Existence of Lévy term structure models
EEXISTENCE OF L´EVY TERM STRUCTURE MODELS
DAMIR FILIPOVI´C AND STEFAN TAPPE
Abstract.
L´evy driven term structure models have become an importantsubject in the mathematical finance literature. This paper provides a compre-hensive analysis of the L´evy driven Heath–Jarrow–Morton type term structureequation. This includes a full proof of existence and uniqueness in particular,which seems to have been lacking in the finance literature so far.
Key Words: forward curve spaces; L´evy term structure models, stochas-tic integration in Hilbert spaces; strong, weak and mild solutions of infinitedimensional SDE’s. Introduction
A zero coupon bond with maturity T is a financial asset which pays the holderone unit of cash at T . Its price at t ≤ T can be written as P ( t, T ) = exp (cid:32) − (cid:90) Tt f ( t, u ) du (cid:33) where f ( t, T ) is the forward rate for date T . The classical continuous frameworkfor the evolution of the forward rates goes back to Heath, Jarrow and Morton(HJM) [32]. They assume that, under the risk-neutral measure, for every date T ,the forward rates f ( t, T ) follow an Itˆo process of the form df ( t, T ) = (cid:18) n (cid:88) i =1 σ i ( t, T ) (cid:90) Tt σ i ( t, s ) ds (cid:19) dt + n (cid:88) i =1 σ i ( t, T ) dW it , t ∈ [0 , T ] , (1.1)where W = ( W , . . . , W n ) is a standard Brownian motion in R n . The dynamics(1.1) guarantee that the discounted zero coupon bond price processes e − (cid:82) t f ( s,s ) ds P ( t, T ) , t ∈ [0 , T ] , are local martingales for all maturities T . This is the well known condition for theabsence of arbitrage in the bond market model.Empirical studies have revealed that models based on Brownian motion onlyprovide a poor fit to observed market data. We refer to [51, Chap. 5], where it isargued that empirically observed log returns of zero coupon bonds are not normallydistributed, a fact, which has long before been known for the distributions of stockreturns. Bj¨ork et al. [5, 6], Eberlein et al. [22, 21, 16, 19, 20, 18] and others ([55,37, 33]) thus proposed to replace the classical Brownian motion W in (1.1) by amore general process X = ( X , . . . , X n ), also taking into account the occurrenceof jumps. If X is a L´evy process, this leads to df ( t, T ) = α HJM ( t, T ) dt + n (cid:88) i =1 σ i ( t, T ) dX it , t ∈ [0 , T ] . (1.2) Key words and phrases. a r X i v : . [ q -f i n . M F ] J u l DAMIR FILIPOVI´C AND STEFAN TAPPE
The HJM drift in (1.1) accordingly is replaced by some appropriate α HJM ( t, T ),which is determined by σ ( t, T ) and the cumulant generating function of X , see(2.4) below.Equation (1.2) constitutes a generic description of the forward rate process f ( t, T, ω ) in terms of a stochastic volatility process σ ( t, T, ω ). From a financial mod-elling point of view one would rather consider σ ( t, T, ω ), and thus α HJM ( t, T, ω ), tobe a function of the prevailing forward curve T (cid:55)→ f ( t − , T, ω ) = lim s ↑ t f ( s, T, ω ),that is σ ( t, T, ω ) = σ ( t, T, f ( t − , · , ω )) , α HJM ( t, T, ω ) = α HJM ( t, T, f ( t, · , ω )) . This makes f ( t, T ) being a solution of the stochastic equation (cid:40) df ( t, T ) = α HJM ( t, T, f ( t, · )) dt + (cid:80) ni =1 σ i ( t, T, f ( t − , · )) dX it , t ∈ [0 , T ] ,f (0 , T ) = h ( T )(1.3)for some given initial forward curve h ( T ). Term structure models of the type (1.3) are frequently considered in the liter-ature. The typical assumption is that drift α HJM and volatility σ depend on thecurrent state of the short rate, σ ( t, T, ω ) = σ ( t, T, f ( t − , t, ω )), as in [38], [52], [4],[34] and [33] (the latter studies models driven by jump-diffusions). A model, wherethe volatility σ is allowed to depend on a finite number of benchmark forward rates,is considered in [11] and [12]. We emphasize that these papers, whose setups arespecial cases of our present framework, assume that the forward rates f ( t, T ) evolveaccording to an equation of the kind (1.3). To our knowledge, there has not beenyet an explicit proof for the existence of a solution to (1.3) in the mathematicalfinance literature. We thus provide such a proof in our paper (Theorem 4.6 andCorollary 4.7).Note that (1.3) is an infinite-dimensional and therefore non-trivial problem. Infact, (1.3) is not simply a system of infinitely many univariate stochastic equationsfor f ( t, T ), t ∈ [0 , T ], indexed by T . Indeed, these equations are coupled as α HJM and σ depend on the entire forward curve f ( t − , · ), say e.g. on the short rate f ( t − , t ),which is a functional of f ( t − , · ). To express this functional dependence, one switchesbest to the alternative parametrization r t ( x ) = f ( t, t + x ) , x ≥ , which is due to Musiela [46]. We then write S t f ( x ) := f ( x + t ) for the shift operator S t . Equation (1.3) becomes in integrated form(1.4) r t ( x ) = S t h ( x ) + (cid:90) t S t − s α HJM ( s, s + x, r s ) ds + n (cid:88) i =1 (cid:90) t S t − s σ i ( s, s + x, r s − ) dX is , where S t − s operates on the functions x (cid:55)→ α HJM ( s, s + x, r s ) and x (cid:55)→ σ i ( s, s + x, r s − ). Hence, in the spirit of Da Prato and Zabczyk [13], the process r t is a socalled mild solution of the stochastic differential equation (cid:40) dr t = (cid:0) ddx r t + α HJM ( t, r t ) (cid:1) dt + (cid:80) ni =1 σ i ( t, r t − ) dX it ,r = h (1.5)in some appropriate Hilbert space H of forward curves, where ddx becomes thegenerator of the strongly continuous semigroup of shifts S t . Note the slight abuseof notation α HJM ( t, t + · , r ) (cid:32) α HJM ( t, r ) and σ i ( t, t + · , r ) (cid:32) σ i ( t, r ). In the sequel,we are therefore concerned with the L´evy HJMM (Heath–Jarrow–Morton–Musiela)equation (1.5) in various choices of the state space H . XISTENCE OF L´EVY TERM STRUCTURE MODELS 3
Several authors have dealt with the existence issue for (1.5) for the Brownianmotion case X = W . Bj¨ork and Svensson [7] chose the state space H β,γ smallenough such that ddx : H β,γ → H β,γ becomes a bounded linear operator. In thiscase, the methods from finite dimension essentially carry over to (1.5). It turns out,however, that the Bj¨ork–Svensson space H β,γ is too small and does not containsome important classical term structure models (see [25]). In [24], we thus analyzedand solved (1.5) for X = W on a larger space H w , where ddx becomes unbounded.In this paper we provide the existence proof for (1.5) for the L´evy case. Weproceed as follows. Using an existence result for general Hilbert space valued sto-chastic differential equations from the appendix, we first show existence for (1.5)in the Bj¨ork–Svensson space H β,γ . However, often it turns out that H β,γ is toosmall to assert that α HJM lies in H β,γ , even for the very simple case where σ is con-stant and the driver X is a compound Poisson process (Example 3.4). Afterwards,we thus consider (1.5) in the larger state space H w from [24] where ddx becomesunbounded.Term structure models based on infinite dimensional driving processes X arediscussed e.g. in [36] and [47] for the L´evy case. Again, in these papers it is typicallyassumed that the forward curve evolution satisfies a stochastic differential equation,but the authors do not treat existence and uniqueness of solutions.The remainder of the paper is organized as follows. In Section 2 we introducesome notation and specify the HJM drift α HJM , which ensures that the bond marketis free of arbitrage. In Section 3 we treat the existence of strong solutions to (1.5) onthe Bj¨ork–Svensson space H β,γ . Afterwards, Section 4 is devoted to the existenceof mild and weak solutions to (1.5) on the larger space H w where ddx becomesunbounded. Section 5 concludes.For our results of Section 3 and Section 4 we apply an existence result for gen-eral Hilbert space valued stochastic differential equations, which is derived in theappendix. The ground for this result, Theorem C.1, is prepared by two works ofvan Gaans [26, 27]. In addition to his result [27, Thm. 4.1] we prove that the mildsolution to (1.5) has a c`adl`ag modification, and that there exists a unique weaksolution.The c`adl`ag property of the solution is an important feature for financial applica-tions. Indeed, general arbitrage theory [15] requires that the basic financial instru-ments, here the implied zero coupon bond prices P ( t, T ) = exp( − (cid:82) T − t r t ( x ) dx ),are real semimartingales and therefore have c`adl`ag paths. This essentially requiresc`adl`ag paths of the weak solution ( r t ), which is satisfied in our framework.As it turns out, the stochastic integral of van Gaans [27] is not consistent with theusual Itˆo-integral, which is used for financial modelling. Therefore, after giving anoverview and the required notation in Appendix A, we show in Appendix B that thestochastic integral of van Gaans always has a c`adl`ag modification and analyze whenit coincides with the Itˆo-integral. Then, in Appendix C, we prove Theorem C.1, theexistence and uniqueness result for Hilbert space valued stochastic equations. Atthe end of Appendix C we give an overview of related literature.2. The HJM drift condition
Throughout this text, X , . . . , X n denote independent real-valued L´evy processeson a filtered probability space (Ω , F , ( F t ) t ≥ , P ) satisfying the usual conditions.Let H be a separable Hilbert space representing the space of forward curves andlet σ , . . . , σ n : R + × H → H be the volatilities. Recall that a L´evy term structuremodel of the form (1.5) is free of arbitrage if the probability measure P is a localmartingale measure, that is all discounted bond prices are local martingales. DAMIR FILIPOVI´C AND STEFAN TAPPE
In order to provide a condition which ensures that P is a local martingale mea-sure, we assume that there are compact intervals [ a , b ] , . . . , [ a n , b n ] having zero asan inner point such that the L´evy measures F , . . . , F n of X , . . . , X n , respectively,satisfy for i = 1 , . . . , n (cid:90) | x | > e zx F i ( dx ) < ∞ , z ∈ [ a i , b i ] . (2.1)Condition (2.1) ensures that the cumulant generating functionsΨ i ( z ) := ln E [exp( zX i )] , i = 1 , . . . , n (2.2)exist on [ a i , b i ] and that they are of class C ∞ (see [54, Lemma 26.4]). Moreover,the L´evy processes X i possess moments of arbitrary order. Let [ c i , d i ] ⊂ ( a i , b i ) befurther compact intervals having zero as an inner point.For any continuous function h : R + → R we define T h : R + → R as T h ( x ) := (cid:90) x h ( η ) dη. (2.3)For i = 1 , . . . , n denote A Ψ i H := { h ∈ H : − T h ( R + ) ⊂ [ c i , d i ] } . Provided σ i ( R + × H ) ⊂ A Ψ i H for i = 1 , . . . , n , the HJM drift (2.4) α HJM ( t, r )( x ) = n (cid:88) i =1 ddx Ψ i (cid:18) − (cid:90) x σ i ( t, r )( η ) dη (cid:19) = − n (cid:88) i =1 σ i ( t, r )( x )Ψ (cid:48) i (cid:18) − (cid:90) x σ i ( t, r )( η ) dη (cid:19) is well defined pointwise for all x . The HJM drift condition (2.4) implies that P isa local martingale measure. It is derived in [21, Sec. 2.1] for the present L´evy case,using the results of the more general setup in [5]. For an analogous drift conditionin the infinite dimensional L´evy setting, see [36].The HJM drift specification (2.4) causes some problems for an immediate ap-plication of Theorem C.1. First of all, we have to ensure that α HJM ( t, r ) ∈ H forall ( t, r ) ∈ R + × H . Furthermore, we have to establish for an application of Theo-rem C.1 that for Lipschitz functions σ , . . . , σ n the drift α HJM is again a Lipschitzfunction.These demandings emphasize that we have to be careful about the choice of thespace H of forward curves. Another desirable feature of H is that for every x ∈ R + the point evaluation h (cid:55)→ h ( x ) : H → R is a continuous linear functional. Becausethen the variation of constants formula (1.4) is satisfied for all x ∈ R + , whenever( r t ) is a mild solution of (1.5).In the upcoming Section 3 we deal with the existence of strong solutions to (1.5),and Section 4 is devoted to the existence of mild and weak solutions to (1.5).3. Forward curve evolutions as strong solutions of infinitedimensional stochastic differential equations
In this section, where we deal with the existence of strong solutions to (1.5), weconsider the spaces H β,γ of forward curves, which have been used by Bj¨ork andSvensson in [7]. XISTENCE OF L´EVY TERM STRUCTURE MODELS 5
We fix real numbers β > γ >
0. Let H β,γ be the linear space of all h ∈ C ∞ ( R + ; R ) satisfying ∞ (cid:88) n =0 (cid:18) β (cid:19) n (cid:90) ∞ (cid:18) d n h ( x ) dx n (cid:19) e − γx dx < ∞ , We define the inner product (cid:104) g, h (cid:105) β,γ := ∞ (cid:88) n =0 (cid:18) β (cid:19) n (cid:90) ∞ (cid:18) d n g ( x ) dx n (cid:19) (cid:18) d n h ( x ) dx n (cid:19) e − γx dx and denote the corresponding norm by (cid:107) · (cid:107) β,γ .3.1. Proposition.
