TERM STRUCTURE MODELS DRIVEN BY WIENER PROCESSES AND POISSON MEASURES: EXISTENCE AND POSITIVITY

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1 TM STUCTU MODLS DIVN BY WIN POCSSS AND POISSON MASUS: XISTNC AND POSITIVITY DAMI FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN Abstract. In the spirit of [3], we investigate term structure models driven by Wiener processes and Poisson measures with forward curve dependent volatilities. This includes a full existence and uniqueness proof for the corresponding Heath-Jarrow-Morton type term structure equation. Furthermore, we characterize positivity preserving models by means of the characteristic coefficients, which was open for ump-diffusions. A key role in our investigation is played by the method of the moving frame, which allows to transform term structure equations to time-dependent SDs. Key Words: term structure models driven by Wiener processes and Poisson measures, Heath-Jarrow-Morton-Musiela equation, positivity preserving models. 91B8, 6H15 1. Introduction Interest rate theory is dealing with zero-coupon bonds, which are subect to a stochastic evolution due to daily trading of related products like coupon bearing bonds, swaps, caps, floors, swaptions, etc. Zero-coupon bonds, which are financial assets paying the holder one unit of cash at maturity time T, are conceptually important products, since one can easily write all other products as derivatives on them. We always assume default-free bonds, i.e. there are no counterparty risks in the considered markets. The Heath-Jarrow-Morton methodology takes the bond market as a whole as today s aggregation of information on interest rates and one tries to model future flows of information by a stochastic evolution equation on the set of possible scenarios of bond prices. For the set of possible scenarios of bond prices the forward rate proved to be a flexible and useful parametrization, since it maps possible states of the bond market to open subsets of (Hilbert spaces of forward rate curves. Under some regularity assumptions the price of a zero coupon bond at t T can be written as ( T P (t, T = exp f(t, udu, t where f(t, T is the forward rate for date T. We usually assume the forward rate to be continuous in maturity time T. The classical continuous framework for the evolution of the forward rates goes back to Heath, Jarrow and Morton (HJM [3]. They assume that, for every date T, the forward rates f(t, T follow an Itô process Filipović gratefully acknowledges the support from WWTF (Vienna Science and Technology Fund and Swissquote. Tappe gratefully acknowledges the support from WWTF. Teichmann gratefully acknowledges the support from the FWF-grant Y 38 (STAT prize from the Austrian Science Fund. The authors thank two anonymous referees for their helpful remarks and suggestions.

2 DAMI FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN of the form (1.1 d df(t, T = α(t, T dt + σ (t, T dw t, t [, T ] =1 where W = (W 1,..., W d is a standard Brownian motion in d. There are several reasons for generalizing the HJM framework (1.1 by introducing umps. Namely, this allows us to model the impact of unexpected news about the economy, such as interventions by central banks, credit events, or (natural disasters. Indeed, there is strong statistical evidence in the finance literature that empirical features of the data cannot be captured by continuous models. We also mention that for the particular case of deterministic integrands α and σ in (1.1, as it is for the Vasiček model, the log returns for discounted zero coupon bonds are normally distributed, which, however, is not true for empirically observed log returns, see the discussion in [37, Chap. 5]. Börk et al. [3, 4], berlein et al. [1, 11, 1, 13, 14, 15] and others ([38, 6, 4] thus proposed to replace the classical Brownian motion W in (1.1 by a more general driving noise, also taking into account the occurrence of umps. Carmona and Tehranchi [6] proposed models based on infinite dimensional Wiener processes, see also [16]. In the spirit of Börk et al. [3] and Carmona and Tehranchi [6], we focus on term structure models of the type (1. df(t, T = α(t, T dt + σ(t, T dw t + γ(t, x, T (µ(dt, dx F (dxdt, where W denotes a (possibly infinite dimensional Wiener process and, in addition, µ is a homogeneous Poisson random measure on with compensator dt F (dx, where denotes the mark space. For what follows, it will be convenient to switch to the alternative parametrization r t (ξ := f(t, t + ξ, ξ which is due to Musiela [3]. Then, we may regard (r t t as one stochastic process with values in H, that is r : Ω H, where H denotes a Hilbert space of forward curves h : to be specified later. ecall that we always assume that forward rate curves are continuous. Denoting by (S t t the shift semigroup on H, that is S t h = h(t +, equation (1. becomes in integrated form (1.3 r t (ξ = S t h (ξ + + t t S t s α(s, s + ξds + t S t s σ(s, s + ξdw s S t s γ(s, x, s + ξ(µ(ds, dx F (dxds, t where h H denotes the initial forward curve and S t s operates on the functions ξ α(s, s + ξ, ξ σ(s, s + ξ and ξ γ(s, x, s + ξ. From a financial modeling point of view, one would rather consider drift and volatilities to be functions of the prevailing forward curve, that is α : H H, σ : H H, γ : H H. for all

3 TM STUCTU MODLS DIVN BY POISSON MASUS 3 For example, the volatilities could be of the form σ (h = φ (l 1 (h,..., l p (h for some p N with φ : p H and l i : H. We may think of l i (h = 1 ξi ξ i h(ηdη (benchmark yields or l i(h = h(ξ i (benchmark forward rates. (1.4 The implied bond market P (t, T = exp ( T t r t (ξdξ is free of arbitrage if we can find an equivalent (local martingale measure Q P such that the discounted bond prices ( t (1.5 exp r s (ds P (t, T, t [, T ] are local Q-martingales for all maturities T. In the sequel, we will directly specify the HJM equation under a martingale measure. More precisely, we will assume that the drift α = α HJM : H H is given by α HJM (h := ( (1.6 σ (hσ (h γ(h, x e Γ(h,x 1 F (dx for all h H, where we have set (1.7 (1.8 Σ (h(ξ := ξ Γ(h, x(ξ := ξ σ (h(ηdη, γ(h, x(ηdη. for all According to [3] (if the Brownian motion is infinite dimensional, see also [16], condition (1.6 guarantees that the discounted zero coupon bond prices (1.5 are local martingales for all maturities T, whence the bond market (1.4 is free of arbitrage. In the classical situation, where the model is driven by a finite dimensional standard Brownian motion, (1.6 is the well-known HJM drift condition derived in [3]. Our requirements lead to the forward rates (r t t in (1.3 being a solution of the stochastic equation (1.9 r t = S t h + + t t S t s α HJM (r s ds + t S t s σ(r s dw s S t s γ(r s, x(µ(ds, dx F (dxds, t and it arises the question whether this equation possesses a solution. To our knowledge, there has not yet been an explicit proof for the existence of a solution to the Poisson measure driven equation (1.9. We thus provide such a proof in our paper, see Theorem 3.4. For term structure models driven by a Brownian motion, the existence proof has been provided in [16] and for the Lévy case in [19]. We also refer to the related papers [36] and [8]. In the spirit of [8] and [35], an H-valued stochastic process (r t t satisfying (1.9 is a so-called mild solution for the (semi-linear stochastic partial differential equation (1.1 { drt = ( d dξ r t + α HJM (r t dt + σ(r t dw t + γ(r t, x(µ(dt, dx F (dxdt r = h, where d dξ becomes the infinitesimal generator of the strongly continuous semigroup of shifts (S t t.

