EXISTENCE OF INVARIANT MANIFOLDS FOR STOCHASTIC EQUATIONS IN INFINITE DIMENSION

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1 EXISTENCE OF INVARIANT MANIFOLDS FOR STOCHASTIC EQUATIONS IN INFINITE DIMENSION DAMIR FILIPOVIĆ AND JOSEF TEICHMANN Abstract. We provie a Frobenius type existence result for finite-imensional invariant submanifols for stochastic equations in infinite imension, in the spirit of Da Prato an Zabczyk [5]. We recapture an make use of the convenient calculus on Fréchet spaces, as evelope by Kriegl an Michor [16]. Our main result is a weak version of the Frobenius theorem on Fréchet spaces. As an application we characterize all finite-imensional realizations for a stochastic equation which escribes the evolution of the term structure of interest rates. 1. Introuction In this article we investigate the existence of finite-imensional invariant manifols for a stochastic equation of the type r t = (Ar t + α(r t )) t + σ j (r t ) W j t (1.1) r 0 = h 0 on a separable Hilbert space H, in the spirit of Da Prato an Zabczyk [5]. The operator A : D(A) H H generates a strongly continuous semigroup on H. Here N, an W = (W 1,..., W ) enotes a stanar -imensional Brownian motion efine on a fixe reference probability space (see [5]). The mappings α : H H an σ = (σ 1,..., σ ) : H H satisfy a smoothness conition, to be efine precisely in what follows (Section 4). We istinguish, in ecreasing orer of generality, between (local) mil, weak an strong solutions of equation (1.1). The reaer is referre to [5] or [8] for the precise efinitions. Our motivation is coming from the theory of interest rates. The basic interest rate contracts are the zero coupon bons. The price at time t of a zero coupon bon with maturity T t is given by ( ) P (t, T ) = exp T t 0 r t (x) x Date: December 11, 2000 (first raft); November 16, 2001 (this raft). We thank T. Björk, A. Kriegl, P. Malliavin, P. Michor, W. Schachermayer, L. Svensson an J. Zabczyk for interesting an helpful iscussions. D. Filipović gratefully acknowleges the kin hospitality of the Department of Financial an Actuarial Mathematics at Vienna University of Technology an the Department of Mathematics at ETH Zurich, an the financial support from Creit Suisse. J. Teichmann gratefully acknowleges the stimulating atmosphere at the Department of Financial an Actuarial Mathematics at Vienna University of Technology, an the support from grant Wittgestein Preis (Austrian Science Founation Z 36-MAT) aware to Walter Schachermayer. 1,

2 2 DAMIR FILIPOVIĆ AND JOSEF TEICHMANN where r t (x) enotes the instantaneous forwar rate at time t for ate t + x (this notion has been introuce by Musiela [19]). Within the framework of Heath, Jarrow an Morton (henceforth HJM) [14], for every T 0, the real-value process (r t (T t)) 0 t T is an Itô processes satisfying the so calle HJM rift conition, which assures the absence of arbitrage. It is shown in [8] that the stochastic evolution of the entire forwar curve, x r t (x) : R 0 R, can be escribe by a stochastic equation of the above type (1.1), where H consists of real-value continuous functions on R 0, the operator A = /x is the generator of the shift-semigroup S t h = h(t+ ), an α = α HJM is completely etermine by σ accoring to the HJM rift conition. We will be more precise about the HJM setup in Section 4 below. There are several reasons why in practice one is intereste in such HJM moels which amit a finite-imensional realization (FDR) at every initial curve r 0 H, see [1, 7, 8, 12]. The formal efinition of an FDR is as follows. Definition 1.1. Let m N an h 0 H. An m-imensional realization for (1.1) at h 0 is a pair (V, φ), where V R m is open, φ : V H is a smooth immersion, such that h 0 φ(v ) an, for every h φ(v ), there exists a V -value Itô process Z such that φ(z) is a local weak solution to (1.1) with r 0 = h. The notion of a smooth immersion is recapture in Section 3 (see Lemma 3.1). By convention, smooth is a synonym for C (see Section 2 for a thorough iscussion on ifferential calculus). Definition 1.2. A subset U of H is calle locally invariant for (1.1) if, for every initial point h 0 U, there exists a continuous local weak solution r to (1.1) with lifetime τ such that r t τ U, for all t 0. For the notion of a finite-imensional submanifol M of a Hilbert space an its tangent spaces T h M, h M, we refer to Section 3. Finite-imensional locally invariant submanifols for (1.1) have been characterize in [10], see also [8]. Here we restate [10, Theorem 3]. Theorem 1.3. Suppose that α is locally Lipschitz continuous an locally boune, an σ is C 1. Let M be an m-imensional submanifol of H. Then the following conitions are equivalent: i) M is locally invariant for (1.1) ii) M D(A) an for all h M. µ(h) := Ah + α(h) 1 2 Dσ j (h)σ j (h) T h M (1.2) σ j (h) T h M, j = 1,...,, (1.3) Hence the stochastic invariance problem to (1.1) is equivalent to the eterministic invariance problems relate to the vector fiels µ, σ 1,..., σ. An FDR is essentially equivalent to a finite-imensional invariant submanifol in the following sense. If (V, φ) is an m-imensional realization for (1.1) at some h 0 H, then there exists an open neighborhoo V 0 of φ 1 (h 0 ) in R m such that φ(v 0 ) is an m-imensional submanifol, which is locally invariant for (1.1). The converse is given by the following result, which is a restatement of [8, Theorem 6.4.1].