The space ( H β,γ , (cid:104)· , ·(cid:105) β,γ ) is a separable Hilbert space and foreach x ∈ R + , the point evaluation h (cid:55)→ h ( x ) : H β,γ → R is a continuous linearfunctional.Proof. This is a consequence of [7, Prop. 4.2]. (cid:3)
The fact that each point evaluation is a continuous linear functional ensures thatforward curves ( r t ) solving (1.5) satisfy the variation of constants formula (1.4).3.2. Proposition.
We have ddx ∈ L ( H β,γ ) , i.e. ddx is a bounded linear operator on H β,γ .Proof. The assertion is a consequence of [7, Prop. 4.2]. (cid:3)
Theorem.
Let σ i : R + × H β,γ → H β,γ be continuous and satisfying σ i ( R + × H β,γ ) ⊂ A Ψ i H β,γ for i = 1 , . . . , n . Assume that α HJM ( t, r ) ∈ H β,γ for all ( t, r ) ∈ R + × H β,γ . Furthermore, assume that α HJM ( t, r ) : R + × H β,γ → H β,γ is continuousand that there is a constant L ≥ such that for all t ∈ R + and h , h ∈ H β,γ wehave (cid:107) α HJM ( t, h ) − α HJM ( t, h ) (cid:107) β,γ ≤ L (cid:107) h − h (cid:107) β,γ , (cid:107) σ i ( t, h ) − σ i ( t, h ) (cid:107) β,γ ≤ L (cid:107) h − h (cid:107) β,γ , i = 1 , . . . , n. Then, for each h ∈ H β,γ , there exists a unique strong adapted c`adl`ag solution ( r t ) t ≥ to (1.5) with r = h satisfying E (cid:20) sup t ∈ [0 ,T ] (cid:107) r t (cid:107) β,γ (cid:21) < ∞ for all T > . (3.1) Proof.
Taking into account Proposition 3.2, the result is a consequence of CorollaryC.2. (cid:3)
Unfortunately, Theorem 3.3 has some shortcomings, namely it is demanded thatthe drift term α HJM according to the HJM drift condition (2.4) maps again into thespace H β,γ . The following simple counter example shows that this condition maybe violated.3.4. Example.
Let σ = − and X be a compound Poisson process with intensity λ = 1 and jump size distribution N (0 , . Notice that the compound Poisson processsatisfies the exponential moments condition (2.1) for all z ∈ R , because its L´evymeasure is given by F ( dx ) = 1 √ π e − x dx. DAMIR FILIPOVI´C AND STEFAN TAPPE
But we have α HJM / ∈ H β,γ , because (cid:90) ∞ α HJM ( x ) e − γx dx = (cid:90) ∞ (cid:18) ddx Ψ( x ) (cid:19) e − γx dx = (cid:90) ∞ (cid:18) ddx (cid:16) e x − (cid:17)(cid:19) e − γx dx = (cid:90) ∞ x e x − γx dx = ∞ . The phenomena that the drift α HJM may be located outside the space of forwardcurves H β,γ has to do with the fact that the space H β,γ is a very small space in asense, in particular, every function must necessarily be real-analytic (see [7, Prop.4.2]).The small size of this space arises from the requirement that ddx should be abounded operator, because we deal with the existence of strong solutions. Whendealing with mild and weak solutions in the next Section 4, problems of this kindwill not occur.Nevertheless, for certain types of term structure models, we can apply Theorem3.3. For this purpose, we proceed with a lemma. For a given real-analytic function h : R + → R it is, in general, difficult to decide whether h belongs to H β,γ or not.For the following functions this can be provided.3.5. Lemma.
Every polynomial p belongs to H β,γ , and for δ ∈ R satisfying δ < β and δ < γ , the function h ( x ) = e δx belongs to H β,γ .Proof. The first statement is clear. For h ( x ) = e δx we obtain ∞ (cid:88) n =0 (cid:18) β (cid:19) n (cid:90) ∞ (cid:18) d n h ( x ) dx n (cid:19) e − γx dx = ∞ (cid:88) n =0 (cid:18) β (cid:19) n (cid:90) ∞ (cid:0) δ n e δx (cid:1) e − γx dx = ∞ (cid:88) n =0 (cid:18) δ β (cid:19) n (cid:90) ∞ e − ( γ − δ ) x dx = 11 − δ β · γ − δ = β ( β − δ )( γ − δ ) , whence h ∈ H β,γ . (cid:3) Let n = 3, that is we have three independent driving processes. We denote by X , X two standard Wiener processes, and X is a Poisson process with intensity λ >
0. We specify the volatilities as σ ( r )( x ) = ϕ ( r ) p ( x ) , σ ( r )( x ) = ϕ ( r ) e δx and σ ( r )( x ) = − η, where p is a polynomial, δ, η ∈ R satisfy 4 δ < β , δ < γ and η < β , η < γ , andwhere ϕ i : H γ,β → R for i = 1 ,
2. Note that σ i ( H β,γ ) ⊂ H β,γ for i = 1 , , α HJM ( r )( x ) = ddx (cid:34) ϕ ( r ) q ( x ) + 12 ϕ ( r ) (cid:18) e δx − δ (cid:19) + λ ( e ηx − (cid:35) , where q ( x ) = (cid:82) x p ( η ) dη is again a polynomial. From Lemma 3.5 and Proposition3.2 we infer α HJM ( H β,γ ) ⊂ H β,γ .3.6. Proposition.
Assume there is a constant L ≥ such that for all h , h ∈ H β,γ we have | ϕ i ( h ) − ϕ i ( h ) | ≤ L (cid:107) h − h (cid:107) β,γ , i = 1 , | ϕ i ( h ) − ϕ i ( h ) | ≤ L (cid:107) h − h (cid:107) β,γ , i = 1 , . Then, for each h ∈ H β,γ , there exists a unique strong adapted c`adl`ag solution ( r t ) t ≥ to (1.5) with r = h satisfying (3.1). XISTENCE OF L´EVY TERM STRUCTURE MODELS 7
Proof.
We have for all h , h ∈ H β,γ (cid:107) σ ( h ) − σ ( h ) (cid:107) ≤ L (cid:107) p (cid:107) β,γ (cid:107) h − h (cid:107) β,γ , (cid:107) σ ( h ) − σ ( h ) (cid:107) ≤ L (cid:107) e δ • (cid:107) β,γ (cid:107) h − h (cid:107) β,γ . Using Proposition 3.2, we obtain for all h , h ∈ H β,γ (cid:107) α HJM ( h ) − α HJM ( h ) (cid:107) β,γ ≤ L (cid:107) A (cid:107) L ( H β,γ ) (cid:0) (cid:107) q (cid:107) β,γ + (cid:107) δ ( e δ • − (cid:107) β,γ (cid:1) (cid:107) h − h (cid:107) β,γ . Applying Theorem 3.3 completes the proof. (cid:3)
In order to generalize Proposition 3.6, by allowing that η may depend on thepresent state of the forward curve, we prepare two auxiliary results.3.7. Lemma.
Let γ > and g, h ∈ C ( R + ; R ) . Assume there are c > , ε ∈ ( −∞ , γ ) and x ∈ R + such that | g ( x ) h ( x ) | ≤ ce εx for all x ≥ x .Then we have (cid:90) ∞ g ( x ) h ( x ) e − γx dx = 1 γ (cid:20) g (0) h (0) + (cid:90) ∞ g (cid:48) ( x ) h ( x ) e − γx dx + (cid:90) ∞ g ( x ) h (cid:48) ( x ) e − γx dx (cid:21) . Proof.