4 4 DAMI FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN As in [], we understand stochastic partial differential equations as time-dependent transformations of time-dependent stochastic differential equations with infinite dimensional state space. More precisely, on an enlarged space H of forward curves h :, which are indexed by the whole real line, equipped with the strongly continuous group (U t t of shifts, we solve the stochastic differential equation (1.11 df t = U t lα HJM (πu t f t dt + U t lσ(πu t f t dw t + U tlγ(πu t f t, x(µ(dt, dx F (dxdt f = lh, where l : H H is an isometric embedding and π : H H is the orthogonal proection on H, and afterwards, we transform the solution process (f t t by r t := πu t f t in order to obtain a mild solution for (1.1. Notice that (1.11 ust corresponds to the original HJM dynamics in (1., where, of course, the forward rate f t (T has no economic interpretation for T < t. Thus, we will henceforth refer to (1.11 as the HJM (Heath-Jarrow-Morton equation. We emphasize that knowledge about the HJM equation (1.11 is not necessary in order to deduce existence and uniqueness for the forward curve evolution (1.1. The only thing we require in order to apply the existence result from [] is that we can embed the space H of forward curves into a larger Hilbert space H, on which the shift semigroup extends to a group. The point of the method of the moving frame from [] is to transform an SPD into a time-dependent SD on H, but we do not need the particular structure of this extension. Our existence result of this paper generalizes that for pure diffusion models from [16] and for term structure models driven by Lévy processes from [19] (we also mention the papers [36] and [8] concerning Lévy term structure models, see Corollary 3.6. As described above, in order to establish the proof we apply a result from [], which contains existence and uniqueness results for general SPDs on Hilbert spaces. Other references for SPDs on Hilbert spaces are [1] and [9]. Since the HJM drift term α HJM is of the particular form (1.6, we have to work out sufficient conditions which permit an application of this existence result. In practice, we are interested in term structure models producing positive forward curves, since negative forward rates are very rarely observed. After establishing the existence issue, we shall therefore focus on positivity preserving term structure models, and give a characterization of such models. The HJM equation (1.11 on the enlarged function space will be the key for analyzing positivity of forward curves. Indeed, the method of the moving frame, see [], allows us to use standard stochastic analysis (see [5] for our investigations. It will turn out that the conditions σ (h(ξ =, for all ξ (,, h P ξ and all h + γ(h, x P, for all h P and F -almost all x γ(h, x(ξ =, for all ξ (,, h P ξ and F -almost all x where P denotes the convex cone of all nonnegative forward curves, and P ξ is the edge consisting of all nonnegative forward curves h with h(ξ =, are necessary and sufficient for the positivity preserving property, see Theorem 4.4. For this purpose, we provide a general positivity preserving result, see Theorem 4., which is of independent interest and can also be applied on other function spaces. Positivity results for the diffusion case have been worked out in [7] and [3]. In particular, we would like to mention the important and beautiful work [34], where, through an application of a general support theorem, positivity is proved. We shall also apply this general argument for our reasonings.

5 TM STUCTU MODLS DIVN BY POISSON MASUS 5 The remainder of this text is organized as follows. In Section we introduce the space H β of forward curves. Using this space, we prove in Section 3, under appropriate regularity assumptions, the existence of a unique solution for the HJMM equation (1.1. The positivity issue of term structure models is treated in Section 4. There, we show first the necessary conditions with a general semimartingale argument. The sufficient conditions are proved to hold true by switching on the umps slowly. This allows for a reduction to results from [34]. For the sake of lucidity, we postpone the proofs of some auxiliary results to Appendix A.. The space of forward curves In this section, we introduce the space of forward curves, on which we will solve the HJMM equation (1.1 in Section 3. We fix an arbitrary constant β >. Let H β be the space of all absolutely continuous functions h : such that ( 1/ h β := h( + h (ξ e βξ dξ <. Let (S t t be the shift semigroup on H β defined by S t h := h(t + for t. Since forward curves should flatten for large time to maturity ξ, the choice of H β is reasonable from an economic point of view. Moreover, let H β be the space of all absolutely continuous functions h : such that ( 1/ h β := h( + h (ξ e dξ β ξ <. Let (U t t be the shift group on H β defined by U t h := h(t + for t. The linear operator l : H β H β defined by { h(, ξ < l(h(ξ := h H β h(ξ, ξ, is an isometric embedding with adoint operator π := l : H β H β given by π(h = h +, h H β..1. Theorem. Let β > be arbitrary. (.1 (. (.3 (.4 (1 The space (H β, β is a separable Hilbert space. ( For each ξ, the point evaluation h h(ξ : H β is a continuous linear functional. (3 (S t t is a C -semigroup on H β with infinitesimal generator d dξ : D( d dξ H β H β, d dξ h = h, and domain D( d dξ = {h H β h H β }. (4 ach h H β is continuous, bounded and the limit h( := lim ξ h(ξ exists. (5 H β := {h H β h( = } is a closed subspace of H β. (6 There are universal constants C 1, C, C 3, C 4 >, only depending on β, such that for all h H β we have the estimates h L 1 ( C 1 h β, h L ( C h β, h h( L1 ( C 3 h β, (h h( 4 e β L 1 ( C 4 h 4 β.

6 6 DAMI FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN (.5 (7 For each β > β, we have H β H β, the relation h β h β, h H β and there is a universal constant C 5 >, only depending on β and β, such that for all h H β we have the estimate (.6 (h h( e β L1 ( C 5 h β. (8 The space (H β, β is a separable Hilbert space, (U t t is a C -group on H β and, for each ξ, the point evaluation h h(ξ, H β is a continuous linear functional. (9 The diagram H β l U t Hβ π (.7 S H t β Hβ commutes for every t, that is Proof. See Appendix A. πu t l = S t for all t. 3. xistence of term structure models driven by Wiener processes and Poisson measures In this section, we establish existence and uniqueness of the HJMM equation (1.1 with diffusive and ump components on the Hilbert spaces introduced in the last section. Let < β < β be arbitrary real numbers. We denote by H β and H β the Hilbert spaces of the previous section, equipped with the strongly continuous semigroup (S t t of shifts, which has the infinitesimal generator d dξ. In the sequel, (Ω, F, (F t t, P denotes a filtered probability space satisfying the usual conditions. Let U be another separable Hilbert space and let Q L(U be a compact, selfadoint, strictly positive linear operator. Then there exist an orthonormal basis {e } of U and a bounded sequence λ of strictly positive real numbers such that Qu = λ u, e e, u U namely, the λ are the eigenvalues of Q, and each e is an eigenvector corresponding to λ, see, e.g., [41, Thm. VI.3.]. The space U := Q 1/ (U, equipped with the inner product u, v U := Q 1/ u, Q 1/ v U, is another separable Hilbert space and { λ e } is an orthonormal basis. Let W be a Q-Wiener process [8, p. 86,87]. We assume that tr(q = λ <. Otherwise, which is the case if W is a cylindrical Wiener process, there always exists a separable Hilbert space U 1 U on which W has a realization as a finite trace class Wiener process, see [8, Chap. 4.3]. We denote by L (H β := L (U, H β the space of Hilbert-Schmidt operators from U into H β, which, endowed with the Hilbert-Schmidt norm Φ L (H β := λ Φe, Φ L (H β itself is a separable Hilbert space.