3 EXISTENCE OF INVARIANT MANIFOLDS FOR STOCHASTIC EQUATIONS 3 Theorem 1.4. Let α, σ an M be as in Theorem 1.3. Suppose M is locally invariant for (1.1). Then, for any h 0 M, there exists an m-imensional realization (V, φ) for (1.1) at h 0 such that φ(v ) = U M, where U is an open set in H. Theorem 1.3 provies conitions for the invariance of a given submanifol M. However, it oes not say anything about the existence of an FDR for (1.1). This issue will be exploite in the present article. The FDR-problem consists of fining sufficient conitions on µ, σ 1,..., σ for the existence of FDRs. Björk et al [1], [3] translate this into an appropriate geometric language. In [3] they completely solve the FDR-problem for equations (1.1) of HJM type on a very particular Hilbert space. Their key argument is the classical Frobenius theorem (see for example [17]), since they are looking for foliations (which is the appropriate notion for the FDR-problem on Hilbert spaces). Therefore they efine a Hilbert space, H, on which A = /x is a boune linear operator. As a consequence H consists solely of entire analytic functions (see [3, Proposition 4.2]). It is well known however that the forwar curves implie by a Cox Ingersoll Ross (CIR) [4] short rate moel are of the form r t = g 0 + r t (0)g 1 where g 0 (x) = eax 1 e ax + c an g 1 (x) = be ax (e ax + c) 2, for some a, b, c > 0 an 0 (see e.g. [8, Section 7.4.1]). Since both g 0 an g 1, when extene to C, have a singularity at x = (log(c)+iπ)/a, they cannot be entire analytic. Hence the CIR forwar curves o not belong to H. Since the CIR moel is one of the basic HJM moels, the Björk-Svensson [3] setting is too narrow for the HJM framework, even though all geometric ieas are alreay formulate there. To overcome this ifficulty we have to choose a larger forwar curve space. But we cannot o without the Frobenius theorem. The problem is that A is typically an unboune operator on H, so µ is not continuous an not even efine on the whole space H (the choice of H = H in [3] is exactly mae to overcome this problem). The appropriate framework for an extene version of the Frobenius theorem is thus given by the Fréchet space D(A ) := n N D(A n ), equippe with the family of seminorms n p n (h) = A i h H, n N 0. i=0 We prove the existence of FDRs on this space uner aitional technical assumptions on α an σ 1,..., σ. They have to map D(A ) into itself an generate local flows on D(A ). However, as typical for Frécht spaces, smoothness of α an σ on D(A ) is not enough to guarantee the existence of local flows. Thus we shall provie sufficient conitions on the coefficients, which can be foun in Hamilton [13] (α an σ have to be so calle Banach maps). Then the existence of FDRs on an open subset U in D(A ) is essentially equivalent to the bouneness of the imension of the Lie algebra generate by µ, σ 1,..., σ on U. We o not obtain a true foliation of U as in the finite-imensional case, which is ue to the fact that µ merely amits a local semiflow on U an not a local flow. So we are le to the notion of a weak foliation.

4 4 DAMIR FILIPOVIĆ AND JOSEF TEICHMANN We then exemplify the use of these results with the HJM framework. Here we eventually obtain a striking global result. HJM moels that amit an FDR at any initial curve r 0 are necessarily affine term structure moels, in a sense to be explaine in Section 4 (see Remark 4.14). The remainer of the paper is organize as follows. In Section 2 we provie a convenient ifferential calculus on Fréchet spaces (an more general locally convex spaces), as eveloppe in [16]. We iscuss the existence of local (semi)flows relate to smooth vector fiels on a Fréchet space, base on the Banach map principle (Theorems 2.10 an 2.13). In Section 3 we recapture the notion of a finiteimensional submanifol, an the Lie bracket of two smooth vector fiels in a Fréchet space. We point out the crucial fact that the Lie bracket of a Banach map with a boune linear operator is a Banach map (Lemma 3.4). After the efinition of a finite-imensional weak foliation (Definition 3.7) we prove a Frobenius theorem on Fréchet spaces (Theorem 3.9). In Section 4 we provie the rigorous setup for HJM moels. Uner the appropriate assumptions we solve the FDR-problem an give a global characterization of all finite-imensional weak foliations. 2. Analysis on Fréchet Spaces For the purposes of analysis on open subsets of Fréchet spaces we shall follow two equivalent approaches. The classical Gateaux-approach as outline in [13] an so calle convenient analysis as in [16]. On Fréchet spaces these two notions of smoothness coincie an convenient calculus is an appropriate extension of analysis to more general locally convex spaces. Furthermore these methos allow simple an elegant calculations. The main avantage of convenient calculus is however, that one can give a precise analytic meaning (in simple terms) to geometric objects on Fréchet spaces as for example vector fiels, ifferential forms (see [16]). Definition 2.1. Let E, F be Fréchet spaces an U E an open subset. A map P : U F is calle Gateaux-C 1 if P (f + th) P (f) DP (f)h := lim t 0 t exists for all f U an h E an DP : U E F is a continuous map. For the efinition of Gateaux-C 2 -maps the ambiguities of calculus on Fréchet spaces alreay appear. Since there is no Fréchet space topology on the vector space of continuous linear mappings L(E, F ) one has to work by point evaluations: Definition 2.2. Let E, F be Fréchet spaces an U E an open subset. A map P : U F is calle Gateaux-C 2 if D 2 P (f)(h 1, h 2 ) := lim t 0 DP (f + th 2 )h 1 DP (f)h 1 t exists for all f U an h 1, h 2 E an D 2 P : U E E F is a continuous map. Higher erivatives are efine in a similar way. A map is calle Gateaux-smooth or Gateaux-C if it is Gateaux-C n for all n 0. The next Theorem collects all essential results of Gateaux-Calculus for our purposes (see [13], pp , pp ): Theorem 2.3. Let E, F, G be Fréchet spaces an U E be open in E. Let P : U E F an Q : V F G be continuous maps:

5 EXISTENCE OF INVARIANT MANIFOLDS FOR STOCHASTIC EQUATIONS 5 i) If P an Q are Gateaux-C n, then Q P is Gateaux-C n an the usual chain rule hols. ii) Let U be convex: P is Gateaux-C 1 if an only if there exists a continuous map L : U E E F, linear in the last variable, such that for all f 1, f 2 U P (f 1 ) P (f 2 ) = L(f 1, f 2 )(f 1 f 2 ). iii) If P is Gateaux-C 1, then for f 0 U an a continuous seminorm q on F, there is a continuous seminorm p on E an ε > 0 such that for p(f i f 0 ) < ε, i = 1, 2. q(p (f 1 ) P (f 2 )) p(f 1 f 2 ) For the construction of ifferential calculus on locally convex spaces we nee the concept of smooth curves into locally convex spaces an the concept of smooth maps on open subsets of locally convex spaces. We remark that alreay on Fréchet spaces the situation concerning analysis was complicate an unclear until convenient calculus was invente (see [16], pp , for extensive historical remarks). The reason for inconsistencies can be foun in the funamental ifference between boune an open subsets. We enote the set of continuous linear functionals on a locally convex space E by E c. A subset B E is calle boune if l(b) is a boune subset of R for all l E c. A multilinear map m : E 1... E n F is calle boune if boune sets B 1... B n are mappe onto boune subsets of F. Continuous linear functionals are clearly boune. The locally convex vector space of boune linear operators with uniform convergence on boune sets is enote by L(E, F ), the ual space forme by boune linear functionals by E. These spaces are locally convex vector spaces we shall nee for analysis (see [16], 3.17). Definition 2.4. Let E be a locally convex space, then c : R E is calle smooth if all erivatives exist as limits of ifference quotients. The set of smooth curves is enote by C (R, E). A subset U E is calle c -open if c 1 (U) is open in R for all c C (R, E). The generate topology on E is calle c -topology an E equipe with this topology is enote by c E. If U is c -open, a map f : U E R is calle smooth if f c C (R, R) for all c C (R, E). These efinitions work for any locally convex vector space, but for the following theorem we nee a weak completeness assumption. A locally convex vector space E is calle convenient if the following property hols: a curve c : R E is smooth if an only if it is weakly smooth, i.e. l c C (R, R) for all l E. This is equivalent to the assertion that any smooth curve c : R E can be (Riemann-) integrate in E on compact intervals (see [16], 2.14). The spaces L(E, F ) an E are convenient vector spaces (see [16], 3.17), if E an F are convenient. Theorem 2.5. Let E, G, H be convenient vector spaces, U E, V G c -open subsets: i) Smooth maps are continuous with respect to the c -topology. ii) Multilinear maps are smooth if an only if they are boune.