Performing partial integration with three factors, we obtain (cid:104) g ( x ) h ( x ) e − γx (cid:105) ∞ = (cid:90) ∞ g (cid:48) ( x ) h ( x ) e − γx dx + (cid:90) ∞ g ( x ) h (cid:48) ( x ) e − γx dx − γ (cid:90) ∞ g ( x ) h ( x ) e − γx dx. By hypothesis, we have lim x →∞ g ( x ) h ( x ) e − γx = 0, and so the stated formula fol-lows. (cid:3) Lemma.
Let γ > and h ∈ C ( R + ; R ) be such that h, h (cid:48) , h (cid:48)(cid:48) ≥ . Assumethere are c > , ε ∈ ( −∞ , γ ) and x ∈ R + such that | h ( x ) | ≤ ce εx and | h (cid:48) ( x ) | ≤ ce εx for all x ≥ x .Then we have (cid:90) ∞ h (cid:48) ( x ) e − γx dx ≤ γ (cid:90) ∞ h ( x ) e − γx dx. Proof.
Using two times Lemma 3.7, we obtain (cid:90) ∞ h ( x ) e − γx dx = 2 γ (cid:90) ∞ h ( x ) h (cid:48) ( x ) e − γx dx + 1 γ h (0) = 2 γ (cid:20)(cid:90) ∞ h (cid:48) ( x ) e − γx dx + (cid:90) ∞ h ( x ) h (cid:48)(cid:48) ( x ) e − γx dx (cid:21) + 1 γ (cid:20) h (0) + 2 γ h (0) h (cid:48) (0) (cid:21) . Since h, h (cid:48) , h (cid:48)(cid:48) ≥ (cid:3) Now we generalize Proposition 3.6 by assuming that η : H β,γ → R is allowed todepend on the current state of the forward curve. The rest of our present frameworkis exactly as in Proposition 3.6.3.9. Proposition.
Assume that, in addition to the hypothesis of Proposition 3.6,we have γ ≤ √ , η ( H β,γ ) ⊂ [0 , γ ) ∩ [0 , √ β ) and | η ( h ) − η ( h ) | ≤ L (cid:107) h − h (cid:107) β,γ for all h , h ∈ H β,γ . Then, for each h ∈ H β,γ , there exists a unique strong adaptedc`adl`ag solution ( r t ) t ≥ to (1.5) with r = h satisfying (3.1). DAMIR FILIPOVI´C AND STEFAN TAPPE
Proof.
It suffices to show that Γ : H β,γ → H β,γ defined as Γ( r )( x ) := e η ( r ) x isLipschitz continuous. So let h , h ∈ H β,γ be arbitrary. Without loss of generalitywe assume that η ( h ) ≤ η ( h ). Observe that all derivatives of Γ( h ) − Γ( h ) are non-negative. So we obtain by applying Lemma 3.8 (notice that γ ≤ √ | e x − e y | ≤ e x | x − y | for y ≤ x that (cid:107) Γ( h ) − Γ( h ) (cid:107) β,γ = ∞ (cid:88) n =0 (cid:18) β (cid:19) n (cid:90) ∞ (cid:16) η ( h ) n e η ( h ) x − η ( h ) n e η ( h ) x (cid:17) e − γx dx ≤ ββ − (cid:90) ∞ (cid:16) e η ( h ) x − e η ( h ) x (cid:17) e − γx dx ≤ ββ − (cid:90) ∞ (cid:16) e η ( h ) x ( η ( h ) − η ( h )) x (cid:17) e − γx dx ≤ ββ − (cid:18)(cid:90) ∞ (cid:16) xe η ( h ) x (cid:17) e − γx dx (cid:19) L (cid:107) h − h (cid:107) β,γ . The integral is finite, because we have η ( h ) ∈ [0 , γ ) by assumption. ApplyingTheorem 3.3 finishes the proof. (cid:3) Forward curve evolutions as mild and weak solutions of infinitedimensional stochastic differential equations
In this section, where we deal with the existence of mild and weak solutions to(1.5), we consider the spaces H w of forward curves, which have been introduced in[24, Chap. 5].Let w : R + → [1 , ∞ ) be a non-decreasing C -function such that w − ∈ L ( R + ).4.1. Example. w ( x ) = e αx , for α > . Example. w ( x ) = (1 + x ) α , for α > . Let H w be the linear space of all absolutely continuous functions h : R + → R satisfying (cid:90) R + | h (cid:48) ( x ) | w ( x ) dx < ∞ , where h (cid:48) denotes the weak derivative of h . We define the inner product( g, h ) w := g (0) h (0) + (cid:90) R + g (cid:48) ( x ) h (cid:48) ( x ) w ( x ) dx and denote the corresponding norm by (cid:57) · (cid:57) w . Since forward curves flatten for largetime to maturity x , the choice of H w is reasonable from an economic point of view.4.3. Proposition.
The space ( H w , ( · , · ) w ) is a separable Hilbert space. Each h ∈ H w is continuous, bounded and the limit h ( ∞ ) := lim x →∞ h ( x ) exists. Moreover, foreach x ∈ R + , the point evaluation h (cid:55)→ h ( x ) : H w → R is a continuous linearfunctional.Proof. All of these statements can be found in the proof of [24, Thm. 5.1.1]. (cid:3)
The fact that each point evaluation is a continuous linear functional ensures thatforward curves ( r t ) solving (1.5) satisfy the variation of constants formula (1.4).Defining the constants C , . . . , C > C := (cid:107) w − (cid:107) L ( R + ) , C := 1 + C , C := (cid:107) w − (cid:107) L ( R + ) , C := (cid:107) w − (cid:107) L ( R + ) , XISTENCE OF L´EVY TERM STRUCTURE MODELS 9 we have for all h ∈ H w the estimates (cid:107) h (cid:48) (cid:107) L ( R + ) ≤ C (cid:57) h (cid:57) w , (4.1) (cid:107) h (cid:107) L ∞ ( R + ) ≤ C (cid:57) h (cid:57) w , (4.2) (cid:107) h − h ( ∞ ) (cid:107) L ( R + ) ≤ C (cid:57) h (cid:57) w , (4.3) (cid:107) ( h − h ( ∞ )) w (cid:107) L ( R + ) ≤ C (cid:57) h (cid:57) w , (4.4)which also follows by inspecting the proof of [24, Thm. 5.1.1].Since for an application of Theorem C.1 we require that the shift semigroup( S t ) t ≥ defined by S t h = h ( t + · ) for t ∈ R + is pseudo-contractive in a closedsubspace of H w , we perform an idea, which is due to Tehranchi [57], namely wechange to the inner product (cid:104) g, h (cid:105) w := g ( ∞ ) h ( ∞ ) + (cid:90) R + g (cid:48) ( x ) h (cid:48) ( x ) w ( x ) dx and denote the corresponding norm by (cid:107) · (cid:107) w . The estimates (4.1)–(4.4) are alsovalid with the norm (cid:107) · (cid:107) w for all h ∈ H w , which is proven exactly as for the originalnorm (cid:57) · (cid:57) w . Therefore we conclude, by using (4.2),1(1 + C ) (cid:107) h (cid:107) w ≤ (cid:57) h (cid:57) w ≤ (1 + C ) (cid:107) h (cid:107) w , h ∈ H w showing that (cid:107) · (cid:107) w and (cid:57) · (cid:57) w are equivalent norms on H w . From now on, we shallwork with the norm (cid:107) · (cid:107) w .4.4. Proposition. ( S t ) is a C -semigroup in H w with generator ddx : D ( ddx ) ⊂ H w → H w , ddx h = h (cid:48) , and domain D ( ddx ) = { h ∈ H w | h (cid:48) ∈ H w } . The subspace H w := { h ∈ H w | h ( ∞ ) = 0 } is a closed subspace of H w and ( S t ) iscontractive in H w with respect to the norm (cid:107) · (cid:107) w .Proof. Except for the last statement, we refer to the proof of [24, Thm. 5.1.1]. Bythe monotonicity of w we have (cid:107) S t h (cid:107) w = (cid:90) R + | h (cid:48) ( x + t ) | w ( x ) dx ≤ (cid:107) h (cid:107) w for all t ∈ R + and h ∈ H w , showing that ( S t ) is contractive in H w . (cid:3) We define for any h = ( h , . . . , h n ) ∈ Π ni =1 A Ψ i H w Σ h ( x ) := − n (cid:88) i =1 h i ( x )Ψ (cid:48) i (cid:18) − (cid:90) x h i ( η ) dη (cid:19) , x ∈ R + . (4.5)4.5. Proposition.