7 TM STUCTU MODLS DIVN BY POISSON MASUS 7 According to [8, Prop. 4.1], the sequence of stochastic processes {β } defined as β := 1 λ W, e is a sequence of real-valued independent (F t -Brownian motions and we have the expansion (3.1 W = λ β e, where the series is convergent in the space M (U of U-valued square-integrable martingales. Let Φ : Ω L (H β be an integrable process, i.e. Φ is predictable and satisfies ( T P Φ t L (H β dt < = 1 for all T. Setting Φ := λ Φe for each, we have (3. t Φ s dw s = t Φ sdβ s, t where the convergence is uniform on compact time intervals in probability, see [8, Thm. 4.3]. Let (, be a measurable space which we assume to be a Blackwell space (see [9, ]. We remark that every Polish space with its Borel σ-field is a Blackwell space. Furthermore, let µ be a homogeneous Poisson random measure on, see [5, Def. II.1.]. Then its compensator is of the form dt F (dx, where F is a σ-finite measure on (,. Let measurable vector fields σ : H β L (Hβ and γ : H β Hβ be given, where the subspace Hβ was defined in Theorem.1. For each we define σ : H β Hβ as σ (h := λ σ(he. We shall now focus on the HJMM equation ( Assumption. We assume there exists a measurable function Φ : satisfying (3.3 a constant L > such that (3.4 (3.5 ( Γ(h, x(ξ Φ(x, h H β, x and ξ σ(h 1 σ(h L (H β L h 1 h β e Φ(x γ(h 1, x γ(h, x β F (dx 1/ L h 1 h β for all h 1, h H β, and a constant M > such that (3.6 (3.7 σ(h L (H β M e Φ(x ( γ(h, x β γ(h, x 4 β F (dx M for all h H β. Furthermore, we assume that for each h H β the map ( (3.8 α (h := γ(h, x e Γ(h,x 1 F (dx is absolutely continuous with weak derivative d (3.9 dξ α (h = γ(h, x e Γ(h,x F (dx d ( dξ γ(h, x e Γ(h,x 1 F (dx. 3.. emark. The proof of Proposition 3.3 below gives rise to the following remarks concerning conditions (3.8, (3.9 from Assumption 3.1.

8 8 DAMI FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN For each h H β the map α (h is well-defined (that is, for every ξ the integral in (3.8 exists and continuous (but not necessarily absolutely continuous with α (h( = and lim ξ α (h(ξ =. If α (h is absolutely continuous for some h H β, then we have α (h Hβ. For Lévy driven term structure models we can verify conditions (3.8 and (3.9 directly, see Corollary 3.6 below. Let h H β be such that γ(h, x C 1 ( for all x. Suppose for each ξ there exist δ = δ(h, ξ > and C = C(h, ξ > such that d γ(h, x(η dη C γ(h, x β for all x and η (ξ δ, ξ + δ. Then, we even have α (h C 1 ( with derivative ( Proposition. Suppose Assumption 3.1 is fulfilled. Then we have α HJM (H β Hβ and there is a constant K > such that (3.1 α HJM (h 1 α HJM (h β K h 1 h β for all h 1, h H β. Proof. Note that α HJM = α 1 + α, where α 1 (h := σ (hσ (h, h H β and α is given by (3.8. By [16, Cor. 5.1.] we have σ (hσ (h Hβ, h H β for all. For an arbitrary h H β we obtain, by using [16, Cor. 5.1.] again, σ (hσ (h β 3(C3 + C 4 σ (h β = 3(C3 + C 4 σ(h L (H β, and hence, we deduce α 1 (H β H β. Let h H β be arbitrary. For all x and ξ we have by (. and (.5 (3.11 γ(h, x(ξ C γ(h, x β C γ(h, x β, and for all x and ξ we have by (3.3, (.3 and (.5 (3.1 e Γ(h,x(ξ 1 e Φ(x Γ(h, x(ξ e Φ(x γ(h, x L1 ( C 3 e Φ(x γ(h, x β. stimates (3.11, (3.1 and (3.7 show that lim ξ α (h(ξ =. From (3.3, (3.11, (3.7 and (.6 it follows that ( CM CM γ(h, x(ξ e Γ(h,x(ξ F (dx e βξ dξ C MC 5 ( γ(h, x(ξ e Γ(h,x(ξ F (dx e βξ dξ e Φ(x γ(h, x(ξ e βξ dξf (dx e Φ(x γ(h, x β F (dx C M C 5.

9 TM STUCTU MODLS DIVN BY POISSON MASUS 9 We obtain by (3.1, Hölder s inequality, (3.7 and (.5 ( d ( dξ γ(h, x(ξ e Γ(h,x(ξ 1 F (dx e βξ dξ ( C3 d γ(h, x(ξ dξ e 1 Φ(x e 1 Φ(x γ(h, x β F (dx e βξ dξ C3M e Φ(x d γ(h, x(ξ dξ e βξ dξf (dx C3M e Φ(x γ(h, x β F (dx C 3M. In view of (3.9, we conclude that α (H β H β, and hence α HJM(H β H β. Let h 1, h H β be arbitrary. By [16, Cor. 5.1.], Hölder s inequality, (3.4 and (3.6 we have α 1 (h 1 α 1 (h β 3(C3 + C 4 ( σ (h 1 β + σ (h β σ (h 1 σ (h β 3(C3 + C 4 ( σ (h 1 β + σ (h β σ (h 1 σ (h β 6(C3 + C 4( σ(h 1 L (H β + σ(h L (H β σ(h 1 σ(h L (H β ML 6(C3 + C 4 h 1 h β. Furthermore, by (3.9, α (h 1 α (h β 4(I 1 + I + I 3 + I 4, where we have put I 1 := I := I 3 := I 4 := ( ( ( ( γ(h 1, x(ξ ( e Γ(h1,x(ξ e Γ(h,x(ξ F (dx e βξ dξ, e Γ(h,x(ξ (γ(h 1, x(ξ γ(h, x(ξ F (dx e βξ dξ, d ( dξ γ(h 1, x(ξ e Γ(h1,x(ξ e Γ(h,x(ξ F (dx e βξ dξ, ( ( e Γ(h,x(ξ d 1 dξ γ(h 1, x(ξ d dξ γ(h, x(ξ F (dx e βξ dξ. We get for all x and ξ by (3.3, (.3 and (.5 (3.13 e Γ(h1,x(ξ e Γ(h,x(ξ e Φ(x Γ(h 1, x(ξ Γ(h, x(ξ e Φ(x γ(h 1, x γ(h, x L 1 ( C 3 e Φ(x γ(h 1, x γ(h, x β.