6 6 DAMIR FILIPOVIĆ AND JOSEF TEICHMANN iii) If P : U G is smooth, then DP : U L(E, G) is smooth an boune linear in the secon component, where DP (f)h := t t=0p (f + th). iv) The chain rule hols. v) Let [f, f + h] := {f + sh for s [0, 1]} U, then Taylor s formula is true at f U, where higher erivatives are efine as usual (see iii.), P (f + h) = n i=0 1 i! Di P (f)h (i) (1 t) n D n+1 P (f + th) (h (n+1) )t n! for all n N. vi) There are natural convenient locally convex structures on C (U, F ) an we have cartesian closeness C (U V, H) C (U, C (V, H)). via the natural map f ˇf : U C (V, H) for f C (U V, H). This natural map is well efine an a smooth linear isomorphism. vii) The evaluation an the composition ev : C (U, F ) U F, (P, f) P (f).. : C (F, G) C (U, F ) C (U, G), (Q, R) Q R are smooth maps. viii) A map P : U E L(G, H) is smooth if an only if (ev g P ) is smooth for all g G. Proof. For the proofs see [16] in Subsections 3.12, 3.13, 3.18, 5.11, 5.12, Convenient Calculus is an extension of the Gateaux-Calculus to locally convex spaces, where all necessary tools for analysis are preserve. Since typically vector spaces like C (U, F ) or L(E, F ) are not Fréchet spaces, this extension is very useful for the analysis of the geometric objects in Section 3. Theorem 2.6. Let E, F be Fréchet spaces an U E a c -open subset, then U is open an P : U E F is Gateaux-smooth if an only if P is smooth (in the convenient sense). Proof. By Theorem 4.11 of [16] we get that U is open since c E = E. Assume that P is Gateaux-smooth, then by the chain rule for Gateaux-C n maps (see Theorem 2.3, i.) the composition P c is Gateaux-C n for all n 0 an all smooth curves c C (R, E), so P is smooth in the convenient sense. If P is smooth in the convenient sense, then the first erivative DP as efine in Theorem 2.5 exists an is continuous as map DP : U E F by cartesian closeness an the fact that c E = E (see Theorem 2.5, i.). The same reasoning hols for higher erivatives, so we obtain that P is Gateaux-C n for all n 0. Since we shall calculate with semiflows an semigroups of boune linear operators, we shall nee convenient calculus on omains of the form [0, ε[ U, where U is open in a Fréchet space E. The efinition of smooth maps is straightforwar ue to the simple strucure of convenient calculus. Let K be a convex set with non-voi interior K in a Fréchet space E an F a convenient vector space, then f : K F is calle smooth if f c : R F is

7 EXISTENCE OF INVARIANT MANIFOLDS FOR STOCHASTIC EQUATIONS 7 smooth for all smooth curves c C (R, E) with c(r) K. We have the following properties (for a proof see [16], pp ): Theorem 2.7. Let K be a convex subset with non-voi interior K in a Fréchet space E, F a convenient space an P : K F a map. i) P is smooth if an only if P is smooth on K an all erivatives D n (P K ) exten continuously (with respect to the c -topology to K) to K. ii) If P is smooth an DP : K L(E, F ) a continuous extension of D(P K ), then the chain rule hols, i.e. for c C (R, E) with c(r) K we have (P c) (t) = DP (c(t)) c (t). iii) There exists a boune linear extension operator C ([0, ε[, F ) C (R, F ). Consequently we can reformulate all assertions of Theorem 2.5 for maps on [0, ε[ U with U open in a Fréchet space, since ([0, ε[ U) =]0, ε[ U, in particular, the chain rule an cartesian closeness hol. The time erivatives at 0 can be calculate as right erivatives by the boune linear extension operator. This convenient approach is through its generality an simplicity much more practical than the equivalent Gateaux approach. In the sequel we shall apply concepts from both approaches: Gateaux-smoothness for existence theorems an convenient analysis for the sake of generality, simplicity an elegance. Notice that convenient calculus provies a very powerful tool for analysis in concrete calculations, too (see [16] for many examples an [23] for a particularly simple proof of a general Frobenius Theorem). Concerning ifferential equations, there are possible counterexamples on nonnormable Fréchet spaces in all irections, which causes some problems in the founations of ifferential geometry (see [16] an the excellent review article [18]). Nevertheless a useful generalization of the existence theorem for ifferential equations on Banach spaces is given by the following Banach map principle (see [13] for etails, compare also [16], for weaker results in a more general situation). If not otherwise state, E an F enote Fréchet spaces an B a Banach space in what follows. Given P : U E E a smooth map. We are looking for solutions of the orinary ifferential equation with initial value g U f :] ε, ε[ U smooth f(t) = P (f(t)) t f(0) = g U. If for any initial value g in a small neighborhoo V of f 0 U there is a unique smooth solution t f g (t) for t ] ε, ε[ epening smoothly on the initial value g, then F l(t, g) := f g (t) efines a local flow, i.e. a smooth map F l :] ε, ε[ V E F l(0, g) = g F l(t, F l(s, g)) = F l(s + t, g) if s, t, s+t ] ε, ε[ an F l(s, g) V. If there is a local flow aroun f 0 U (this shall mean once an for all: in an open, convex neighborhoo of f 0 ), the ifferential equation is uniquely solvable aroun f 0 U an the epenence on initial values is smooth (see Lemma 2.11 for the proof). Notice at this point that it is irrelevant if