There is a constant C > such that for all g, h ∈ Π ni =1 A Ψ i H w wehave (cid:107) Σ g − Σ h (cid:107) w ≤ C n (cid:88) i =1 (cid:0) (cid:107) h i (cid:107) w + (cid:107) g i (cid:107) w + (cid:107) g i (cid:107) w (cid:1) (cid:107) g i − h i (cid:107) w . (4.6) Furthermore, for each h ∈ Π ni =1 A Ψ i H w we have Σ h ∈ H w , and the map Σ : Π ni =1 A Ψ i H w → H w is continuous.Proof. We define K i := sup x ∈ [ c i ,d i ] | Ψ (cid:48) i ( x ) | , L i := sup x ∈ [ c i ,d i ] | Ψ (cid:48)(cid:48) i ( x ) | and M i := sup x ∈ [ c i ,d i ] | Ψ (cid:48)(cid:48)(cid:48) i ( x ) | for i = 1 , . . . , n . By the boundedness of the derivatives Ψ (cid:48) i on [ c i , d i ], the definition(4.5) of Σ yields that for each h ∈ Π ni =1 A Ψ i H w the limit Σ h ( ∞ ) := lim x →∞ Σ h ( x )exists and Σ h ( ∞ ) = 0 , h ∈ Π ni =1 A Ψ i H w . (4.7)By using (4.7) and the universal inequality | x + . . . + x k | ≤ k (cid:0) | x | + . . . + | x k | (cid:1) , k ∈ N we get for arbitrary g, h ∈ Π ni =1 A Ψ i H w the estimation (cid:107) Σ g − Σ h (cid:107) w = (cid:90) R + (cid:12)(cid:12)(cid:12) n (cid:88) i =1 h (cid:48) i ( x )Ψ (cid:48) i (cid:18) − (cid:90) x h i ( η ) dη (cid:19) − n (cid:88) i =1 g (cid:48) i ( x )Ψ (cid:48) i (cid:18) − (cid:90) x g i ( η ) dη (cid:19) + n (cid:88) i =1 g i ( x ) Ψ (cid:48)(cid:48) i (cid:18) − (cid:90) x g i ( η ) dη (cid:19) − n (cid:88) i =1 h i ( x ) Ψ (cid:48)(cid:48) i (cid:18) − (cid:90) x h i ( η ) dη (cid:19) (cid:12)(cid:12)(cid:12) w ( x ) dx ≤ n ( I + I + I + I ) , where we have put I := n (cid:88) i =1 (cid:90) R + | h (cid:48) i ( x ) | (cid:12)(cid:12)(cid:12) Ψ (cid:48) i (cid:18) − (cid:90) x h i ( η ) dη (cid:19) − Ψ (cid:48) i (cid:18) − (cid:90) x g i ( η ) dη (cid:19) (cid:12)(cid:12)(cid:12) w ( x ) dx,I := n (cid:88) i =1 (cid:90) R + Ψ (cid:48) i (cid:18) − (cid:90) x g i ( η ) dη (cid:19) | h (cid:48) i ( x ) − g (cid:48) i ( x ) | w ( x ) dx,I := n (cid:88) i =1 (cid:90) R + g i ( x ) (cid:20) Ψ (cid:48)(cid:48) i (cid:18) − (cid:90) x g i ( η ) dη (cid:19) − Ψ (cid:48)(cid:48) i (cid:18) − (cid:90) x h i ( η ) dη (cid:19)(cid:21) w ( x ) dx,I := n (cid:88) i =1 (cid:90) R + Ψ (cid:48)(cid:48) i (cid:18) − (cid:90) x h i ( η ) dη (cid:19) ( g i ( x ) − h i ( x ) ) w ( x ) dx. Using (4.3) yields I ≤ n (cid:88) i =1 L i (cid:107) h i (cid:107) w (cid:107) g i − h i (cid:107) L ( R + ) ≤ C n (cid:88) i =1 L i (cid:107) h i (cid:107) w (cid:107) g i − h i (cid:107) w , and I is estimated as I ≤ n (cid:88) i =1 K i (cid:107) g i − h i (cid:107) w . Taking into account (4.3) and (4.4), we get I ≤ n (cid:88) i =1 M i (cid:107) g i w (cid:107) L ( R + ) (cid:107) g i − h i (cid:107) L ( R + ) ≤ C C n (cid:88) i =1 M i (cid:107) g i (cid:107) w (cid:107) g i − h i (cid:107) w , and by using H¨older’s inequality and (4.4), we obtain I ≤ n (cid:88) i =1 L i (cid:90) R + ( g i ( x ) + h i ( x )) w ( x ) ( g i ( x ) − h i ( x )) w ( x ) dx ≤ n (cid:88) i =1 L i (cid:107) ( g i + h i ) w (cid:107) L ( R + ) (cid:107) ( g i − h i ) w (cid:107) L ( R + ) ≤ C n (cid:88) i =1 L i ( (cid:107) g i (cid:107) w + (cid:107) h i (cid:107) w ) (cid:107) g i − h i (cid:107) w , XISTENCE OF L´EVY TERM STRUCTURE MODELS 11 which gives us the desired estimation (4.6). For all h ∈ Π ni =1 A Ψ i H w we have Σ h ∈ H w by (4.6) and (4.7), and the map Σ : Π ni =1 A Ψ i H w → H w is locally Lipschitz continuousby (4.6). (cid:3) By Proposition 4.5 we can, for given volatilities σ i : R + × H w → H w satisfying σ i ( R + × H w ) ⊂ A Ψ i H w for i = 1 , . . . , n , define the drift term α HJM according to theHJM drift condition (2.4) by α HJM := Σ ◦ σ : R + × H w → H w , (4.8)where σ = ( σ , . . . , σ n ).Now, we are ready to establish the existence of L´evy term structure models onthe space H w of forward curves.4.6. Theorem.
Let σ i : R + × H w → H w be continuous and satisfying σ i ( R + × H w ) ⊂ A Ψ i H w for i = 1 , . . . , n . Assume there are M, L ≥ such that for all i = 1 , . . . , n and t ∈ R + we have (cid:107) σ i ( t, h ) (cid:107) w ≤ M, h ∈ H w (cid:107) σ i ( t, h ) − σ i ( t, h ) (cid:107) w ≤ L (cid:107) h − h (cid:107) w , h , h ∈ H w . Then, for each h ∈ H w , there exists a unique mild and a unique weak adaptedc`adl`ag solution ( r t ) t ≥ to (1.5) with r = h satisfying E (cid:20) sup t ∈ [0 ,T ] (cid:107) r t (cid:107) w (cid:21) < ∞ for all T > . (4.9) Proof.
By Proposition 4.5, α HJM maps into H w , see (4.8). Since σ = ( σ , . . . , σ n ) : R + × H w → Π ni =1 A Ψ i H w is continuous by assumption and Σ : Π ni =1 A Ψ i H w → H w iscontinuous by Proposition 4.5, it follows that α HJM = Σ ◦ σ is continuous. Moreover,by estimate (4.6), we obtain for all t ∈ R + and h , h ∈ H w the estimation (cid:107) α HJM ( t, h ) − α HJM ( t, h ) (cid:107) w ≤ C (1 + M ) n (cid:88) i =1 (cid:107) σ i ( t, h ) − σ i ( t, h ) (cid:107) w ≤ C (1 + M ) nL (cid:107) h − h (cid:107) w . Taking also into account Proposition 4.4, applying Theorem C.1 finishes the proof. (cid:3)
As an immediate consequence, we get the existence of L´evy term structure modelswith constant direction volatilities.4.7.
Corollary.
Let σ i : R + × H w → H w be defined by σ i ( t, r ) = σ i ( r ) = ϕ i ( r ) λ i ,where λ i ∈ A Ψ i H w and ϕ i : H w → [0 , for i = 1 , . . . , n . Assume there is L ≥ suchthat for all i = 1 , . . . , n we have | ϕ i ( h ) − ϕ i ( h ) | ≤ L (cid:107) h − h (cid:107) w , h , h ∈ H w . Then, for each h ∈ H w , there exists a unique mild and a unique weak adaptedc`adl`ag solution ( r t ) t ≥ to (1.5) with r = h satisfying (4.9).Proof. For all h , h ∈ H w and all i = 1 , . . . , n we get (cid:107) σ i ( h ) − σ i ( h ) (cid:107) w ≤ L (cid:107) λ i (cid:107) w (cid:107) h − h (cid:107) w . Also observing that (cid:107) σ i ( h ) (cid:107) w ≤ (cid:107) λ i (cid:107) w for all h ∈ H w and i = 1 , . . . , n , the proof isa straightforward consequence of Theorem 4.6. (cid:3) The only assumption on the driving L´evy processes X , . . . , X n , in order toapply the previous results, is the exponential moments condition (2.1). It is clearlysatisfied for Brownian motions and Poisson processes.There are also several purely discontinuous L´evy processes fulfilling (2.1), forinstance generalized hyperbolic processes, which have been introduced by Barndorff-Nielsen [2], and their subclasses, namely the normal inverse Gaussian and hyperbolicprocesses. They have been applied to finance by Eberlein and co-authors in a seriesof papers, e.g. in [17].Other purely discontinuous L´evy processes satisfying (2.1) are the generalizedtempered stable processes, see [10, Sec. 4.5], which include Variance Gamma pro-cesses [43], CGMY processes [9] and bilateral Gamma processes [42].Consequently, Theorem 4.6 applies to term structure models driven by any ofthe above types of L´evy processes.5. Conclusion
We have established the existence of L´evy term structure models on two spacesof forward curves, namely in Section 3 on the Bj¨ork–Svensson space H β,γ , on which ddx is a bounded linear operator, and in Section 4 on the larger space H w , where ddx becomes unbounded.In Section 3 it turned out that H β,γ is too small to assert that α HJM given by theHJM drift-condition (2.4) lies in H β,γ . However, for certain jump-diffusion modelswe have established existence and uniqueness on this space, see Proposition 3.6 andProposition 3.9.Our main results of Section 4 (Theorem 4.6 and Corollary 4.7), where we workon the larger space H w , are applicable for a large range of driving L´evy processes,including mixtures of Brownian motion and Poisson processes, and purely discon-tinuous L´evy processes such as generalized hyperbolic processes and generalizedtempered stable processes as well as several subclasses.The existence results for L´evy term structure models are based on a generalresult for Hilbert space valued stochastic equations, see Theorem C.1 from theappendix. This result relies on two works of van Gaans [26, 27]. In order to make [27,Thm. 4.1] applicable for financial applications, where one is in particular interestedin a solution with c`adl`ag trajectories, we have shown in the appendix that thestochastic integral constructed in van Gaans [27] has a c`adl`ag modification and wehave analyzed when it coincides with the usual Itˆo-integral. Appendix A. Overview and notation
The goal of Appendix A – Appendix C is to provide an existence result forsolutions of infinite dimensional stochastic differential equations, which is requiredin order to establish the existence of L´evy term structure models.We intend to apply a result of van Gaans [27, Thm. 4.1]. However, as we shallsee in Section B, the stochastic integral (G-) (cid:82) t Φ s dX s defined in van Gaans [27]is not consistent with the usual Itˆo-integral (cid:82) t Φ s dX s , which is used for financialmodelling. This matters in view of applications to finance, because, as we haveargued at the end of Section 1, we are in particular interested in a solution processwith c`adl`ag paths.In order to make [27, Thm. 4.1] applicable, we review the stochastic integral,which is defined in van Gaans [27], in Appendix B, show that it always possessesa c`adl`ag modification and analyze when it coincides with the usual Itˆo-integral.In Appendix C, we obtain the desired existence result concerning mild solutions,Theorem C.1, by applying [27, Thm. 4.1]. Using our findings of Appendix B, we XISTENCE OF L´EVY TERM STRUCTURE MODELS 13 additionally show that the solution has a c`adl`ag modification and that it is also aweak solution.Let H denote a separable Hilbert space with inner product (cid:104)· , ·(cid:105) H and associatednorm (cid:107) · (cid:107) H . If there is no ambiguity, we shall simply write (cid:104)· , ·(cid:105) and (cid:107) · (cid:107) .Let T > C ad ([0 , T ]; L (Ω; H )) the spaceof all continuous mappings Φ : [0 , T ] → L (Ω; H ) which are also adapted.For two stochastic processes (Φ t ) t ∈ [0 ,T ] and (Ψ) t ∈ [0 ,T ] we say that Ψ is a modi-fication of Φ if P (Φ t = Ψ t ) = 1 for all t ∈ [0 , T ].An adapted H -valued process (Φ t ) t ∈ [0 ,T ] is called a martingale if • E [ (cid:107) Φ t (cid:107) ] < ∞ for all t ∈ [0 , T ]; • E [Φ t | F s ] = Φ s ( P – a.s.) for all 0 ≤ s ≤ t ≤ T .For the notion of conditional expectation of random variables having values in aseparable Banach space, we refer to [13, Sec. 1.3].An indispensable tool will be Doob’s martingale inequality E (cid:20) sup t ∈ [0 ,T ] (cid:107) Φ t (cid:107) (cid:21) ≤ t ∈ [0 ,T ] E (cid:2) (cid:107) Φ t (cid:107) (cid:3) = 4 E (cid:2) (cid:107) Φ T (cid:107) (cid:3) , (A.1)valid for every H -valued c`adl`ag martingale Φ, which is a consequence of Thm. 3.8and Prop. 3.7 in [13]. Appendix B. Stochastic integration
Let M be a real-valued L´evy martingale satisfying E [ M ] < ∞ . We recall howin this case the stochastic integral (G-) (cid:82) t Φ s dM s , in the sense of van Gaans [27,Sec. 3], is defined for Φ ∈ C ad ([0 , T ]; L (Ω; H )).B.1. Lemma.