10 1 DAMI FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN elations (3.13, Hölder s inequality, (3.5, (.4, (.5 and (3.7 give us ( I 1 C3 γ(h 1, x(ξ e 1 Φ(x e 1 Φ(x γ(h 1, x γ(h, x β F (dx e βξ dξ C3L h 1 h β e Φ(x γ(h 1, x(ξ 4 e βξ dξf (dx + C3L C 4 h 1 h β e Φ(x γ(h 1, x 4 β F (dx C 3L C 4 M h 1 h β. For every ξ we obtain by (3.11 and (3.7 (3.14 e Φ(x (γ(h 1, x(ξ + γ(h, x(ξ F (dx e Φ(x (γ(h 1, x(ξ + γ(h, x(ξ F (dx ( C e Φ(x γ(h 1, x β F (dx + e Φ(x γ(h, x β F (dx 4CM. Using (3.3, Hölder s inequality, (3.14, (.6 and (3.5 we get ( I (γ(h 1, x(ξ + γ(h, x(ξe 1 Φ(x e 1 Φ(x (γ(h 1, x(ξ γ(h, x(ξf (dx e βξ dξ 4CM e Φ(x (γ(h 1, x(ξ γ(h, x(ξ e βξ dξf (dx + 4C MC 5 4C MC 5 L h 1 h β. e Φ(x γ(h 1, x(ξ γ(h, x(ξ β F (dx Using (3.13, Hölder s inequality, (3.5, (.5 and (3.7 gives us ( I 3 C3 d dξ γ(h 1, x(ξ e 1 Φ(x e 1 Φ(x γ(h 1, x γ(h, x β F (dx e βξ dξ C3L h 1 h β C3L h 1 h β e Φ(x d dξ γ(h 1, x(ξ e βξ dξf (dx e Φ(x γ(h 1, x β F (dx C 3L M h 1 h β. We obtain by (3.1, Hölder s inequality, (3.7, (.5 and (3.5 ( I 4 C3 γ(h, x β e 1 Φ(x e 1 Φ(x d dξ γ(h 1, x(ξ d dξ γ(h, x(ξ F (dx e βξ dξ C3M e Φ(x C3M d dξ γ(h 1, x(ξ d dξ γ(h, x(ξ e βξ dξf (dx e Φ(x γ(h 1, x γ(h, x β F (dx C 3ML h 1 h β. Summing up, we deduce that there is a constant K > such that (3.1 is satisfied for all h 1, h H β.

11 TM STUCTU MODLS DIVN BY POISSON MASUS Theorem. Suppose Assumption 3.1 is fulfilled. Then, for each initial curve h L (Ω, F, P; H β there exists a unique adapted, càdlàg, mean-square continuous H β -valued solution (f t t for the HJM equation (1.11 with f = lh satisfying [ ] (3.15 sup t [,T ] f t β < for all T, and there exists a unique adapted, càdlàg, mean-square continuous mild and weak H β -valued solution (r t t for the HJMM equation (1.1 with r = h satisfying [ ] (3.16 sup t [,T ] r t β < for all T, which is given by r t := πu t f t, t. Moreover, the implied bond market (1.4 is free of arbitrage. Proof. By virtue of Theorem.1, Proposition 3.3 and (3.4, (3.5, (3.7, (.5, all assumptions from [, Cor. 1.9] are fulfilled, which therefore applies and establishes the claimed existence and uniqueness result. For all h H β, x and ξ we have by (3.3, (.3 and (.5 e Γ(h,x(ξ 1 Γ(h, x(ξ 1 eφ(x Γ(h, x(ξ ( eφ(x γ(h, x L 1 ( C 3 + eφ(x γ(h, x β. Integrating (1.6 we obtain, by using [16, Lemma 4.3.] and (3.17, (3.7 α HJM (h(ηdη = 1 ( Σ (h + e Γ(h,x 1 Γ(h, x F (dx for all h H β. Combining [3, Prop. 5.3] and [16, Lemma 4.3.3] (the latter result is only required if W is infinite dimensional, the probability measure P is a local martingale measure, and hence the bond market (1.4 is free of arbitrage. The case of Lévy-driven HJMM models is now a special case. We assume that the mark space is = e for some positive integer e N, equipped with its Borel σ-algebra = B( e. The measure F is given by e (3.18 F (B := 1 B (xe k F k (dx, B B( e k=1 where F 1,..., F e are Lévy measures on (, B( satisfying (3.19 (3. F k ({} =, k = 1,..., e ( x 1F k (dx <, k = 1,..., e and where the (e k k=1,...,e denote the unit vectors in e. Note that Definition (3.18 implies e (3.1 g(xf (dx = g(xe k F k (dx e k=1 for any nonnegative measurable function g : e. In particular, the support of F is contained in e k=1 span{e k}, the union of the coordinate axes in e. For each k = 1,..., e let δ k : H β Hβ be a vector field. We define γ : H β e Hβ as e (3. γ(h, x := δ k (hx k. k=1

12 1 DAMI FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN Then, equation (1.1 corresponds to the situation where the term structure model is driven by several real-valued, independent Lévy processes with Lévy measures F k. For all h H β and ξ we set k (h(ξ := ξ δ k (h(ηdη, k = 1,..., e Assumption. We assume there exist constants N, ɛ > such that for all k = 1,..., e we have (3.3 e zx F k (dx <, z [ (1 + ɛn, (1 + ɛn] (3.4 { x >1} k (h(ξ N, h H β, ξ a constant L > such that (3.4 and (3.5 δ k (h 1 δ k (h β L h 1 h β, k = 1,..., e are satisfied for all h 1, h H β, and a constant M > such that (3.6 and (3.6 are satisfied for all h H β. δ k (h β M, k = 1,..., e Now, we obtain the statement of [19, Thm. 4.6] as a corollary Corollary. Suppose Assumption 3.5 is fulfilled. Then, for each initial curve h L (Ω, F, P; H β there exists a unique adapted, càdlàg, mean-square continuous H β -valued solution (f t t for the HJM equation (1.11 with f = lh satisfying (3.15, and there exists a unique adapted, càdlàg, mean-square continuous mild and weak H β -valued solution (r t t for the HJMM equation (1.1 with r = h satisfying (3.16, which is given by r t := πu t f t, t. Moreover, the implied bond market (1.4 is free of arbitrage. Proof. Using (3.4, the measurable function Φ : e defined as e Φ(x := N x k, x e satisfies (3.3. For each k = 1,..., e and every m N with m we have (3.7 x m e zx F k (dx <, z ( (1 + ɛn, (1 + ɛn. k=1 Indeed, let z ( (1 + ɛn, (1 + ɛn be arbitrary. There exists η (, ɛ such that z (1 + ηn. By (3., (3.3 and the basic inequality x m m!e x for x we obtain x m e zx F k (dx x m e (1+ηN x F k (dx { x ln (1+ηN } x m F k (dx + m! ((ɛ ηn m { x > ln (1+ηN } e (1+ɛN x F k (dx <, proving (3.7. Taking into account (3.1, (3.7, relations (3.5, (3.6 imply (3.5, (3.7. Furthermore, (3.7, the elementary inequalities e x 1 x 1 x e x, x e x 1 x e x, x and Lebesgue s theorem show that the cumulant generating functions Ψ k (z = (e zx 1 zxf k (dx, k = 1,..., e