8 8 DAMIR FILIPOVIĆ AND JOSEF TEICHMANN we efine smooth epenence on initial values via maps V C (] ε, ε[, E) or V ] ε, ε[ E by cartesian closeness. We shall enote f g (t) = F l t (g) = F l(t, g). Definition 2.8. Given a Fréchet space E, a smooth map P : U E E is calle a Banach map if there are smooth (not necessarily linear) maps R : U E B an Q : V B E such that P = Q R U E E R Q V B where B is a Banach space an V B is an open set. A vector fiel P on an open subset U E is a smooth map P : U E. We enote by B(U) the set of Banach map vector fiels an by X(U) the convenient space of all vector fiels on an open subset of a Fréchet space E. Theorem 2.9. B(U) is a C (U, R)-submoule of X(U). Proof. We have to show that for ψ, η C (U, R) an P 1, P 2 B(U) the linear combination ψp 1 + ηp 2 B(U). Given P i = Q i R i for i = 1, 2 with intermeiate Banach spaces B i, then ψp 1 +ηp 2 = Q R with Q : R 2 V 1 V 2 R 2 B 1 B 2 E an R : U R 2 B 1 B 2 such that Q(r, s, v 1, v 2 ) = rq 1 (v 1 ) + sq 2 (v 2 ) R(f) = (ψ(f), η(f), R 1 (f), R 2 (f)) So the sum ψp 1 + ηp 2 is a Banach map an therefore the set of all Banach map vector fiels carries the asserte submoule structure. Theorem 2.10 (Banach map principle). Let P : U E E be a Banach map, then P amits a local flow aroun any point g U. Proof. For the proof see [13], Theorem Parameters an time-epenence are treate in the following way. Given an open subset of parameters Z V of a Banach space V an P : I Z U E, where I is a open set in R an U is open in E, such that P t,p = Q t,p R t,p, where Q an R epen smoothly on time an parameters, P amits a unique smooth solution for any initial value f 0 U at any time point t 0 I epening smoothly on parameters, time an initial values. For the proof of this assertion we look at the extene space G := R V E with P (t, p, f) = (1, 0, P t,p (f)) an Q(t, p, z) = (1, 0, Q t,p (f)) R(t, p, f) = (t, p, R t,p (f)) with Banach space B := R V B. We can replace in the above efinition of a local flow the interval ] ε, ε[ by [0, ε[ to obtain local semiflows, see Theorem 2.7 for etails in calculus. The initial value P

9 EXISTENCE OF INVARIANT MANIFOLDS FOR STOCHASTIC EQUATIONS 9 problem f : [0, ε[ U smooth f(t) = P (f(t)) t f(0) = g U. amits unique solutions aroun an initial value epening smoothly on the initial values if an only if a local semiflow exists. The notion of a local semiflow is reunant on Banach spaces. Lemma Let F l be a local semiflow on [0, ε[ U E, then the map P (f) := t t=0f l(t, f) is a well efine smooth vector fiel. We obtain DF l t (f)p (f) = P (F l t (f)) an the initial value problem has unique solutions for small times which coincie with the given semiflow. Proof. The equation follows by the flow property an the efinition of P immeiately: DF l t (f)p (f) = s F l t(f l s (f)) s=0 = s F l t+s(f) s=0 = P (F l t (f)). Given a solution f : [0, δ[ E of the initial value problem associate to P with f(0) = f 0 U, then s F l t s(f(s)) = P (F l t s (f(s))) + P (F l t s (f(s))) = 0 F l t s (f(s)) = f(t) for all 0 s t, whence uniqueness for the solutions of the initial value problem. We are in particular intereste in special types of ifferential equations on Fréchet spaces E, namely Banach map perturbe boune linear equations. Given a boune linear operator A : E E, the abstract Cauchy problem associate to A is given by the initial value problem associate to A. We assume that there is a smooth semigroup of boune linear operators S : R 0 L(E, E) such that S t i lim = A t 0 t which is a global semiflow for the linear vector fiel f Af. Notice that the theory of boune linear operators on Fréchet spaces contains as a special case Hille-Yosia-Theory of unboune operators on Banach spaces (see for example [22]). Given a strongly continuous semigroup S t for t 0 of boune linear operators on a Banach space B, then D(A n ) with the respective operator norms p n (f) := n i=0 Ai f for n 0 an f D(A n ) is a Banach space, where the semigroup restricts to a strongly continuous semigroup S (n). Consequently the semigroup restricts to the Fréchet space D(A ). This semigroup is now smooth, since it is sufficient by Theorem 2.5, viii. to show smoothness of t S t f for all f D(A ). This is true since A n S t f = S t A n f for t 0 an f D(A ). For the purposes of classification in Section 4 we shall nee the following result.

10 10 DAMIR FILIPOVIĆ AND JOSEF TEICHMANN Lemma Let A be the generator of a strongly continuous semigroup S on a Banach space B, then the operator A : D(A ) D(A ) is a Banach map if an only if A : B B is boune. Proof. For the properties of D(A ) see [20], in particular it is a Fréchet space with seminorms p n (f) = n i=0 Ai f. If A : D(A ) D(A ) is a Banach map in a neighborhoo U of a point f 0, then there are smooth maps R : U D(A ) X an Q : V X D(A ) such that A = Q R an X is a Banach space. By ifferentiation at f 0 we obtain A = DQ(f 0 ) DR(f 0 ) which means in particular by continuity that there exists n 0 such that DR(f 0 ) can be extene continuously to a linear mapping DR(f 0 ) : D(A n ) X (see Theorem 2.3, iii.). So A : D(A n ) D(A n ) is a continuous mapping. We recall the Sobolev Hierarchy for strongly continuous semigroups (see [20]) efine by the following commutative iagram B R(λ) D(A) R(λ) D(A 2 ) S (0) t S (1) t S(2) t B R(λ) D(A) R(λ) D(A 2 ) Here R(λ) := (λ A) 1 enotes the resolvent at a point of the resolvent set, which efines an isomorphism from D(A n ) to D(A n+1 ). The semigroups S (n) are efine by restriction an are strongly continuous in the respective topologies. The generator of S (n) is given through A restricte to D(A n+1 ). If A is continuous on D(A n ) then S (n) is a smooth group, so by climbing up through the isomorphisms S (0) is a smooth group an therefore the infinitesimal generator is continuous, since it is everywhere efine, by the close graph theorem. Given a Banach map P : U E E, we want to investigate the solutions of the initial value problem t f(t) = Af(t) + P (f(t)), f(0) = f 0. Theorem Let E be a Fréchet space an A be the generator of a smooth semigroup S : R L(E) of boune linear operators on E. Let P : U E E be a Banach map. For any f 0 U there is ε > 0 an an open set V containing f 0 an a local semiflow F l : [0, ε[ V U satisfying for all (t, f) [0, ε[ V. F l(t, f) = AF l(t, f) + P (F l(t, f)) t F l(0, f) = f