Let Φ ∈ C ad ([0 , T ]; L (Ω; H )) . For each t ∈ [0 , T ] , there exists aunique random variable Y t ∈ L (Ω; H ) such that for every ε > there exists δ > such that E (cid:34)(cid:13)(cid:13)(cid:13)(cid:13) Y t − n − (cid:88) i =0 ( M t i +1 − M t i )Φ t i (cid:13)(cid:13)(cid:13)(cid:13) (cid:35) < ε (B.1) for every partition t < t < . . . < t n = t with sup i =0 ,...,n − | t i +1 − t i | < δ .Proof. The assertion is a consequence of [27, Prop. 3.2.1]. (cid:3)
B.2.
Definition.
Let Φ ∈ C ad ([0 , T ]; L (Ω; H )) . Then the stochastic integral Y t =(G-) (cid:82) t Φ s dM s , t ∈ [0 , T ] is the stochastic process Y = ( Y t ) t ∈ [0 ,T ] where every Y t isthe unique element from L (Ω; H ) such that (B.1) is valid. We observe that for every t ∈ [0 , T ] the stochastic integral (G-) (cid:82) t Φ s dM s is onlydetermined up to a P -null set. With regard to our applications to finance it arisesthe question if we can find a modification of the stochastic integral with c`adl`agpaths, a question which is not treated in [27].Let Φ ∈ C ad ([0 , T ]; L (Ω; H )). We define I t (Φ) := (G-) (cid:90) t Φ s dM s , t ∈ [0 , T ](B.2)and the sequence of c`adl`ag adapted processes I n (Φ) := n − (cid:88) i =0 ( M t ni +1 − M t ni )Φ t ni , n ∈ N (B.3)where we set for n ∈ N and i ∈ { , . . . , n } t ni := i − n T, (B.4) that is, we have a sequence of dyadic decompositions of the interval [0 , T ]. Notethat each I n (Φ) is a martingale and that for each t ∈ [0 , T ] we have I nt (Φ) → I t (Φ)in L (Ω; H ) by Lemma B.1.We let M be the linear space of all c`adl`ag H -valued martingales (Φ t ) t ∈ [0 ,T ] ,which are square-integrable, i.e. E (cid:2) (cid:107) Φ t (cid:107) (cid:3) < ∞ for all t ∈ [0 , T ], equipped with thenorm (cid:107) Φ (cid:107) = E (cid:20) sup t ∈ [0 ,T ] (cid:107) Φ t (cid:107) (cid:21) . Note that by Doob’s martingale inequality (A.1), (cid:107) Φ (cid:107) is finite for every Φ ∈ M ,and therefore (cid:107)·(cid:107) defines a norm on the linear space M . For the next result, we canalmost literally follow the proof of [13, Prop. 3.9], which considers the continuoustime case. For convenience of the reader, we provide the proof here.B.3. Proposition.
The normed space ( M , (cid:107) · (cid:107) ) is a Banach space.Proof. Let (Φ n ) be a Cauchy sequence in M , i.e. for every ε > n ∈ N such that E (cid:20) sup t ∈ [0 ,T ] (cid:107) Φ nt − Φ mt (cid:107) (cid:21) < ε for all n, m ≥ n .(B.5)By the Markov inequality, there exists a subsequence (Φ n k ) such that P (cid:18) sup t ∈ [0 ,T ] (cid:107) Φ n k +1 t − Φ n k t (cid:107) ≥ − k (cid:19) ≤ − k for all k ∈ N .The Borel-Cantelli lemma implies that for almost all ω ∈ Ω the sequence (Φ n k ( ω ))is a Cauchy sequence in the space of c`adl`ag functions on [0 , T ] equipped withthe supremum-norm. Therefore, (Φ n k ) converges P –a.s. to an adapted process Φ,uniformly on [0 , T ]. Hence, Φ is c`adl`ag.For each t ∈ [0 , T ], the convergence Φ n k t → Φ t is valid in L (Ω; H ), because (Φ nt )is a Cauchy sequence in L (Ω; H ) by (B.5). For 0 ≤ s ≤ t ≤ T and k ∈ N we have E [Φ n k t | F s ] = Φ n k s ( P –a.s.), implying that E [Φ t | F s ] = Φ s ( P –a.s.). Consequently, Φis a martingale, and by Doob’s martingale inequality (A.1), we get E (cid:20) sup t ∈ [0 ,T ] (cid:107) Φ t − Φ nt (cid:107) (cid:21) ≤ t ∈ [0 ,T ] E (cid:2) (cid:107) Φ t − Φ nt (cid:107) (cid:3) = 4 E (cid:2) (cid:107) Φ T − Φ nT (cid:107) (cid:3) → L (Ω; H ), i.e. Φ n → Φ in M . (cid:3) In the following auxiliary result, (cid:104)
M, M (cid:105) denotes the predictable quadratic co-variation of the real-valued square-integrable martingale M , see [35, Thm. I.4.2].B.4. Lemma.
Let t < . . . < t n = T and Z i : Ω → H be F t i -measurable for i = 0 , . . . , n − . Then we have E (cid:34)(cid:13)(cid:13)(cid:13)(cid:13) n − (cid:88) i =0 ( M t i +1 − M t i ) Z i (cid:13)(cid:13)(cid:13)(cid:13) (cid:35) = E (cid:34) n − (cid:88) i =0 ( (cid:104) M, M (cid:105) t i +1 − (cid:104) M, M (cid:105) t i ) (cid:107) Z i (cid:107) (cid:35) . XISTENCE OF L´EVY TERM STRUCTURE MODELS 15
Proof.
By using the identity (cid:107) x (cid:107) = (cid:104) x, x (cid:105) H , x ∈ H we obtain that (cid:13)(cid:13)(cid:13)(cid:13) n − (cid:88) i =0 ( M t i +1 − M t i ) Z i (cid:13)(cid:13)(cid:13)(cid:13) − n − (cid:88) i =0 ( (cid:104) M, M (cid:105) t i +1 − (cid:104) M, M (cid:105) t i ) (cid:107) Z i (cid:107) = 2 n − (cid:88) i,j =0 i Theorem. Let Φ ∈ C ad ([0 , T ]; L (Ω; H )) . Then I (Φ) has a modification whichbelongs to M and, moreover, I n (Φ) → I (Φ) in M .Proof. Let ε > , T ] → L (Ω; H ) is uniformly continuouson the compact interval [0 , T ], there exists δ > E (cid:2) (cid:107) Φ t − Φ s (cid:107) (cid:3) < ε T (cid:0) c + (cid:82) R x F ( dx ) (cid:1) (B.6)for all s, t ∈ [0 , T ] with | t − s | < δ , where c denotes the Gaussian part and F theL´evy measure of M . Choose n ∈ N such that 2 − n T < δ . For all n, m ∈ N with n > m ≥ n we obtain I n (Φ) − I m (Φ) = n − (cid:88) i =0 (cid:0) M t ni +1 − M t ni (cid:1)(cid:0) Φ t ni − Φ t nj ( i ) (cid:1) with j ( i ) ∈ { , . . . , i } such that | t ni − t nj ( i ) | < − n T < δ for all i = 0 , . . . , n − n, m ∈ N with n > m ≥ n E (cid:20) sup t ∈ [0 ,T ] (cid:107) I nt (Φ) − I mt (Φ) (cid:107) (cid:21) ≤ E (cid:34)(cid:13)(cid:13)(cid:13)(cid:13) n − (cid:88) i =0 (cid:0) M t ni +1 − M t ni (cid:1)(cid:0) Φ t ni − Φ t nj ( i ) (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) (cid:35) = 4 n − (cid:88) i =0 E (cid:104)(cid:0) (cid:104) M, M (cid:105) t ni +1 − (cid:104) M, M (cid:105) t ni (cid:1) (cid:107) Φ t ni − Φ t nj ( i ) (cid:107) (cid:105) = 4 (cid:18) c + (cid:90) R x F ( dx ) (cid:19) n − (cid:88) i =0 ( t ni +1 − t ni ) E (cid:104) (cid:107) Φ t ni − Φ t nj ( i ) (cid:107) (cid:105) < ε. The latter identity is valid, because (cid:104) M, M (cid:105) is the compensator of [ M, M ] by [35,Prop. I.4.50.b] and because the relation [ M, M ] t = ct + (cid:80) s ≤ t ∆ M s is valid accordingto [35, Thm. I.4.52].Thus, the sequence ( I n (Φ)) is a Cauchy sequence in M . Proposition B.3 andLemma B.1 complete the proof. (cid:3) For Φ ∈ C ad ([0 , T ]; L (Ω; H )), the integral with respect to dt can, according to[27, Lemma 3.6], be defined as a Riemann integral. More precisely: B.6. Lemma. Let Φ ∈ C ad ([0 , T ]; L (Ω; H )) . For each t ∈ [0 , T ] , there exists aunique random variable Y t ∈ L (Ω; H ) such that for every ε > there exists δ > such that E (cid:34)(cid:13)(cid:13)(cid:13)(cid:13) Y t − n − (cid:88) i =0 ( t i +1 − t i )Φ t i (cid:13)(cid:13)(cid:13)(cid:13) (cid:35) < ε (B.7) for every partition t < t < . . . < t n = t with sup i =0 ,...,n − | t i +1 − t i | < δ .Proof. Fix t ∈ [0 , T ] and let ε > , t ] → L (Ω; H ) isuniformly continuous on the compact interval [0 , t ], there exists δ > E (cid:2) (cid:107) Φ s − Φ r (cid:107) (cid:3) < εt (B.8)for all r, s ∈ [0 , t ] with | s − r | < δ .Let Z = { t < t < . . . < t n = t } and Z = { s < s <. . . < s m = t } be two decompositions satisfying sup i =0 ,...,n − | t i +1 − t i | < δ andsup i =0 ,...,m − | s i +1 − s i | < δ . Then there is a unique decomposition Z = { r 1. We obtain by theCauchy-Schwarz inequality and (B.8) E (cid:34)(cid:13)(cid:13)(cid:13)(cid:13) n − (cid:88) i =0 ( t i +1 − t i )Φ t i − m − (cid:88) i =0 ( s i +1 − s i )Φ s i (cid:13)(cid:13)(cid:13)(cid:13) (cid:35) ≤ E (cid:20)(cid:18) p − (cid:88) i =0 ( r i +1 − r i ) (cid:107) (Φ a i − Φ b i ) (cid:107) (cid:19) (cid:21) ≤ E (cid:20)(cid:18) p − (cid:88) i =0 ( r i +1 − r i ) (cid:19)(cid:18) p − (cid:88) i =0 ( r i +1 − r i ) (cid:107) Φ a i − Φ b i (cid:107) (cid:19)(cid:21) = t p − (cid:88) i =0 ( r i +1 − r i ) E (cid:2) (cid:107) Φ a i − Φ b i (cid:107) (cid:3) < ε. By the completeness of L (Ω; H ), the lemma is