REGULARITY OF AFFINE PROCESSES ON GENERAL STATE SPACES

Transcription

1 REGULARITY OF AFFINE PROCESSES ON GENERAL STATE SPACES MARTIN KELLER-RESSEL, WALTER SCHACHERMAYER, AN JOSEF TEICHMANN Abstract. We consider a stochastically continuous, affine Markov process in the sense of uffie, Filipovic, and Schachermayer [9], with càdlàg paths, on a general state space, i.e. an arbitrary Borel subset of R d. We show that such a process is always regular, meaning that its Fourier-Laplace transform is differentiable in time, with derivatives that are continuous in the transform variable. As a consequence, we show that generalized Riccati equations and Lévy-Khintchine parameters for the process can be derived, as in the case of = R m Rn studied in uffie et al. [9]. Moreover, we show that when the killing rate is zero, the affine process is a semi-martingale with absolutely continuous characteristics up to its time of explosion. Our results generalize the results of Keller-Ressel, Schachermayer, and Teichmann [15] for the state space R m Rn and provide a new probabilistic approach to regularity. 1. Introduction A time-homogeneous, stochastically continuous Markov process X on the state space R d is called affine, if its transition kernel p t x, dξ has the following property: There exist functions Φ and ψ, taking values in C and C d respectively, such that e ξ,u p t x, dξ = Φt, u exp x, ψt, u for all t R, x and u in the set U = { u C d : sup x Re u, x < }. The class of stochastic processes resulting from this definition is a rich class that includes Brownian motion, Lévy processes, squared Bessel processes, continuousstate branching processes with and without immigration [14], Ornstein-Uhlenbecktype processes [18, Ch. 17], Wishart processes [1] and several models from mathematical finance, such as the affine term structure models of interest rates [1] and the affine stochastic volatility models [13] for stock prices. For a state space of the form = R m Rn the class of affine processes has been originally defined and systematically studied by uffie et al. [9], under a regularity ate: November 9, Mathematics Subject Classification. 6J25. Key words and phrases. affine process, regularity, semimartingale, generalized Riccati equation. The first and third author gratefully acknowledge the support by the ETH foundation. The second author gratefully acknowledges financial support from the Austrian Science Fund FWF under grant P19456, from the European Research Council ERC under grant FA5641 and from the Vienna Science and Technology Fund WWTF under grant MA9-3. Furthermore this work was financially supported by the Christian oppler Research Association CG. The authors would like to thank Enno Veerman and Maurizio Barbato for comments on an earlier draft. 1

2 2 MARTIN KELLER-RESSEL, WALTER SCHACHERMAYER, AN JOSEF TEICHMANN condition. In this context, regularity means that the time-derivatives Φt, u ψt, u F u =, Ru = t t t=+ t=+ exist for all u U and are continuous on subsets U k of U that exhaust U. Once regularity is established, the process X can be described completely in terms of the functions F and R. The problem of showing that regularity of a stochastically continuous affine process X always holds true was originally considered for processes on the state space = R m Rn, and was proven giving a positive answer by Keller-Ressel et al. [15], building on results by awson and Li [8] and Keller-Ressel [16]. Already uffie et al. [9] remarked that affine processes can be considered on other state spaces R m Rn, where also no reduction to the canonical case by embedding or linear transformation is possible. One such example is given by the Wishart process for d 2, which is an affine process taking values in S + d, the cone of positive semidefinite d d-matrices. Recently, Cuchiero, Filipovic, Mayerhofer, and Teichmann [6] gave a full characterization of all affine processes with state space S + d and Cuchiero, Keller-Ressel, Mayerhofer, and Teichmann [7] consider the even more general case, when is an irreducible symmetric cone in the sense of Faraut and Korányi [11], which includes the S + d case.1 In both articles, regularity of the process remains a crucial ingredient, and the authors give direct proofs showing that regularity follows from the definition of the process, as in the case of = R m Rn. Even though the affine processes on R m Rn and on symmetric cones are regular and have been completely classified, it is known that this does not amount to a full classification of all affine processes on a general state space. A simple example is given by the process X x,x2 t = B t + x, B t + x 2 t, where B is a standard Brownian motion. This process is an affine process that lives on the parabola = { y, y 2, y R } R 2, and can be characterized by the functions 1 Φt, u = exp 1 2tu2 u 2 1t 21 2tu 2, ψt, u = u 1, u 2 /1 2tu 2. It can even be extended into an affine process on the parabola s epigraph { y, z : z y 2, y R } see uffie et al. [9, Sec. 12.2], but not into a process on the state space R m Rn, or on any symmetric cone. For more general results in this direction we refer to Spreij and Veerman [19], who provide a classification of affine diffusion processes on polyhedral cones and state spaces which are level sets of quadratic functions quadratic state spaces. They start from a slightly different definition of an affine process through a stochastic differential equation, which also immediately implies the regularity of the process. The contribution of this article is to show regularity of an affine process on a general state space R d under the only assumptions that is a non-empty Borel set whose affine span is R d and that the affine process has càdlàg paths. All existing regularity proofs, with the notable exception of Cuchiero and Teichmann [4] which has been prepared in parallel to this article use some particular properties 1 A symmetric cone is a self-dual convex cone, such that for any two points x, y a linear automorphism f of exists, which maps x into y. It is called irreducible if it cannot be written as a non-trivial direct sum of two other symmetric cones.