13 TM STUCTU MODLS DIVN BY POISSON MASUS 13 belong to class C on the open interval ( (1 + ɛn, (1 + ɛn with derivatives Ψ k(z = x(e zx 1F k (dx, Ψ (m k (z = x m e zx F k (dx, m. Therefore, and because of (3.1, we can, for an arbitrary h H β, write α (h, which is defined in (3.8, as e ( α (h = δ k (hψ k δ k (ηdη. k=1 Hence, α (h is absolutely continuous with weak derivative (3.9. Consequently, Assumption 3.1 is fulfilled and Theorem 3.4 applies. Note that the boundedness assumptions (3.6, (3.7 of Theorem 3.4 resp. (3.6, (3.6 of Corollary 3.6 cannot be weakened substantially. For example, for arbitrage free term structure models driven by a single Brownian motion, it was shown in [31, Sec. 4.7] that for the simple case of proportional volatility, that is σ(h = σ h for some constant σ >, solutions necessarily explode. We mention, however, that [36, Sec. 6] contains some existence results for Lévy term structure models with linear volatility. 4. Positivity preserving term structure models driven by Wiener processes and Poisson measures In applications, we are often interested in term structure models producing positive forward curves. In this section, we characterize HJMM forward curve evolutions of the type (1.1, which preserve positivity, by means of the characteristics of the SPD. In the case of short rate models this can be characterized by the positivity of the short rate, a one-dimensional Markov process. In case of an infinite-factor evolution, as described by a generic HJMM equation (see for instance [], this problem is much more delicate. Indeed, one has to find conditions such that a Markov process defined by the HJMM equation (on a Hilbert space of forward rate curves stays in a small set of curves, namely the convex cone of positive curves bounded by a non-smooth set. Our strategy to solve this problem is the following: First we show by general semimartingale methods necessary conditions for positivity. These necessary conditions are basically described by the facts that the Itô drift is inward pointing and that the volatilities are parallel at the boundary of the set of non-negative functions. Taking those conditions we can also prove that the Stratonovich drift is inward pointing, since parallel volatilities produce parallel Stratonovich corrections (a fact which is not true for general closed convex sets, but holds true for the set of non-negative functions P. Then we reduce the sufficiency proof to two steps: First we essentially apply results from [34] in order to solve the pure diffusion case and then we slowly switch on the umps to see the general result. Let H β be the space of forward curves introduced in Section for some fixed β >. We introduce the half spaces and define the closed, convex cone H + ξ := {h H β h(ξ }, ξ P := ξ H + ξ

14 14 DAMI FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN consisting of all nonnegative forward curves from H β. In what follows, we shall use that, by the continuity of the functions from H β, we can write P as P = H + ξ. Furthermore, we define the edges ξ (, P ξ := {h P h(ξ = }, ξ (,. First, we consider the positivity problem for general forward curve evolutions, where the HJM drift condition (1.6 is not necessarily satisfied, and afterwards we apply our results to the arbitrage free situation. We emphasize that, in the sequel, we assume the existence of solutions. Sufficient conditions for existence and uniqueness are provided in [] (we also mention the related articles [1] and [9] for general stochastic partial differential equations and in the previous Section 3 for the HJMM term structure equation (1.1. As in the previous section, we work on the space H β of forward curves from Section for some β >. At first glance, it looks reasonable to treat the positivity problem by working with weak solutions on H β. However, this is unfeasible, because the point evaluations at ξ (,, i.e., a linear functional ζ H β such that h(ξ = ζ, h for all h H β, do never belong to the domain D(( d dξ of the adoint operator. Indeed, a well-known mollifying technique shows that for each ξ (, the linear functional h h (ξ : D( d dξ is unbounded. Therefore treating the positivity problem with weak solutions does not bring an immediate advantage, hence we shall work with mild solutions on H β. Let measurable vector fields α : H β H β, σ : H β L (H β and γ : H β H β be given. Currently, we do not assume that the drift term α is given by the HJM drift condition (1.6. For each we define σ : H β H β as σ (h := λ σ(he. We assume that for each h P the HJM equation (4.1 df t = U t lα(πu t f t dt + U t lσ(πu t f t dw t + U tlγ(πu t f t, x(µ(dt, dx F (dxdt f = lh, has at least one H β -valued solution (f t t. Then, because of (.7, the transformation r t := πu t f t, t is a mild H β -valued solution of the HJMM equation (4. { drt = ( d dξ r t + α(r t dt + σ(r t dw t + γ(r t, x(µ(dt, dx F (dxdt r = h Definition. The HJMM equation (4. is said to be positivity preserving if for all h L (Ω, F, P; H β with P(h P = 1 there exists a solution (f t t of (4.1 with f = lh such that P( t {r t P } = 1, where r t := πu t f t, t. 4.. emark. Note that the seemingly weaker condition P(r t P = 1 for all t is equivalent to the condition of the previous definition due to the càdlàg property of the traectories Definition. The HJMM equation (4. is said to be locally positivity preserving if for all h L (Ω, F, P; H β with P(h P = 1 there exists a solution (f t t of (4.1 with f = lh and a strictly positive stopping time τ such that P( t {r t τ P } = 1, where r t := πu t f t, t Lemma. Let h P be arbitrary and let (f t t be a solution for (4.1 with f = lh. Set r t := πu t f t, t. The following two statements are equivalent:

15 TM STUCTU MODLS DIVN BY POISSON MASUS 15 (1 We have P( t {r t P } = 1. ( We have P( t [,T ] {f t(t } = 1 for all T (,. Proof. The claim follows, because the processes (r t t and (f t (T t [,T ] for an arbitrary T (, are càdlàg, and because the functions from H β are continuous Assumption. We assume that the vector fields α : H β H β and σ : H β L (H β are continuous and that h B γ(h, xf (dx is continuous on H β for all B with F (B < emark. Notice that, by Hölder s inequality, Assumption 4.5 is implied by Assumptions 4.1, 4.11 below, and therefore in particular by Assumption Proposition. Suppose Assumption 4.5 is fulfilled. If equation (4. is positivity preserving, then we have (4.3 γ(h, x(ξf (dx <, for all ξ (,, h P ξ (4.4 α(h(ξ γ(h, x(ξf (dx, for all ξ (,, h P ξ (4.5 (4.6 σ (h(ξ =, for all ξ (,, h P ξ and all h + γ(h, x P, for all h P and F -almost all x emark. Observe that condition (4.6 implies (4.7 γ(h, x(ξ, for all ξ (,, h P ξ and F -almost all x. Therefore, condition (4.3 is equivalent to γ(h, x(ξ F (dx <, for all ξ (,, h P ξ. Consequently, conditions (4.3 and (4.4 can be unified to γ(h, x(ξ F (dx α(h(ξ for all ξ (, and h P ξ. Proof. Let h P be arbitrary and let (f t t be a solution for (4.1 with f = lh such that P( t {r t P } = 1, where r t := πu t f t, t. By Lemma 4.4, for each T (, and every stopping time τ T we have (4.8 P(f τ (T = 1. Let φ U be a linear functional such that φ := φe for only finitely many, and let ψ : be a measurable function of the form ψ = c1 B with c > 1 and B satisfying F (B <. Let Z be the Doléans-Dade xponential ( Z t = φ β + ψ(x(µ(ds, dx F (dxds, t. t By [5, Thm. I.4.61] the process Z is a solution of Z t = 1 + t t φ Z s dβs + Z s ψ(x(µ(ds, dx F (dxds, t and, since ψ > 1, the process Z is a strictly positive local martingale. There exists a strictly positive stopping time τ 1 such that Z τ1 is a martingale. Due to the method of the moving frame, see [], we can use standard stochastic analysis, to