proven. (cid:3) B.7. Definition. Let Φ ∈ C ad ([0 , T ]; L (Ω; H )) . Then the integral Y t = (G-) (cid:82) t Φ s ds , t ∈ [0 , T ] is the stochastic process Y = ( Y t ) t ∈ [0 ,T ] where every Y t is the unique ele-ment from L (Ω; H ) such that (B.7) is valid. Again, for every t ∈ [0 , T ] the integral (G-) (cid:82) t Φ s ds is only determined up to a P –null set. We shall prove the existence of a continuous modification.Let Φ ∈ C ad ([0 , T ] , L (Ω; H )). We define J t (Φ) := (G-) (cid:90) t Φ s ds, t ∈ [0 , T ](B.9)and the sequence of continuous adapted processes J nt (Φ) := n − (cid:88) i =0 ( t ni +1 ∧ t − t ni ∧ t )Φ t ni , n ∈ N (B.10)where the t ni are defined in (B.4). Note that for each t ∈ [0 , T ] we have J nt (Φ) → J t (Φ) in L (Ω; H ) by Lemma B.6. XISTENCE OF L´EVY TERM STRUCTURE MODELS 17 B.8. Theorem. Let Φ ∈ C ad ([0 , T ]; L (Ω; H )) . Then J (Φ) has a continuous modi-fication and, moreover, sup t ∈ [0 ,T ] (cid:107) J nt (Φ) − J t (Φ) (cid:107) → P –a.s.Proof. Let ε > , T ] → L (Ω; H ) is uniformly continuouson the compact interval [0 , T ], there exists δ > E (cid:2) (cid:107) Φ t − Φ s (cid:107) (cid:3) < εT (B.11)for all s, t ∈ [0 , T ] with | t − s | < δ . Choose n ∈ N such that 2 − n T < δ . For all n, m ∈ N with n > m ≥ n we obtain J n (Φ) − J m (Φ) = n − (cid:88) i =0 (cid:0) t ni +1 ∧ t − t ni ∧ t (cid:1)(cid:0) Φ t ni − Φ t nj ( i ) (cid:1) with j ( i ) ∈ { , . . . , i } such that | t ni − t nj ( i ) | < − n T < δ for all i = 0 , . . . , n − 1. Weobtain by the Cauchy-Schwarz inequality and (B.11) for all n, m ≥ n with n > m E (cid:20) sup t ∈ [0 ,T ] (cid:107) J nt (Φ) − J mt (Φ) (cid:107) (cid:21) = E (cid:34) sup t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13)(cid:13) n − (cid:88) i =0 (cid:0) t ∧ t ni +1 − t ∧ t ni (cid:1)(cid:0) Φ t ni − Φ t nj ( i ) (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) (cid:35) ≤ T E (cid:20) sup t ∈ [0 ,T ] 2 n − (cid:88) i =0 (cid:0) t ∧ t ni +1 − t ∧ t ni (cid:1) (cid:107) Φ t ni − Φ t nj ( i ) (cid:107) (cid:21) = T n − (cid:88) i =0 (cid:0) t i +1 − t i (cid:1) E (cid:104) (cid:107) Φ t ni − Φ t nj ( i ) (cid:107) (cid:105) < ε. By the Markov inequality, there exists a subsequence ( J n k (Φ)) such that P (cid:18) sup t ∈ [0 ,T ] (cid:107) J n k +1 t (Φ) − J n k t (Φ) (cid:107) ≥ − k (cid:19) ≤ − k for all k ∈ N .The Borel-Cantelli lemma implies that for almost all ω ∈ Ω the sequence ( J n k (Φ)( ω ))is a Cauchy sequence in the space of continuous functions on [0 , T ] equipped withthe supremum-norm. Therefore, ( J n k (Φ)) converges P –a.s. to an adapted process,uniformly on [0 , T ], which is therefore continuous.According to Lemma B.6, this limit process is a modification of the integralprocess J (Φ). (cid:3) Now let X be a real-valued L´evy process with E [ X ] < ∞ . Then it admits aunique decomposition X t = M t + bt , where M is a L´evy martingale satisfying E [ M ] < ∞ and b = E [ X ]. According to [27, Def. 3.7], we set(G-) (cid:90) t Φ s dX s := (G-) (cid:90) t Φ s dM s + b · (G-) (cid:90) t Φ s ds. We shall also use the notation G (Φ) t = (G-) (cid:90) t Φ s dX s , t ∈ [0 , T ] . Note that G (Φ) = I (Φ)+ b · J (Φ), where I (Φ) is defined in (B.2) and J (Φ) is definedin (B.9). We also introduce G n (Φ) = I n (Φ) + b · J n (Φ) for n ∈ N , where I n (Φ) isdefined in (B.3) and J n (Φ) is defined in (B.10).For a predictable H -valued process Φ and a real-valued semimartingale X , we candefine the usual Itˆo-integral (developed e.g. in Jacod and Shiryaev [35] or Protter [48]) (cid:90) t Φ s dX s , which is used for financial modelling. The construction is just as for real-valuedintegrands, namely by defining the integral first for simple integrands and thenextending it via the Itˆo-isometry. In order to get the Itˆo-isometry, it is vital thatthe state space H is a Hilbert space.The construction of the stochastic integral in the more general situation, wherethe driving semimartingale may also be infinite dimensional, can be found in M´etivier[44]. Da Prato and Zabczyk [13] and Carmona and Tehranchi [8] treat the case withinfinite dimensional Brownian motion as integrator, in [8] also with a focus on in-terest rate models.We also remark that the stochastic integral can still be defined on appropriateBanach spaces, so-called M-type 2 spaces. Then the integral is still a bounded linearoperator, but no isometry, in general. We refer to [53] for further details.We now observe that the integral (G-) (cid:82) t Φ s dX s of van Gaans [27] is not consis-tent with the usual stochastic integral (cid:82) t Φ s dX s used in financial modelling. As anexample, let X be a standard Poisson process with values in R . In Ex. 3.9 in [27]it is derived that (G-) (cid:90) t X s dX s = 12 (cid:0) X t − X t (cid:1) . Apparently, this does not coincide with the pathwise Lebesgue-Stieltjes integral (cid:90) t X s dX s = 12 (cid:0) X t + X t (cid:1) , but we have (G-) (cid:90) t X s dX s = (cid:90) t X s − dX s , showing that inconsistencies occur as soon as integrands with jumps are used. In-deed, we have the following general result about the relation between the integralof van Gaans and the usual Itˆo-integral:B.9. Theorem. Let Φ ∈ C ad ([0 , T ]; L (Ω; H )) be left-continuous or c`adl`ag. Thenwe have for all t ∈ [0 , T ](G-) (cid:90) t Φ s dX s = (cid:90) t Φ s − dX s P –a.s. (B.12) Proof. If Φ is left-continuous, we have sup t ∈ [0 ,T ] (cid:107) G nt − G t (cid:107) → t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13) G nt − (cid:82) t Φ s dX s (cid:13)(cid:13)(cid:13) → t ∈ [0 , T ](G-) (cid:90) t Φ s dX s = (cid:90) t Φ s dX s P –a.s.(B.13)and, since Φ is left-continuous, also relation (B.12).If Φ is c`adl`ag, we show that Φ − is a modification of Φ, because then (B.12) isa consequence of (B.13) and Lemma B.11 below. Let t ∈ (0 , T ] be arbitrary and( t n ) be a sequence such that t n ↑ t . Since Φ : [0 , T ] → L (Ω; H ) is continuous, wededuce E [ (cid:107) Φ t − Φ t n (cid:107) ] → 0. Thus there is a subsequence ( n k ) with (cid:107) Φ t − Φ t nk (cid:107) → P (Φ t = Φ t − ) = 1. (cid:3) XISTENCE OF L´EVY TERM STRUCTURE MODELS 19 B.10. Remark. If the driving process X is a (possibly infinite dimensional) Brow-nian motion, the equivalence of the van Gaans integral with the usual stochasticintegral (see Da Prato and Zabczyk [13] for the infinite dimensional case) is pro-vided in [26, Sec. 3] . It remains to show the following auxiliary result, which we have used in the proofof Theorem B.9.B.11. Lemma. Let Φ , Ψ ∈ C ad ([0 , T ]; L (Ω; H )) be such that Ψ is a modificationof Φ . Then G (Ψ) is a modification of G (Φ) .Proof. Let t ∈ [0 , T ] be arbitrary. By hypothesis, we have P ( G nt (Φ) = G nt (Ψ)) = 1for all n ∈ N . Since G nt (Φ) → G t (Φ) and G nt (Ψ) → G t (Ψ) in L (Ω; H ) by LemmaB.1 and Lemma B.6, there is a subsequence ( n k ) such that G n k t (Φ) → G t (Φ) almostsurely, and another subsequence n k l such that G n kl t (Ψ) → G t (Ψ) almost surely,showing that P ( G t (Φ) = G t (Ψ)) = 1. (cid:3) Appendix C. Stochastic differential equations Now let ( S t ) t ≥ be a C -semigroup in the separable Hilbert space H , i.e. a familyof bounded linear operators S t : H → H such that • S = Id; • S s + t = S s S t for all s, t ≥ • lim t → S t h = h for all h ∈ H ;with generator A : D ( A ) ⊂ H → H . By (cid:107) · (cid:107) L ( H ) we denote the operator norm ofa bounded linear operator. The semigroup ( S t ) is called contractive in H if (cid:107) S t (cid:107) L ( H ) ≤ , t ≥ pseudo-contractive in H if there is a constant ω ≥ (cid:107) S t (cid:107) L ( H ) ≤ e ωt , t ≥ . In this section, we intend to find mild solutions of stochastic differential equationsof the type (cid:40) dr t = ( Ar t + α ( t, r t )) dt + (cid:80) ni =1 σ i ( t, r t − ) dX it ,r = h (C.1)driven by real-valued L´evy processes X , . . . , X n satisfying E [( X i ) ] < ∞ , i =1 , . . . , n , for each initial condition h ∈ H , that is, a process ( r t ) t ≥ satisfying r t = S t h + (cid:90) t S t − s α ( s, r s ) ds + n (cid:88) i =1 (cid:90) t S t − s σ i ( s, r s − ) dX is , t ∈ R + . (C.2)We also intend to establish the existence of a weak solution ( r t ) t ≥ to (C.1), i.e.