11 EXISTENCE OF INVARIANT MANIFOLDS FOR STOCHASTIC EQUATIONS 11 Proof. The arguments follow a proof for the case A = 0 in [13]. We prove the theorem by constructing a solution to the integral equation arising from variation of constants: F l(t, f) = S t f + t 0 S t s P (F l(s, f))s for small positive time intervals an an open neighborhoo of a given initial value f 0. Given f 0 U there exists a seminorm p on E an δ > 0 such that R(f 1 ) R(f 2 ) p(f 1 f 2 ) for p(f i f 0 ) < δ an i = 1, 2, where. enotes the norm on B. Furthermore given g 0 B, then for any seminorm q on F there are constants C q an δ q such that q(q(g 1 ) Q(g 2 )) C q g 1 g 2 for g i g 0 < δ q an i = 1, 2. Both assertions follow from Theorem 2.3,iii. By the uniform bouneness principle the set of continuous linear operators {S t } 0 t T is uniformly boune for any fixe T 0, i.e. for any seminorm p on E there is a seminorm q p such that p(s t f) q p (f) for t T an for all f E. We enote by C([0, ε], B) continuous curves on the interval [0, ε] to B, g 0 := R(f 0 ). Without any restriction we can assume that f 0 = 0 an g 0 = 0 by translations. We can then efine a mapping M : U V E C([0, ε], B) V such that M(f, h)(t) = R(S t f + t 0 S t sq(h(s))s) for t [0, ε]. Given h C([0, ε], B) such that h(t) θ for 0 t ε with {h sup t h(t) θ} V, we have p(s t f + t 0 S t s Q(h(s))s) q p (f) + ε(q p (Q(g 0 )) + C qp θ) provie θ δ qp. This can be mae smaller than θ if ε is appropriately small an U := {f E with q p (f) < η} with η appropriately small. In particular C qp ε < 1. If we assume these conitions, then M is well efine, continuous an furthermore sup t M(f, h 1 )(t) M(f, h 2 )(t) ε sup q p (Q(h 1 (t)) Q(h 2 (t))) t h 1 (t) h 2 (t) C qp ε sup t Consequently M(f,.) is a contraction in V with contraction constant boune uniformly in f U by a constant strictly smaller than 1. It follows that there is a unique h(t, f) for any f U epening continuously on f, such that M(f, h) = h by the contraction mapping theorem. We efine an obtain F l(t, f) := S t f + F l(t, f) = S t f + t 0 t 0 S t s Q(h(s, f))s S t s P (F l(s, f))s

12 12 DAMIR FILIPOVIĆ AND JOSEF TEICHMANN since R(F l(t, f)) = h(t, f) by construction. Any solution of the initial value problem is therefore unique by the Banach contraction principle. By inuction with Theorem 2.3, i.), smoothness with respect to time is easily establishe. For the first erivative one can calculate the limits irectly. Concerning smoothness with respect to the initial value, we procee in the following way. We show that there exist irectional erivatives an calculate them. By Taylor s formula we obtain P (f 1 ) P (f 2 ) = L(f 1, f 0 ) (f 1 f 2 ) where L(x 1, x 2 ) h := 1 0 DP (x 0 + s(x 1 x 0 )) hs is a Banach map in all three variables (see Theorem 2.3, ii.). So we can solve the system given by (f 0, f 1, h) (Af 0 + P (f 0 ), Af 1 + P (f 1 ), Ah + L(f 1, f 2 ) h) with the flow -construction from above F l(t, f 0, f 1, h) = (F l(t, f 0 ), F l(t, f 1 ), M(t, f 0, f 1, h)), smooth in time an continuous in initial values, where the epenence on h is homogenous, so the flow can be efine everywhere in h. By uniqueness of the flow the ientity t (F l(t, f 0) F l(t, f 1 )) = A(F l(t, f 0 ) F l(t, f 1 )) + L(F l(t, f 0 ), F l(t, f 1 )) (F l(t, f 0 ) F l(t, f 1 )) leas to M(t, f 0, f 1, f 0 f 1 ) = F l(t, f 0 ) F l(t, f 1 ). By homogenity in h we obtain the existence of the irectional erivatives an its continuity in point an irection at the omain of efinition, so the solution is Gateaux-C 1, by inuction we can procee since we can write own an initial value problem for the erivative DF l t which has the same form as the treate equation on an extene phase space. 3. Submanifols an Weak Foliations in Fréchet Spaces We are intereste in the geometry generate by a finite number of vector fiels given on an open subset of a Fréchet space E. Therefore we nee the notions of finite-imensional submanifols (with bounary) of a Fréchet space (see [16] for all etails an more). Here an subsequent E enotes a Fréchet space. A chart on a set M is a bijective mapping u : U u(u) E U, where E U is a Fréchet space an U M, u(u) E U is open. We shall enote a chart by (U, u) or (u, u(u)). For two charts (U α, u α ), (U β, u β ) the chart changing are given by u αβ := u α u 1 β : u β (U αβ ) u α (U αβ ), where U αβ := U α U β. An atlas is a collection of charts such that the U α form a cover of M an the chart changings are efine on open subsets of the respective Fréchet spaces. A C -atlas is an atlas with smooth chart changings. Two C -atlases are equivalent is their union is an C -atlas. A maximal C -atlas is calle a C -structure on M (maximal is unerstoo with respect to some carefully chosen universe of sets). A (smooth) manifol is a set together with a C -structure. A smooth mapping F : M N between smooth manifols is efine in the canonical way, i.e. for any m M there is a chart (V, v) with F (m) V, a chart (U, u) of M with m U an F (U) V, such that v F u 1 is smooth. This is

13 EXISTENCE OF INVARIANT MANIFOLDS FOR STOCHASTIC EQUATIONS 13 the case if an only if F c is smooth for all smooth curves c : R M, where the concept of a smooth curve is easily set upon. The final topology with respect to smooth curves or equivalently the final topology with respect to all inverses of chart mappings is the canonical topology of the smooth manifol. We assume manifols to be smoothly Hausorff (see the iscussion in [16], p. 265), i.e. the real value smooth functions on M separate points. A submanifol N of a Fréchet manifol M is given by a subset N M, such that for each n N there is a chart (u, u(u)), a splitting E = E E an u(u) = V W with u(n) = V {u(n) }. By a splitting we shall always unerstan E an E as close subspaces of E. An n-imensional manifol with bounary is efine as orinary manifol except that we take open subsets in a halfspace R n + := {x R n with x n 0}. For the notion (without surprises) of smooth mappings on such open sets see any textbook on ifferential geometry, for example [17]. The bounary {x R n with x n = 0} of the subspace moels the bounary N of the manifol N, which is canonically a manifol without bounary of imension n 1. We enote the interior by N := N \ N. A submanifol with bounary is given by the analogue submanifol structure. We restrict ourselves to finite-imensional submanifols with bounary M of Fréchet spaces: A parametrization of M is an injective, smooth mapping φ : U R n + E such that φ(u) M is open in M an Dφ(u) is injective for all u U. In this case φ(u) naturally is a submanifol with bounary again. Given a finite imensional submanifol with bounary M, then the map u 1 V u(n) : V E is a parametrization of M. The tangent space T m M of a finite imensional submanifol with bounary is efine by parametrizations: given a parametrization φ of M with m φ(u), then T φ(u) M := Dφ(u)(R n ) for u U. The tangent space T r M is certainly inepenent of the chosen parametrization, since it is equally given at interior points by the space of all vectors c (0) with c : R E smooth, c(r) M an c(0) = r by the submanifol property. Therefore a smooth map F : M N, where M an N are submanifols efines a linear map T r F : T r M T F (r) N via c (0) (F c) (0), which is given through DF (r) c (0). Lemma 3.1 (Submanifols by Parametrization). Let E be a Fréchet space an φ : U R n + E a smooth immersion, i.e. for u U the map Dφ(u) is injective, then for any u 0 U there is a small open neighborhoo V of u 0 such that φ(v ) is a submanifol with bounary of E an φ V is a parametrization. Proof. We assume by translation φ(u 0 ) = 0, since it is a local result. Given a linear basis e 1,..., e n of R n, we get linearly inepenent vectors Dφ(u 0 )(e i ) =: f i E. We choose l 1,..., l m linearly inepenent linear functionals, such that l i (f j ) = δ ij an get a splitting E = E E with im E = n via E := m i=1 ker l i. The projection on the first variable p 1 inuces a local iffeomorphism p 1 φ on a small open neighborhoo V of u 0 U by the classical inverse function theorem an the extension result in Theorem 2.7. The inverse is enote by ψ : V E V. Now we construct a new iffeomorphism η(u, f ) = (p 1 φ(u), f + p 2 φ(u)) on V W, which is invertible by the above consierations: η 1 (g, g ) = (ψ(g ), g p 2 φ(ψ(g ))),