16 16 DAMI FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN proceed further. For an arbitrary T (,, integration by parts yields (see [5, Thm. I.4.5] (4.9 f t (T Z t = t f s (T dz s + Taking into account the dynamics (4.1 we have (4.11 (4.1 f t (T = lh (T + + t t + s t f s (T Z s, t. t f(t c, Z c t = f s (T Z s = s t U s lα(πu s f s (T ds + Z s df s (T + f(t c, Z c t t U s lσ (πu s f s (T dβ s U s lγ(πu s f s, x(t (µ(ds, dx F (dxds, t t t φ Z s U s lσ (πu s f s (T ds, t Z s ψ(xu s lγ(πu s f s, x(t µ(ds, dx, t. Incorporating (4.1, (4.11 and (4.1 into (4.9, we obtain t ( f t (T Z t = M t + Z s U s lα(πu s f s (T + φ U s lσ (πu s f s (T ( ψ(xu s lγ(πu s f s, x(t F (dx ds, t where M is a local martingale with M =. There exists a strictly positive stopping time τ such that M τ is a martingale. By Assumption 4.5 there exist a strictly positive stopping time τ 3 and a constant α > such that U (t τ3lα(πu t τ3 f (t τ3 (T α, t. Let B := {x : h + γ(h, x / P }. In order to prove (4.6, it suffices, since F is σ-finite, to show that F (B C = for all C with F (C <. Suppose, on the contrary, there exists C with F (C < such that F (B C >. By the continuity of the functions from H β, there exists T (, such that F (B T C >, where B T := {x : h (T + γ(h, x(t < }. We obtain γ(h, x(t F (dx (h (T + γ(h, x(t F (dx <. B T C B T C By Assumption 4.5 and the left-continuity of the process f., there exist η > and a strictly positive stopping time τ 4 T such that U (t τ4lγ(πu (t τ4f (t τ4, x(t F (dx η, t. B T C Let φ :=, ψ := α+1 η 1 B T C and τ := 4 i=1 τ i. Taking expectation in (4.13 we obtain [f τ (T Z τ ] <, implying P(f τ (T < >, which contradicts (4.8. This yields (4.6. From now on, we assume that h P T for an arbitrary T (,. Suppose that σ (h (T for some. By the continuity of σ (see Assumption 4.5 there exist η > and a strictly positive stopping time τ 4 T such that U (t τ4lσ (πu t τ4 f (t τ4 (T η, t.

17 TM STUCTU MODLS DIVN BY POISSON MASUS 17 Let φ U be the linear functional with φ = sign(σ (h (T α+1 η and φ k = for k. Furthermore, let ψ := and τ := 4 i=1 τ i. Taking expectation in (4.13 yields [f τ (T Z τ ] <, implying P(f τ (T < >, which contradicts (4.8. This proves (4.5. Now suppose γ(h, x(t F (dx =. Using Assumption 4.5, relation (4.7 and the σ-finiteness of F, there exist B with F (B < and a strictly positive stopping time τ 4 T such that 1 U (t τ4lγ(πu t τ4 f (t τ4, x(t F (dx ( α + 1, t. B Let φ :=, ψ := 1 1 B and τ := 4 i=1 τ i. Taking expectation in (4.13 we obtain [f τ (T Z τ ] <, implying P(f τ (T < >, which contradicts (4.8. This yields (4.3. Since F is σ-finite, there exists a sequence (B n n N with B n and F (B n <, n N. Next, we show for all n N the relation (4.14 α(h (T + ψ n (xγ(h, x(t F (dx, where ψ n := (1 1 n 1 B n. Suppose, on the contrary, that (4.14 is not satisfied for some n N. Using Assumption 4.5, there exist η > and a strictly positive stopping time τ 4 T such that U (t τ4lα(πu t f (t τ4 (T + ψ n (xu (t τ4lγ(πu t τ4 f (t τ4, x(t F (dx η, t. Let φ := and τ := 4 i=1 τ i. Taking expectation in (4.13 we obtain [f τ (T Z τ ] <, implying P(f τ (T < >, which contradicts (4.8. This yields (4.14. By (4.14, (4.3 and Lebesgue s theorem, we conclude ( emark. The Cox-Ingersoll-oss (CI model [7] is celebrated for its feature to produce nonnegative interest rates. At this point, it is worth pointing out that the Hull-White extension of the CI model (HWCI is not positivity preserving. Indeed, in Musiela parametrization its term structure dynamics are given by { drt = ( d dξ r t + α HJM (r t dt + σ(r t dw t r = h, where W is a one-dimensional Wiener process and the vector fields α HJM, σ : H β Hβ are defined as α HJM (h := ρ h( λλ, σ(h := ρ h( λ, h H β h H β where ρ > is a constant, h h( : H β denotes the evaluation of the short rate, and where λ Hβ is a function with λ(x > for all x such that Λ = λ(ηdη satisfies a certain iccati equation, see [1, Sec. 6.] for more details. Note that Assumption 4.5 is satisfied, because the vector fields α HJM, σ : H β Hβ are continuous, but condition (4.5 from Proposition 4.7 does not hold, because σ only depends on the current state of the short rate. Hence, the HWCI model cannot be positivity preserving.

18 18 DAMI FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN Indeed, we can also verify this directly as follows. According to [1, Sec. 6.], for any initial curve h H β the short rate t = r t (, t has the dynamics { dt = (b(t c t dt + ρ t dw t (4.15 = h ( for some constant c and a time-dependent function b = b(h :. Due to [4], for each starting point h ( the stochastic differential equation (4.15 has a unique strong solution, and, according to [17], this solution is nonnegative if and only if h ( and b(t for all t. By [18, Prop. 5.], the forward rates are given by f(t, T = T which implies for the initial forward curve h (T = T t b(sλ(t sds + λ(t t t, b(sλ(t sds + λ(t h (, T. Having in mind that λ(t > for all t, we see that for certain nonnegative initial curves h P the function b can also reach negative values, which yields negative short rates. For example, take an initial curve h P with h ( > and h (T = for some T >. We shall now present sufficient conditions for positivity preserving term structure models. In the sequel, we will require the following linear growth and Lipschitz conditions Assumption. We assume γ(, x βf (dx < and that there is a constant K > such that ( 1/ γ(h 1, x γ(h, x βf (dx K h 1 h β for all h 1, h H β Assumption. We assume there is a constant L > such that for all h 1, h H β. α(h 1 α(h β L h 1 h β, σ(h 1 σ(h L (H β L h 1 h β This ensures existence and uniqueness of solutions by [, Cor. 1.9] Lemma. Suppose Assumptions 4.1, 4.11 are fulfilled, and for each h P we have P( t {r t P } = 1, where (r t t denotes the mild solution for (4. with r = h. Then, the HJMM equation (4. is positivity preserving. Proof. Let h L (Ω, F, P; H β with P(h P = 1 be arbitrary. There exists a sequence (h n n N L (Ω, F, P; H β such that h n h in L (Ω, F, P; H β and for each n N we have P(h n P = 1 and h n has only a finite number of values. By assumption we have P( t {rt n P } = 1 for all n N. Applying [, Prop. 9.1] yields [ ] sup t [,T ] r t rt n β showing that P( t {r t P } = 1. for all T, For the following Lemma 4.14 we prepare an auxiliary result, which is proven in Appendix A.