3 REGULARITY OF AFFINE PROCESSES ON GENERALSTATE SPACES 3 of the state space: In the case of S + d and the symmetric cones the fact that the set U has open interior, and in the case R m Rn a degeneracy argument that reduces the problem to R m, which is again a symmetric cone. The existence of an non-empty interior of U leads to a purely analytical proof based in broad terms on the theory of differentiable transformation semigroups of Montgomery and Zippin [17]; see Keller-Ressel et al. [15]. For general state spaces it is not true that U has non-empty interior, and the analytic technique ceases to work. An empty interior of U also causes problems when applying the techiques of uffie et al. [9] and Cuchiero et al. [6] to show the Feller property of the process. Therefore, we present in this paper a substantially different probabilistic technique that is independent of the nature of the state space under consideration. It should be considered as an alternative to the approach of Cuchiero and Teichmann [4], where another probabilistic regularity proof for affine processes on general state spaces is given. Our proof is largely self-contained, while the approach of Cuchiero and Teichmann [4] uses the theory of full and complete function classes put forward in Çinlar, Jacod, Protter, and Sharpe [3]. On the other hand, Cuchiero and Teichmann [4] also show the automatic right-continuity of the augmented natural filtration and the existence of a càdlàg modification of the affine process, which results in slightly weaker assumptions than in this article. 2. efinitions and Preliminaries Let be a non-empty Borel subset of the real Euclidian space R d, equipped with the Borel σ-algebra, and assume that the affine hull of is the full space R d. To we add a point δ that serves as a cemetery state, define = {δ}, = σ, {δ}, and equip with the Alexandrov topology, in which any open set with a compact complement in is declared an open neighborhood of δ. 2 Any continuous function f defined on is tacitly extended to by setting fδ =. Let Ω, F, F be a filtered space, on which a family P x x of probability measures is defined, and assume that F is P x -complete for all x and that F is right continuous. Finally let X be a càdlàg process taking values in, whose transition kernel 2.1 p t x, A = P x X t A, t, x, A is a normal time-homogeneous Markov kernel, for which δ is absorbing. That is, p t x,. satisfies the following: a x p t x, A is -measurable for each t, A R. b p x, {x} = 1 for all x, c p t δ, {δ} = 1 for all t d p t x, = 1 for all t, x R, and e the Chapman-Kolmogorov equation p t+s x, dξ = p t y, dξ p s x, dy holds for each t, s and x, dξ. 2 Note that the topology of enters our assumptions in a subtle way: We require later that X is càdlàg on, which is a property for which the topology matters.