14 14 DAMIR FILIPOVIĆ AND JOSEF TEICHMANN η 1 efines a submanifol chart for φ(v ) since for u V by efinition. η 1 (φ(u)) = (u, 0) Definition 3.2. A vector fiel X on a open subset U E of a Fréchet space is a smooth map X : U E. The set of all vector fiels on U is enote by X(U). Given a iffeomorphism F : U V, i.e. F an F 1 are smooth, the map (F Y )(f) := DF (f) 1 (Y (F (f))) is well efine for Y X(V ) an efines a boune linear isomorphism F : X(V ) X(U) by cartesian closeness (see Theorem 2.5). It is calle the pull-back of vector fiels, furthermore F := (F ) 1 is calle the push forwar. The Lie bracket of two vector fiels X, Y X(U) is efine by the following formula: [X, Y ](f) = DX(f) Y (f) DY (f) X(f) an is a boune, skew-symmetric bilinear map from X(U) X(U) into X(U). We can treat the pull back as in finite imensional analysis ue to convenient calculus. In the Gateaux approach we are force to formulate each of these results by point evaluations. Nevertheless it is natural to talk of analytic properties of the objects themselves. Proposition 3.3. Let U E be an open subset. Given two vector fiels X, Y X(U), where X amits a local flow F l X : I U E, then [X, Y ] = t (F lx t) Y t=0. Furthermore for any iffeomorphisms F : U V, G : V W an F [X, Y ] = [F X, F Y ] (G F ) = F G, (G F ) = G F. Consequently the pull back is a boune Lie algebra isomorphism, since vector fiels constitute a Lie algebra with the Lie bracket. Finally we obtain the useful formula for a smooth map H : S U, where S E is open: t F F lx t H = (F X)(F F l X t H), where we only assume that X generates a semiflow F l X : I U E. Proof. (see [16], 32.15) We can calculate irectly with the flow F l X for the vector fiel X t (F lx t) Y (f) t=0 = t DF lx t (F l X t(f)) Y (F l X t(f)) t=0 = DX(f) Y (f) D 2 F l X 0 (f)(x(f), Y (f)) DY (f) X(f) = [X, Y ](f) for f U. We applie the flow property (F l t ) 1 = F l t for small t an the chain rule of convenient analysis. The interchange of t an D is possible ue

15 EXISTENCE OF INVARIANT MANIFOLDS FOR STOCHASTIC EQUATIONS 15 to the symmetry of secon erivatives. The two following properties are clear by calculating both sies irectly. The last equation can is prove by t F F lx t (H(f)) = DF ( F l X t (H(f)) ) X(F l X t (H(f))) = (F X) (F (F l X t (H(f)))) for f U by the efinition of the push forwar. The following crucial lemma collects algebraic properties of Banach map vector fiels. Lemma 3.4. Let U be an open set in a Fréchet space E, then B(U) is a subalgebra with respect to the Lie bracket. Let A be a boune linear operator on E, then [A, B(U)] B(U). Consequently the Lie algebra L(E) acts on B(U) by the Lie bracket. Proof. Given two Banach maps P 1 an P 2, DP 1 (f) P 2 (f) = DQ 1 (R 1 (f)) DR 1 (f) P 2 (f) hols, which can be written as composition of DQ 1 (v) w for v, w B an (R 1 (f), DR 1 (f) P 2 (f)) for f U. So the Lie bracket lies in B(U). Given A L(E), we see that AP 1 (f) DP 1 (f) Af is a Banach map by an obvious ecomposition. We enote by... the generate vector space over the reals R. which means that D f is vector space generate by the set S of local vector fiels at f U: Definition 3.5. Let E be a Fréchet space, U an open subset. A istribution on U is a collection of vector subspaces D = {D f } f U of E. A vector fiel X X(U) is sai to take values in D if X(f) D(f) for f U. A istribution D on U is sai to be involutive if for any two locally given vector fiels X, Y with values in D the Lie bracket [X, Y ] has values in D. A istribution is sai to have constant rank if im R D f is locally constant f U. A istribution is calle smooth if there is a set S of local vector fiels on U such that D f = {X(f) (X : U X E) S an f U X }. We say that the istribution amits local frames on U if for any f U there is an open neighborhoo f V U an n smooth, pointwise linearly inepenent vector fiels X 1,..., X n on V with for g V. X 1 (g),..., X n (g) = D g Remark 3.6. Given a istribution D on U generate by a set of local vector fiels S, such that the imensions of D f are boune by a fixe constant N. Let f U be a point with maximal imension n f = im R D f, then there are n f smooth local vector fiels X 1,..., X nf S with common omain of efinition U such that X 1 (f),..., X n (f) = D f. Choosing n f continuous linear functionals l 1,..., l nf E with l i (X j (f)) = δ ij, then the continuous mapping M : U L(R n f ), g (l i (X j (g))) has range in the invertible matrices in a small neighborhoo of f. Consequently in this neighborhoo the imension of D g is at least n f. It follows by maximality of n f that it is exactly n f. In particular the istribution amits a local frame at f. The concept of weak foliations will be perfectly aapte to the FDR-problem:

16 16 DAMIR FILIPOVIĆ AND JOSEF TEICHMANN Definition 3.7. A weak foliation F of imension n on an open subset U of a Fréchet space E is a collection of submanifols with bounary {M r } r U such that i) For all r U we have r M r an the imension of M r is n. ii) The istribution D(F)(f) := T f M r for all r U with f M r has imension n for all f U, i.e. given f U the tangent spaces T f M r agree for all M r f. This istribution is calle the tangent istribution of F. Given any istribution D we say that D is tangent to F if D(f) D(F)(f) for all f U. Classically one is intereste in the existence of tangent weak foliations for a given istribution of minimal imension m. Therefore we shall nee the following essential lemma. Proposition 3.8. Let D be an involutive, smooth istribution of constant rank n on an open subset U of a Fréchet space E. Let X an Y be vector fiels with values in D an let X amit a local flow, then for f U, where it is efine. (F l X t ) (Y )(f) D f Proof. Given a local frame X 1,..., X n on an open neighborhoo V of f 0, we have by involutivity that [X, X i ] = n k=1 P i kx k. Notice that Pi k are smooth functions locally on V. Given g V an n linear inepenent functionals l m such that l m (X j (g)) = δ mj, then n l m ([X, X i ](f)) = Pi k (f)l m (X k (f)) k=1 for all f V. Since the matrix M(f) := (l m (X k (f))) is invertible at g an has smooth entries, it is invertible on an open neighborhoo of g, an the inverse has smooth entries. The smooth inverse matrix applie to the left han vector proves smoothness of Pi k. With the above formula an Lemma 3.3 we get t (F lx t ) (X i ) = s (F lx t s) (X i ) s=0 = s (F lx s) (F l X t ) (X i ) s=0 = [X, (F l X t ) (X i )] = (F lt X ) [X, X i ] n = Pi k F lt X (F lt X ) (X k ) k=1 which is a linear equation with time-epenent real value coefficients gi k (t) := Pi k(f lx t (f)) on E n for h i (t) := (F lt X ) (X i )(f) at any point f in an open neighborhoo of f 0, namely t h i(t) = n gi k (t)h k (t) k=1

17 EXISTENCE OF INVARIANT MANIFOLDS FOR STOCHASTIC EQUATIONS 17 with h i (0) E. The solution of this ifferential equation is given by the classical time epenent flow associate to the smooth matrix t (gi k (t)) applie to a vector in E n. If we are given a flow for a vector fiel the solutions are unique ue to Lemma Consequently provie the initial values lie in Df n the solution lies in Df n for small times by the subspace property, but (F lx 0 ) (X i )(f) = X i (f) D f for f U. Theorem 3.9. Let D be an smooth istribution of constant rank n on an open subset U of a Fréchet space E. Assume that for any point f 0 the istribution amits a local frame of vector fiels X 1,..., X n, where X 1,..., X n 1 amit local flows F lt Xi an X n amits a local semiflow F lt Xn. Then D is involutive if an only if it is tangent to an n-imensional weak foliation. Proof. We suppose that D is involutive. Let f 0 U be fixe, then there are on some open set V, with f 0 V, n linearly inepenent vector fiels X 1,..., X n generating each D f for f V. Furthermore local flows F l Xi :] ε, ε[ V 0 V for i = 1,..., n 1 an a local semiflow F l Xn : [0, ε[ V 0 V exist for V 0 V open with f 0 V 0 an some ε > 0. We efine the caniate parametrization α(u, r) = F lu X F lu Xn n (r) on W 1 V 1, where V 1 open with f 0 V 1 an W R n + open, convex aroun 0. This is possible ue to continuity of the (semi-)flows. We can calculate the tangent spaces on the canonical basis of R n : by cartesian closeness we obtain the erivative of F lu X F lu Xn n with respect to u i immeiately by Proposition 3.3 ( ) F lu X1 u 1... F lu Xn n = (F lu X1 1 )...(F lu Xi 1 i 1 ) X i F lu X F lu Xn n i with F = F l X1 u 1... F l Xi 1 u i 1 D 1 α(u, r)(e i ) = an H = F l Xi+1 u i+1... F l Xn u n. So we arrive at ( (F l X1 u 1 )...(F l Xi 1 u i 1 ) X i ) (α(u, r)) for 1 i n. By Proposition 3.8 these vectors lie in D α(u,r) an are linearly inepenent for u W 2 an r V 2 with W 2 W 1 open, convex aroun 0 an V 2 V 1 open aroun f 0. They generate the istribution in a small neighborhoo by a imension argument. It is essential that the first n 1 vector fiels amit a local flow. So we obtain a family of tangential manifols for D. Each parametrization for fixe r efines locally a smooth submanifol with bounary α(u 1,.., u n 1, 0, r) by reoing the proof of Lemma 3.1 with continuously parametrize immersions. Consequently we can fin an open set V 2 E an W 2 R n + such that α W2 V 2 efines a weak foliation with tangent istribution D. Suppose now that there is a weak foliation F = {M r } r U of imension n. We apply the above notation on a subset V, where we have a local frame X 1,..., X n with the state properties. Given r V, there exists a finite imensional submanifol with bounary M r an X i (f) T f M r for 1 i n at any interior point f M r. By Lemma 2.11 the local flows F lt Xi restrict locally to M r for 1 i n 1 since the vector fiels X i are tangent to M r an amit local flows aroun any interior point of M r. So for small t an Y X(U) with values in D the pull back (F lt Xi ) Y (f) takes values in D f for f in the interior of M r, since it can be calculate as pull back of the restriction F lt Xi Mr. The smooth map t (F lt Xi ) Y (f) takes values in the finite

18 18 DAMIR FILIPOVIĆ AND JOSEF TEICHMANN imensional space D f, so the erivative lies there by closeness, but the erivative equals [X i, Y ](f) by Proposition 3.3. We o not know whether r M r lies on the bounary of M r or not, but we can approximate r by interior points f m r as m. At r the vector space D r has a basis X 1 (r),..., X n (r): given n linearly inepenent linear functionals l1, r..., ln r with li r(x j(r)) = δ ij. We can choose V sufficiently small such that the smooth matrix f M(f) := (l i (X j (f)) is invertible for f V, hence the inverse matrix N(f) := M(f) 1 efines linear functionals l f i := n k=1 N(f) iklk r for 1 i n, which epen smoothly on f an satisfy l f i (X j(f)) = δ ij for f V. The associate projections p f := i n k=1 lf k X k(f) etect whether a vector fiel on V takes values in D f or not: for Z X(V ) we obviously have p f (Z(f)) = 0 if an only if Z(f) D f, for f V. However, p fm ([X i, Y ](f m )) = 0 for m 1 as calculate above, so by continuity p r ([X i, Y ](r)) = 0 for 1 i n 1. Hence [X i, X j ] takes values in D locally for 1 i, j n an D is therefore involutive since we can ue the proceure everywhere on U. Remark For etails on Frobenius theorems in the classical setting see [15]. The phenomenon that there is no Frobenius chart is ue to the fact that there is one vector fiel among the vector fiels X 1,...,X n (generating the istribution D) amitting only a local semiflow. If all of them amitte flows, there woul exist a Frobenius chart, which can be given by a construction outline in [23]. The nonexistence of a Frobenius-chart means that the leafs cannot be parallelize, since they follow semiflows, which means that gaps between two leafs can occur an leafs can touch. This is an infinite imensional phenomenon, which oes not appear in finite imensions. 4. Finite-imensional Realizations for HJM Moels In this section we apply the preceing results to characterize those HJM moels that satisfy the appropriate Frobenius conition (see conition (F) below), which is essentially equivalent to the existence of FDRs at any inital curve. We will emonstrate that this conition yiels a very particular geometry of the invariant submanifols loosely speaking, each of them is a ban of copies of an affine submanifol. Remark 4.1. Although we subsequently focus on HJM moels, many arguments can be carrie over to more general stochastic equations (1.1) in the spirit of Da Prato an Zabczyk [5]. First we provie the rigorous setup for HJM moels, summarizing [8]. The Hilbert space H of forwar curves is characterize by the properties (H1): H C(R 0 ; R) with continuous embeing (that is, for every x R 0, the pointwise evaluation ev x : h h(x) is a continuous linear functional on H), an 1 H (the constant function 1). (H2): The family of right-shifts, S t f = f(t + ), for t R 0, forms a strongly continuous semigroup S on H. (H3): There exists a close subspace H 0 of H such that S(f, g)(x) := f(x) x 0 g(η) η, efines a continuous bilinear mapping S : H 0 H 0 H.