19 TM STUCTU MODLS DIVN BY POISSON MASUS Lemma. Let τ be a bounded stopping time. We define the new filtration ( F t t by F t := F τ+t, the new U-valued process W by W t := W τ+t W τ and the new random measure µ on by µ(ω; B := µ(ω; B τ(ω, B B(, where B τ := {(t + τ, x : (t, x B}. Then W is a Q-Wiener process with respect to ( F t t and µ is a homogeneous Poisson random measure on with respect to ( F t t having the compensator dt F (dx. Moreover, we have the expansion W = (4.16 λ β e, where β defined as β t := β τ+t β τ is a sequence of real-valued independent ( F t - Brownian motions. Furthermore, if (r t t is a weak solution for (4., then the ( F t -adapted process ( r t t defined by r t := r τ+t is a weak solution for (4.17 { d rt = ( d dξ r t + α( r t dt + σ( r t d W t + γ( r t, x( µ(dt, dx F (dxdt r = r τ. Proof. See Appendix A Lemma. Suppose Assumptions 4.1, 4.11 are fulfilled. If equation (4. is locally positivity preserving and we have (4.6, then equation (4. is positivity preserving. Proof. Let h L (Ω, F, P; H β with P(h P = 1 be arbitrary. Moreover, let (r t t be the mild solution for (4. with r = h. We define the stopping time (4.18 τ := inf{t : r t / P }. By the closedness of P and (4.6 we have r τ P almost surely on {τ < }. We claim that P(τ = = 1. Assume, on the contrary, that (4.19 P(τ < N > for some N N. Let τ be the bounded stopping time τ := τ N. We define the new filtration ( F t t, the new Q-Wiener process W and the new Poisson random measure µ as in Lemma Note that r τ L (Ω, F, P; H β, because, by (3.16, we have [ [ r τ β] sup r t β t [,N] ] <. By Lemma 4.13, the ( F t -adapted process r t := r τ+t is the unique mild solution for (4.17. Since equation (4. is locally positivity preserving and P(r τ P = 1, there exists a strictly positive stopping time τ 1 such that P( t r t τ1 P = 1. Since {τ < N} {τ = τ}, we obtain r τ+t P almost surely on [, τ 1 ] {τ < N}, which is a contradiction because of (4.19 and the Definition (4.18 of τ. Consequently, we have P(τ = = 1, whence equation (4. is positivity preserving Assumption. We assume σ C (H β ; L (H β, and that the vector field (4. is globally Lipschitz on H β. h Dσ (hσ (h

20 DAMI FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN emark. Note that Assumption 4.15 is satisfied if σ C b (H β; L (H β and the series (4. converges for every h H β Lemma. Suppose Assumption 4.15 and relation (4.5 are fulfilled. Then we have ( Dσ (hσ (h (ξ =, for all ξ (,, h P ξ. Proof. Let ξ (, be arbitrary. It suffices to show ( Dσ (hσ (h (ξ = for all h P ξ and all. Therefore let be fixed and denote σ = σ. By assumption for all h with h(ξ = we have that σ(h(ξ =. In other words the volatility vector field σ is parallel to the boundary at boundary elements of P. We denote the local flow of the Lipschitz vector field σ by Fl being defined on a small time interval ] ɛ, ɛ[ around time and a small neighborhood of each element h P. We state first that the flow Fl leaves the set P invariant, i.e., Fl t (h if h, by convexity and closedness of the cone of positive functions due to [39]. Indeed, P is a closed and convex cone, whose supporting hyperplanes l (a linear functional l is called supporting hyperplane of P at h if l(p and l(h = are given by appropriate positive measures µ on via l(h = h(ξµ(dξ, whence condition (4 from [39] is fulfilled due to (4.5. Next we show that even more holds: the solution Fl t (h evaluated at ξ vanishes if h(ξ =, which we show directly. Indeed, let us additionally fix h P ξ, i.e., h and h(ξ =. Looking now at the Picard-Lindelöf approximation scheme c (n+1 (t = h + t σ(c (n (sds with c (n ( = h and c ( (s = h for s, t ] ɛ, ɛ[ and n, we see by induction that under our assumptions c (n (t(ξ = for all n and t ] ɛ, ɛ[ for the given fixed element h. Consequently as n we obtain that Fl t (h(ξ =, which is the limit of c (n (t. Therefore (Dσ(hσ(h(ξ = d ds s=σ(fl s (h(ξ =, since Fl t (h by invariance and Fl t (h(ξ = by the previous consideration lead to σ(fl t (h(ξ = for t ] ɛ, ɛ[. Notice that we did not need the global Lipschitz property of the Stratonovich correction for the proof of this lemma. Before we show sufficiency for the HJMM equation (4. with umps, we consider the pure diffusion case. Notice that, due to Lemma 4.17, the condition (4.4 is in fact equivalent to the very same condition formulated with the Stratonovich drift σ, defined in (4.1 below, instead of α, since the Stratonovich correction vanishes at the boundary of P. In order to treat the pure diffusion case, we apply [34], which, by using the support theorem provided in [33], offers a general characterization of stochastic invariance of closed sets for SPDs. Other results for positivity preserving SPDs, where, in contrast to our framework, the state space is an L -space, can be found in [7] and [3]. The results from [3] have been used in [36] in order to derive some positivity results for Lévy term structure models on L -spaces.