( r t ) satisfies, for all ζ ∈ D ( A ∗ ), (cid:104) ζ, r t (cid:105) = (cid:104) ζ, h (cid:105) + (cid:90) t (cid:16) (cid:104) A ∗ ζ, r s (cid:105) + (cid:104) ζ, α ( s, r s ) (cid:105) (cid:17) ds + n (cid:88) i =1 (cid:90) t (cid:104) ζ, σ i ( s, r s − ) (cid:105) dX is (C.3)for each t ∈ R + . By convention, uniqueness of a solution to (C.1) is meant up to amodification. Here is our main existence and uniqueness result: C.1. Theorem. Let ( S t ) t ≥ be a C -semigroup in H , and H ⊂ H be a closedsubspace such that ( S t ) is pseudo-contractive in H . Let α, σ , . . . , σ n : R + × H → H be continuous. Assume there is constant L ≥ such that (cid:107) α ( t, h ) − α ( t, h ) (cid:107) ≤ L (cid:107) h − h (cid:107) (C.4) (cid:107) σ i ( t, h ) − σ i ( t, h ) (cid:107) ≤ L (cid:107) h − h (cid:107) , i = 1 , . . . , n (C.5) for all t ∈ R + and all h , h ∈ H . Then, for each h ∈ H , there exists a unique mildand a unique weak adapted c`adl`ag solution ( r t ) t ≥ to (C.1) with r = h satisfying E (cid:20) sup t ∈ [0 ,T ] (cid:107) r t (cid:107) (cid:21) < ∞ for all T > . (C.6) Proof. Let h ∈ H be arbitrary. We decompose each L´evy process X it = M it + b i t into its martingale and finite variation part, where we notice that b i = E [ X i ]. By[27, Thm. 4.1] there exists a unique adapted continuous function r : R + → L (Ω; H )such that for all t ≥ r t = S t h + (G-) (cid:90) t S t − s ˜ α ( s, r s ) ds + n (cid:88) i =1 (G-) (cid:90) t S t − s σ i ( s, r s ) dM is , where ˜ α ( t, r ) = α ( t, r ) + (cid:80) ni =1 b i σ i ( t, r ). By assumption, ( S t ) is pseudo-contractivein H . Hence there exists a constant ω ≥ C -semigroup ( T t ) t ≥ defined as T t := e − ωt S t , t ∈ R + (C.7)is contractive in H . By the Szek¨ofalvi-Nagy’s theorem on unitary dilations (seee.g. [56, Thm. I.8.1], or [14, Sec. 7.2]), there exists another separable Hilbert space H and a strongly continuous unitary group ( U t ) t ∈ R in H such that the diagram H U t −−−−→ H (cid:120) (cid:96) (cid:121) π H T t −−−−→ H commutes for every t ∈ R + , where (cid:96) : H → H is an isometric embedding (hencethe adjoint operator π := (cid:96) ∗ is the orthogonal projection from H into H ), that is πU t (cid:96)h = T t h for all t ∈ R + and h ∈ H .(C.8)Using (C.7), (C.8) and [27, Thm. 3.3.3] we obtain for all i = 1 , . . . , n and t ≥ (cid:90) t S t − s σ i ( s, r s ) dM is = (G-) (cid:90) t e ω ( t − s ) T t − s σ i ( s, r s ) dM is = e ωt (G-) (cid:90) t e − ωs πU t − s (cid:96)σ i ( s, r s ) dM is = e ωt πU t (G-) (cid:90) t e − ωs U − s (cid:96)σ i ( s, r s ) dM is . The integral process (G-) (cid:90) t e − ωs U − s (cid:96)σ i ( s, r s ) dM is has a c`adl`ag modification by Theorem B.5. Thus the process(G-) (cid:90) t S t − s σ i ( s, r s ) dM is has a c`adl`ag modification, because ( t, h ) (cid:55)→ U t h is uniformly continuous on compactsubsets, see e.g. [23, Lemma I.5.2]. XISTENCE OF L´EVY TERM STRUCTURE MODELS 21 A similar argumentation, using Theorem B.8, shows that(G-) (cid:90) t S t − s ˜ α ( s, r s ) ds has a continuous modification.Therefore, ( r t ) has a c`adl`ag modification, and, by Theorem B.9, it satisfies r t = S t h + (cid:90) t S t − s ˜ α ( s, r s ) ds + n (cid:88) i =1 (cid:90) t S t − s σ i ( s, r s − ) dM is , t ≥ . Consequently, ( r t ) t ≥ is a mild solution to (C.1), i.e. it satisfies (C.2). Introducingthe processes Φ t := (cid:90) t S t − s ˜ α ( s, r s ) ds, (C.9) Ψ it := (cid:90) t S t − s σ i ( s, r s − ) dM is , i = 1 , . . . , n (C.10)we have by our findings above r t = S t h + Φ t + n (cid:88) i =1 Ψ it , t ≥ . (C.11)We fix an arbitrary T > 0. By (C.7), (C.8) and noting that (cid:107) π (cid:107) L ( H ; H ) = 1 and (cid:107) U t (cid:107) L ( H ) ≤ t ∈ [0 , T ], we obtain for each i = 1 , . . . , n E (cid:20) sup t ∈ [0 ,T ] (cid:107) Ψ it (cid:107) (cid:21) = E (cid:34) sup t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13)(cid:13) (cid:90) t S t − s σ i ( s, r s − ) dM is (cid:13)(cid:13)(cid:13)(cid:13) (cid:35) = E (cid:34) sup t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13)(cid:13) e ωt πU t (cid:90) t e − ωs U − s (cid:96)σ i ( s, r s − ) dM is (cid:13)(cid:13)(cid:13)(cid:13) (cid:35) (C.12) ≤ e ωT E (cid:34) sup t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13)(cid:13) (cid:90) t e − ωs U − s (cid:96)σ i ( s, r s − ) dM is (cid:13)(cid:13)(cid:13)(cid:13) (cid:35) < ∞ . The latter expression is finite by Theorem B.5 and Theorem B.9.We obtain by (C.7), (C.8), H¨older’s inequality and Fubini’s theorem (note that (cid:107) ˜ α ( t, r t ) (cid:107) is c`adl`ag and therefore B [0 , T ] ⊗ F -measurable) E (cid:20) sup t ∈ [0 ,T ] (cid:107) Φ t (cid:107) (cid:21) = E (cid:34) sup t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13)(cid:13) (cid:90) t S t − s ˜ α ( s, r s ) ds (cid:13)(cid:13)(cid:13)(cid:13) (cid:35) = E (cid:34) sup t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13)(cid:13) e ωt πU t (cid:90) t e − ωs U − s (cid:96) ˜ α ( s, r s ) ds (cid:13)(cid:13)(cid:13)(cid:13) (cid:35) ≤ T e ωT E (cid:20) (cid:90) T (cid:107) ˜ α ( t, r t ) (cid:107) dt (cid:21) ≤ T e ωT (cid:90) T E (cid:2) (cid:107) ˜ α ( t, r t ) (cid:107) (cid:3) dt ≤ T e ωT sup t ∈ [0 ,T ] E (cid:2) (cid:107) ˜ α ( t, r t ) (cid:107) (cid:3) < ∞ . The latter supremum is finite, because t (cid:55)→ E (cid:2) (cid:107) ˜ α ( t, r t ) (cid:107) (cid:3) is continuous on thecompact interval [0 , T ], as t (cid:55)→ ˜ α ( t, r t ) is continuous by the continuity of r : [0 , T ] → L (Ω; H ) and (C.4), (C.5). Since the solution process ( r t ) is given by (C.11), weobtain, together with (C.12), that (C.6) is valid.We proceed by showing that ( r t ) t ≥ is also a weak solution to (C.1). Let ζ ∈ D ( A ∗ ) be arbitrary. We define for arbitrary T ∈ R + the B [0 , T ] ⊗ P -measurable functions H i : [0 , T ] × [0 , T ] × Ω → R as H i ( a, t ) := (cid:40) (cid:104) A ∗ ζ, S a − t σ i ( t, r t − ) (cid:105) , a ≥ t , a < t. We obtain by the Cauchy-Schwarz inequality and the pseudo-contractivity of ( S t )in H that | H i ( a, t ) | ≤ e ωT (cid:107) A ∗ ζ (cid:107) · (cid:107) σ i ( t, r t − ) (cid:107) , a, t ∈ [0 , T ] . (C.13)The processes Y it := ( (cid:82) T H i ( a, t ) da ) / are left-continuous by (C.13) and Lebesgue’sdominated convergence theorem, and therefore predictable. The L´evy martingales M i , considered on [0 , T ], belong to H in the sense of the Definition in Protter [48,p. 156], because, by using [35, Thm. I.4.52], (cid:107) M i (cid:107) H ≤ (cid:107) M ci (cid:107) H + (cid:107) M di (cid:107) H = E (cid:2) [ M ci , M ci ] T (cid:3) / + E (cid:2) [ M di , M di ] T (cid:3) / = ( c i T ) / + E (cid:20) (cid:88) s ≤ T (∆ M is ) (cid:21) / = ( c i T ) / + (cid:18) T (cid:90) R x F i ( dx ) (cid:19) / < ∞ , where we have decomposed M i = M ci + M di into its continuous and purely discon-tinuous martingale part, and where c i denotes the Gaussian part and F i the L´evymeasure of M i .There is, by the assumed continuity of σ , . . . , σ n , a constant C T > (cid:107) σ i ( t, (cid:107) ≤ C T for all t ∈ [0 , T ] and i = 1 , . . . , n . Therefore, we get for all t ∈ [0 , T ],all h ∈ H and all i = 1 , . . . , n by (C.5) (cid:107) σ i ( t, h ) (cid:107) ≤ (cid:107) σ i ( t, (cid:107) + (cid:107) σ i ( t, h ) − σ i ( t, (cid:107) ≤ ( L ∨ C T )(1 + (cid:107) h (cid:107) ) . (C.14)By inequalities (C.13) and (C.14) we obtain