, like the CIR model, may be considered as degenerate examples, (see remark 6.6).

Our proofs have provided constructions of the affine realizations (see remarks 6.3 and 6.4) and we have been able to determine the singular set *Σ* (see remark 6.5), where we have also exhibited the dichotomic behaviour of the forward rate process with respect to *Σ*. Moreover, for particular volatility structures, we have supplemented some known existence results.

## Acknowledgements

The author gratefully acknowledges the support from WWTF (Vienna Science and Technology Fund). The author is also grateful to two anonymous referees for their helpful comments and suggestions.

## Footnotes

- Received September 22, 2009.
- Accepted March 23, 2010.

- © 2010 The Royal Society