19 EXISTENCE OF INVARIANT MANIFOLDS FOR STOCHASTIC EQUATIONS 19 We write shortly S(f) for S(f, f). We assume that the volatility coefficients σ j map H into H 0. Then the HJM rift coefficient α = α HJM := S(σ j ) : H H (4.1) is a well-efine map. Hence an HJM moel is uniquely etermine by the specification of its volatility structure σ = (σ 1,..., σ ). As an illustration we shall always have the following example in min (see [8, Section 5]). Example 4.2. Let w : R 0 [1, ) be a non-ecreasing C 1 -function such that w 1/3 L 1 (R 0 ). We may think of w(x) = e αx or w(x) = (1 + x) α, for α > 0 or α > 3, respectively. The space H w consisting of absolutely continuous functions h on R 0 an equippe with the norm 2 h 2 w := h(0) 2 + x h(x) w(x) x R 0 is a Hilbert space satisfying (H1) (H2). Property (H3) is satisfie for H 0 = H w,0 := {h H w lim x h(x) = 0}. The operator A is the generator of the shift semigroup S. It is easy to see that D(A) {h H C 1 (R 0 ; R) (/x)h H} an Ah = (/x)h. Without much loss of generality we shall in fact assume (H4): D(A) = {h H C 1 (R 0 ; R) (/x)h H}. Also (H4) is satisfie for the spaces H w from Example 4.2. Denote by A 0 : D(A 0 ) H 0 H 0 the restriction of A to H 0. That is, D(A 0 ) = {h D(A) H 0 Ah H 0 }. The efinition of the Fréchet space D(A 0 ) := n N D(A n 0 ) is obvious. The next result follows immeiately from (H1), (H3) an (H4). Lemma 4.3. For any f, g D(A 0 ) we have S(f, g) D(A) an AS(f, g) = S(Af, g) + S(f, Ag) + f ev 0 (g). Hence S : D(A 0 ) D(A 0 ) D(A ) is a continuous bilinear mapping. The preceing specifications for σ are still too general for concrete implementations. We actually have the iea of σ being sensitive with respect to functionals of the forwar curve. That is, σ j (h) = φ j (l 1 (h),..., l p (h)), for some p 1, where φ j : R p D(A 0 ) is a smooth map an l 1,..., l p enote continuous linear functionals on H (or even on C(R 0 ; R)). We may think of l i (h) = (1/x i ) x i h(η) η 0 (benchmark yiels) or l i (h) = ev xi (h) (benchmark forwar rates). This iea is (generalize an) expresse in terms of the following regularity an non-egeneracy assumptions: (A1): σ j = φ j l where l L(H, R p ), for some p N, an φ j : R p D(A 0 ) are smooth an pointwise linearly inepenent maps, 1 j. Hence σ : H D(A 0 ) is a Banach map (see Definition 2.8). (A2): For every q 0, the linear map (l, l A,..., l A q ) : D(A ) R p(q+1) is open. (A3): A is unboune; that is, D(A) is a strict subset of H. Equivalently, A : D(A ) D(A ) is not a Banach map (see Lemma 2.12).

20 20 DAMIR FILIPOVIĆ AND JOSEF TEICHMANN We believe that this setup is flexible enough to capture any reasonable HJM moel. Assumption (A2) is essential for the strong characterization result in Theorem 4.10 below. Intuitively, (A2) says that the following interpolation problem is well-pose on D(A ): given a smooth curve g : R 0 R, for any finite number of ata of the form I = (l(g), l((/x)g),..., l((/x) q g)) R p(q+1) we can fin an interpolating function h D(A ) with (l(h),..., l A q (h)) = I. Notice, however, that egenerate examples such as the following are exclue: let p = 3 an l(h) = (ev 0 (h), ev 1 (h), 1 0 h(x) x). Then l A(h) = (ev 0(Ah), ev 1 (Ah), ev 1 (h) ev 0 (h)). Thus the rank of (l, l A) is at most 5, an (l, l A) : D(A ) R 6 cannot be an open map. Combining Lemma 4.3 an (A1) yiels Lemma 4.4. S(σ j ) : H D(A ) is a Banach map, for every 1 j, hence also α HJM. By the iscussion after the proof of Lemma 2.11, the semigroup S leaves D(A ) invariant an is smooth on D(A ). Hence by (A1) an Lemma 4.4 the assumptions of Theorem 2.13 are satisfie, an the vector fiel h µ(h) = Ah + α HJM (h) (1/2) Dσ j(h)σ j (h), see (1.2), amits a local semiflow on D(A ). Lemma 4.5. Let X 1,..., X k be linearly inepenent Banach maps on an open set U in D(A ), for some k N. Then the set is close an nowhere ense in U. N = {h U µ(h) X 1 (h),..., X k (h) } Proof. Clearly, N is close by continuity of µ an X 1,..., X k. Now suppose there exists a set V N which is open in D(A ). For every h V there exist unique numbers c 1 (h),..., c k (h) such that µ(h) = k c j (h)x j (h). (4.2) We can choose linear functionals ξ 1,..., ξ k on D(A ) such that the k k-matrix M ij (h) := ξ i (X j (h)) is smooth an invertible on V (otherwise we choose a smaller open subset V ). Hence c i (h) = k M 1 ij (h)ξ j(µ(h)) are smooth functions on V. Then (4.2) implies that A is a Banach map on V. But this contraicts (A3), whence the claim. The vector fiels µ, σ 1,..., σ inuce two istributions on D(A ): their linear span D = µ, σ 1,..., σ, an the Lie algebra D LA generate by all multiple Lie brackets of these vector fiels. As a consequence of (A1) an Lemma 4.5 there exists a close an nowhere ense set N in D(A ) such that im D LA im D = + 1 on D(A ) \ N. (4.3) Remark 4.6. The preceing observation proves a conjecture in [3], namely that every nontrivial generic short rate moel is of imension 2 (see [3, Remark 7.1]).