21 TM STUCTU MODLS DIVN BY POISSON MASUS Proposition. Suppose Assumptions 4.11, 4.15 are fulfilled and γ. If conditions (4.4, (4.5 are satisfied, then equation (4. is positivity preserving. Proof. In view of Lemma 4.1, it suffices to show that for all h P we have P( t {r t P } = 1, where (r t t denotes the mild solution for (4. with r = h. Moreover, we may assume that the vector fields α, σ and h σ (h := α(h 1 (4.1 Dσ (hσ (h are bounded in order to apply Nakayama s beautiful support theorem from [34]. Indeed, for n N we choose a bump function ψ n C (H β ; [, 1] such that ψ n 1 on B n ( and supp(ψ n B n+1 ( and define the vector fields α n (h := ψ n (hα(h, h H β σ n (h := ψ n (hσ(h, h H β. These vector fields and h σn(h := α n (h 1 Dσ n(hσn(h are bounded by the Lipschitz continuity of α, σ and σ, and Assumptions 4.11, 4.15 as well as conditions (4.4, (4.5 are again satisfied. Now, we show that the semigroup Nagumo s condition (3 from [33, Prop. 1.1] is fulfilled due to conditions (4.4 and (4.5. Introducing the distance d P (h from P as minimal distance of h H β from P, we can formulate Nagumo s condition as (4. lim inf t 1 t d P (S t h + tσ (h + tσ(hu = for all u U and h P. Fix now h P and u U and introduce the abbreviation σ = σ + σ( u, then we obviously have S t h + tσ (h + tσ(hu S Fl σ t t (h β = t Fl σ σ(h S t (h h t t, t > β which means that 1 lim t t S th + tσ (h + tσ(hu S Fl σ t t (h β =. Hence, Nagumo s condition (4. can equivalently be formulated as (4.3 lim inf t 1 t d P (S t Fl σ t (h =, for the particular choice of u U and h P, since the shortest distance proector onto P is a Lipschitz continuous map. Due to conditions (4.4, (4.5 and Lemma 4.17, the semiflow Fl σ leaves P invariant by [39], the semigroup (S t t certainly, too, and therefore we have d P (S t Fl σ t (h =, t whence Nagumo s condition (4.3 is more than satisfied. Finally, the next result states the sufficient conditions under which we can conclude that equation (4. is positivity preserving Proposition. Suppose Assumptions 4.1, 4.11, 4.15 and conditions (4.3 (4.6 are fulfilled. Then, equation (4. is positivity preserving. Proof. Since the measure F is σ-finite, there exists a sequence (B n n N with B n and F (B n < for all n N. Let h L (Ω, F, P; H β be arbitrary. elations (4.4, (4.7, (4.5, Proposition 4.18 and (4.6 together with the closedness

22 DAMI FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN of P yield that, for each n N, the mild solution (r n t t of the stochastic partial differential equation (4.4 drt n = ( d dξ rn t + α(rt n B n γ(rt n, xf (dxdt + σ(rt n dw t + B n γ(rt, n xµ(dt, dx r n = h satisfies P( t r n t τ P = 1, where τ denotes the strictly positive stopping time τ := inf{t : µ([, t] B n = 1}. By virtue of Lemma 4.14, for each n N equation (4.4 is positivity preserving. According to [, Prop. 9.1] we have [ ] sup t [,T ] r t rt n β for all T, proving that equation (4. is positivity preserving. 4.. Theorem. Suppose Assumptions 4.1, 4.11, 4.15 are fulfilled. Then, for each initial curve h L (Ω, F, P; H β there exists a unique adapted, càdlàg, meansquare continuous H β -valued solution (f t t for the HJM equation (4.1 with f = lh satisfying (3.15, and there exists a unique adapted, càdlàg, mean-square continuous mild and weak H β -valued solution (r t t for the HJMM equation (4. with r = h satisfying (3.16, which is given by r t := πu t f t, t. Moreover, equation (4. is positivity preserving if and only if we have (4.3 (4.6. Proof. The statement follows from [, Cor. 1.9], Proposition 4.7 (see also emark 4.6 and Proposition emark. Note that Theorem 4. is also valid on other state spaces. The only requirements are that the Hilbert space H consists of real-valued, continuous functions, on which the point evaluations are continuous linear functionals, and that the shift semigroup extends to a strongly continuous group on a larger Hilbert space H. 4.. emark. For the particular situation where equation (4. has no umps, Theorem 4. corresponds to the statement of [3, Thm. 3], where positivity on weighted L -spaces is investigated. Since point evaluations are discontinuous functionals on L -spaces, the conditions in [3] are formulated by taking other appropriate linear functionals. We shall now consider the arbitrage free situation. Let α = α HJM : H β H β in (4. be defined according to the HJM drift condition ( Proposition. Conditions (4.3 (4.6 are satisfied if and only if we have (4.5, (4.6 and (4.5 γ(h, x(ξ =, ξ (,, h P ξ and F -almost all x. Proof. Provided (4.5, (4.6 are fulfilled, conditions (4.3, (4.4 are satisfied if and only if we have (4.3 and (4.6 γ(h, x(ξe Γ(h,x(ξ F (dx, ξ (,, h P ξ because the drift α is given by (1.6. By (4.7, relations (4.3, (4.6 are fulfilled if and only if we have (4.5. Now let, as in Section 3, measurable vector fields σ : H β L (Hβ and γ : H β Hβ be given, where β > β is a real number.

23 TM STUCTU MODLS DIVN BY POISSON MASUS Theorem. Suppose Assumptions 3.1, 4.15 are fulfilled. Then, the statement of Theorem 3.4 is valid, and, in addition, the HJMM equation (1.1 is positivity preserving if and only if we have (4.5, (4.6, (4.5. Proof. The statement follows from Theorem 3.4, Theorem 4. and Proposition 4.3. Finally, let us consider the Lévy case, treated at the end of Section 3. In this framework, the following statement is valid Proposition. Conditions (4.5, (4.6 and (4.5 are satisfied if and only if we have (4.5 and (4.7 (4.8 h + δ k (hx P, h P, k = 1,..., e and F k -almost all x δ k (h(ξ =, ξ (,, h P ξ and all k = 1,..., e with F k ( >. Proof. The claim follows from the Definition (3.18 of F and the Definition (3. of γ Corollary. Suppose Assumption 3.5, 4.15 are fulfilled. Then, the statement of Corollary 3.6 is valid, and, in addition, the HJMM equation (1.1 is positivity preserving if and only if we have (4.5, (4.7, (4.8. Proof. The assertion follows from Theorem 4.4 and Proposition 4.5. Our above results on arbitrage free, positivity preserving term structure models apply in particular for local state dependent volatilities. The following two results are obvious Proposition. Suppose for each there exists σ :, and there are γ : and φ : (, such that σ (h(ξ = φ( h β σ (ξ, h(ξ, (h, ξ H β, for all γ(h, x(ξ = φ( h β γ(ξ, h(ξ, x, (h, x, ξ H β. Then, conditions (4.5, (4.6, (4.5 are fulfilled if and only if (4.9 (4.3 (4.31 σ (ξ, =, ξ (,, for all y + z γ(ξ, y, x, ξ (,, y, z φ( and F -almost all x γ(ξ,, x =, ξ (, and F -almost all x. Lévy term structure models with local state dependent volatilities have been studied in [36] and [8]. In the framework of Proposition 4.5 we obtain the following result Proposition. Suppose for each there is σ :, for all k = 1,..., e there is δ k : and there exists φ : (, such that σ (h(ξ = φ( h β σ (ξ, h(ξ, (h, ξ H β, for all δ k (h(ξ = φ( h β δ k (ξ, h(ξ, (h, ξ H β, k = 1,..., e. Then, conditions (4.5, (4.7, (4.8 are fulfilled if and only if we have (4.9 and (4.3 (4.33 y + zx δ k (ξ, y, ξ (,, y, z φ(, k = 1,..., e and F k -almost all x δ k (ξ, =, ξ (, and all k = 1,..., e with F k ( >.