E (cid:20) (cid:90) T ( Y it ) d [ M i , M i ] t (cid:21) = (cid:18) c i + (cid:90) R x F i ( dx ) (cid:19) E (cid:20) (cid:90) T (cid:90) T H i ( a, t ) dadt (cid:21) ≤ T ( L ∨ C T ) e ωT (cid:107) A ∗ ζ (cid:107) (cid:18) c i + (cid:90) R x F i ( dx ) (cid:19) E (cid:20) sup t ∈ [0 ,T ] (1 + (cid:107) r t (cid:107) ) (cid:21) . Thus, the processes Y i are ( H , M i ) integrable in the sense of the Definition inProtter [48, p. 165], because from H¨older’s inequality and (C.6) we infer E (cid:20) sup t ∈ [0 ,T ] (1 + (cid:107) r t (cid:107) ) (cid:21) ≤ E (cid:20) sup t ∈ [0 ,T ] (cid:107) r t (cid:107) (cid:21) / + E (cid:20) sup t ∈ [0 ,T ] (cid:107) r t (cid:107) (cid:21) < ∞ . Consequently, we have Y i ∈ L ( M i ), that is each Y i is M i integrable in the senseof Protter [48, p. 165], and therefore we may apply the Fubini Theorem, see Thm.IV.65 in [48], for the integrands H i . Using the Fubini Theorem and [58, LemmaVII.4.5(a)], we obtain for each i = 1 , . . . , n (cid:90) t (cid:104) A ∗ ζ, Ψ is (cid:105) ds = (cid:90) t (cid:90) s (cid:104) A ∗ ζ, S s − u σ i ( u, r u − ) (cid:105) dM iu ds = (cid:90) t (cid:68) A ∗ ζ, (cid:90) tu S s − u σ i ( u, r u − ) ds (cid:69) dM iu = (cid:90) t (cid:68) ζ, A (cid:90) t − u S s σ i ( u, r u − ) ds (cid:69) dM iu = (cid:90) t (cid:104) ζ, S t − u σ i ( u, r u − ) − σ i ( u, r u − ) (cid:105) dM iu = (cid:104) ζ, Ψ it (cid:105) − (cid:90) t (cid:104) ζ, σ i ( s, r s − ) (cid:105) dM is , XISTENCE OF L´EVY TERM STRUCTURE MODELS 23 where the Ψ i are defined in (C.10). An analogous calculation, using the standardFubini theorem, gives us (cid:90) t (cid:104) A ∗ ζ, Φ s (cid:105) ds = (cid:104) ζ, Φ t (cid:105) − (cid:90) t (cid:104) ζ, ˜ α ( s, r s ) (cid:105) ds, where Φ is defined in (C.9), and finally, we get, by taking into account [58, LemmaVII.4.5(a)] again, (cid:90) t (cid:104) A ∗ ζ, S s h (cid:105) ds = (cid:68) ζ, A (cid:90) t S s h ds (cid:69) = (cid:104) ζ, S t h (cid:105) − (cid:104) ζ, h (cid:105) . Together with (C.11), the latter three identities show that (cid:104) ζ, r t (cid:105) = (cid:104) ζ, h (cid:105) + (cid:90) t (cid:16) (cid:104) A ∗ ζ, r s (cid:105) + (cid:104) ζ, ˜ α ( s, r s ) (cid:105) (cid:17) ds + n (cid:88) i =1 (cid:90) t (cid:104) ζ, σ i ( s, r s − ) (cid:105) dM is for all t ∈ [0 , T ]. Since T ∈ R + was arbitrary, ( r t ) t ≥ is a weak solution to (C.1), asit fulfills (C.3).It remains to show that this weak solution is unique. Let ( r t ) t ≥ be any adaptedc`adl`ag weak solution to (C.1), i.e. ( r t ) satisfies (C.3) for all ζ ∈ D ( A ∗ ). Let ζ ∈ D ( A ∗ ) and g ∈ C ([0 , T ]; R ) for an arbitrary T ∈ R + . By the definition of thequadratic co-variation [ X, Y ], see e.g. [35, Def. I.4.45], we obtain (cid:104) g ( t ) ζ, r t (cid:105) = (cid:104) g (0) ζ, h (cid:105) + (cid:90) t g ( s ) d (cid:104) ζ, r s (cid:105) + (cid:90) t (cid:104) ζ, r s (cid:105) dg ( s ) + [ g, (cid:104) ζ, r (cid:105) ] t . Since g ∈ C ([0 , T ]; R ), we have [ g, (cid:104) ζ, r (cid:105) ] = 0 according to [35, Prop. 4.49.d].Therefore and because of (C.3), we get (cid:104) g ( t ) ζ, r t (cid:105) = (cid:104) g (0) ζ, h (cid:105) + (cid:90) t (cid:16) (cid:104) g (cid:48) ( s ) ζ + A ∗ g ( s ) ζ, r s (cid:105) + (cid:104) g ( s ) ζ, α ( s, r s ) (cid:105) (cid:17) ds + n (cid:88) i =1 (cid:90) t (cid:104) g ( s ) ζ, σ i ( s, r s − ) (cid:105) dX is . Since the set { t (cid:55)→ g ( t ) ζ | g ∈ C ([0 , T ]; R ) } is dense in C ([0 , T ]; D ( A ∗ )), we deduce (cid:104) g ( t ) , r t (cid:105) = (cid:104) g (0) , h (cid:105) + (cid:90) t (cid:16) (cid:104) g (cid:48) ( s ) + A ∗ g ( s ) , r s (cid:105) + (cid:104) g ( s ) , α ( s, r s ) (cid:105) (cid:17) ds + n (cid:88) i =1 (cid:90) t (cid:104) g ( s ) , σ i ( s, r s − ) (cid:105) dX is for all g ∈ C ([0 , T ]; D ( A ∗ )), where we recall that T ∈ R + was arbitrary. Defining g ∈ C ([0 , t ]; D ( A ∗ )) for an arbitrary t ∈ R + and an arbitrary ζ ∈ D ( A ∗ ) as g ( s ) := S ∗ t − s ζ , s ∈ [0 , t ], we obtain g (cid:48) ( s ) = − A ∗ g ( s ), and hence (cid:104) ζ, r t (cid:105) = (cid:104) ζ, S t h (cid:105) + (cid:90) t (cid:104) ζ, S t − s α ( s, r s ) (cid:105) ds + n (cid:88) i =1 (cid:90) t (cid:104) ζ, S t − s σ i ( s, r s − ) (cid:105) dX is . Since D ( A ∗ ) is dense in H , the process ( r t ) t ≥ is also a mild solution to (C.1), i.e.it satisfies (C.2), proving the desired uniqueness. (cid:3) In the special situation where A ∈ L ( H ), i.e. A is a bounded linear operator, wecan now easily establish the existence of a strong solution ( r t ) t ≥ to (C.1), that iswe have r t = h + (cid:90) t (cid:16) Ar s + α ( s, r s ) (cid:17) ds + n (cid:88) i =1 (cid:90) t σ i ( s, r s − ) dX is , t ≥ . (C.15) C.2. Corollary. Let A ∈ L ( H ) be a bounded linear operator and let α, σ , . . . , σ n : R + × H → H be continuous. Assume there is constant L ≥ such that (C.4)and (C.5) are satisfied for all t ∈ R + and h , h ∈ H . Then, for each h ∈ H ,there exists a unique strong adapted c`adl`ag solution ( r t ) t ≥ to (C.1) with r = h satisfying (C.6).Proof. The operator A is generated by the semigroup S t = e tA , which is pseudo-contractive, because (cid:107) S t (cid:107) L ( H ) ≤ e t (cid:107) A (cid:107) L ( H ) , t ≥ . By Theorem C.1, for each h ∈ H , there exists a unique weak adapted c`adl`agsolution ( r t ) t ≥ to (C.1) with r = h satisfying (C.6), which also fulfills (C.15) bythe boundedness of A , showing that ( r t ) is a strong solution to (C.1). (cid:3) We close this section with a couple of remarks. Actually, [27, Thm. 4.1] is notexplicitly proven in [27]. We quote [27, p. 19]: ”For a proof of Theorem 4.1 one canfollow almost literally the proofs of Theorem 4.1 and Theorem 4.2 in [26], . . . ”. Thementioned result, [26, Thm. 4.1], is an analogous result for stochastic equationsdriven by an infinite dimensional Brownian motion.Note that the existence result of van Gaans [27, Thm. 4.1] demands no furtherassumptions on the C -semigroup. In contrast, we require the pseudo-contractivityof ( S t ) in a closed subspace in order to prove that the solution possesses a c`adl`agmodification.The idea to use the Szek¨ofalvi-Nagy’s theorem on unitary dilations in order toovercome the difficulties arising from stochastic convolutions, is due to Hausenblasand Seidler, see [31] and [30].Without using the Szek¨ofalvi-Nagy’s theorem, Baudoin and Teichmann [3] con-sider stochastic equations on separable Hilbert spaces equipped with a stronglycontinuous group, in Sec. 3 of their article also with focus on interest rate theory.For every pseudo-contractive semigroup ( S t ), stochastic convolutions (cid:82) t S t − s Φ s dM s with respect to a square-integrable, c`adl`ag martingale M have a c`adl`ag modifica-tion, which is due to Kotelenez [41]. We use the Szek¨ofalvi-Nagy’s theorem onunitary dilations in order to get a c`adl`ag modification, because we deal with thestochastic integral (G-) (cid:82) t S t − s Φ s dM s defined in van Gaans [27, Sec. 3].Recently, there has been growing interest in stochastic differential equations ofthe type (C.1) with jump noise terms. As a result, a few related papers [1, 39, 40,28, 29, 45, 50] and the forthcoming textbook [49] have been written, but mostlywith other fields of applications than finance.During the revision of this paper we became aware of the recent preprint [50],where the authors derived independently similar results. But they work on dif-ferent function spaces where the forward curve is not necessarily continuous andthus the short rate is not well defined. Moreover, they only consider volatilitiesof composition type, that is σ i ( t, r )( x ) = g i ( t, x, r ( x )) with deterministic functions g i : R + × R + × R → R . 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