4 4 MARTIN KELLER-RESSEL, WALTER SCHACHERMAYER, AN JOSEF TEICHMANN We equip R d with the canonical inner product,, and associate to the set U C d defined by { } 2.2 U = u C d : sup Re u, x < x. Note that the set U is the set of complex vectors u such that the exponential function x e u,x is bounded on. It is easy to see that U is a convex cone and always contains the set of purely imaginary vectors ir d. We will also need the sets { } 2.3 U k = u C d : sup Re u, x k x, k N, for which we note that U = k N U k. efinition 2.1 Affine Process. The process X is called affine with state space, if its transition kernel p t x, dξ satisfies the following: i it is stochastically continuous, i.e. lim s t p s x,. = p t x,. weakly for all t, x, and ii its Fourier-Laplace transform depends on the initial state in the following way: there exist functions Φ : R U C and ψ : R U C d, such that 2.4 e ξ,u p t x, dξ = Φt, u exp x, ψt, u for all t R, x and u U. Remark 2.2. Note that this definition does not specify ψt, u in a unique way. However there is a natural unique choice for ψ that will be discussed in Prop. 2.4 below. Also note that as long as Φt, u is non-zero, there exists φt, u such that Φt, u = e φt,u and 2.4 becomes 2.5 e ξ,u p t x, dξ = expφt, u + x, ψt, u. This is the essentially the definition that was used in uffie et al. [9]; with this notation the Fourier-Laplace transform is the exponential of an affine function of x. This is usually interpreted as the reason for the name affine process, even though affine functions also appear in other aspects of affine processes, e.g. in the coefficients of the infinitesimal generator, or in the differentiated semi-martingale characteristics. We use 2.4 instead of 2.5, as it leads to a slightly more general definition that avoids the a-priori assumption that the left hand side of 2.4 is non-zero. Interestingly, in the paper of Kawazu and Watanabe [14] also the big-φ notation is used to define a continuous-state branching process with immigration, which corresponds to an affine process on R in our terminology. Remark 2.3. It has recently been shown by Cuchiero [5] see also Cuchiero and Teichmann [4], that any affine process on a general state space has a càdlàg modification under every P x, x. Moreover, when X is an affine process relative to an arbitrary filtration F, then the P x -augmentation F x of F is right-continuous, for any x. This implies that the assumptions that we make on the path properties of X are in fact automatically satisfied after a suitable modification of the process.

5 REGULARITY OF AFFINE PROCESSES ON GENERALSTATE SPACES 5 Before we explore the first consequences of efinition 2.1, we introduce some additional notation. For any u U define 2.6 σu := inf {t : Φt, u = }, and 2.7 Q k := {t, u R U k : t < σu}, for k N. We set Q := k Q k. Finally on Q let φ be a function such that Φt, u = e φt,u t, u Q. The uniqueness of φ will be discussed. The functions φ and ψ have the following properties: Proposition 2.4. Let X be an affine process on. Then i It holds that σu > for any u U. ii The functions φ and ψ are uniquely defined on Q by requiring that they are jointly continuous on Q k for k N and satisfy φ, = ψ, =. iii The function ψ maps Q into U. iv The functions φ and ψ satisfy the semi-flow property. For any u U and t, s with t + s, u Q and s, ψt, u Q it holds that 2.8 φt + s, u = φt, u + φs, ψt, u, φ, u = ψt + s, u = ψs, ψt, u, ψ, u = u Proof. Choose some x, and for t, u R U define the function 2.9 ft, u = Φt, ue ψt,u,x = e u,ξ p t x, dξ. Fix k N and let t n, u n n N be a sequence in R U k converging to t, u R U k. For any ɛ > we can find a function ρ : [, 1] with compact support, such that 1 ρξp tx, dξ < ɛ. Moreover, there exists a N N such that e un,ξ e u,ξ < ɛ, n N, ξ supp ρ. By stochastic continuity of p t x, dξ we can find N 1 N such that 1 ρξp tn x, dξ < ɛ, n N 1, and also e u,ξ p tn x, dξ e u,ξ p t x, dξ < ɛ, n N 1,

6 6 MARTIN KELLER-RESSEL, WALTER SCHACHERMAYER, AN JOSEF TEICHMANN For n N 1, we now have ft n, u n ft, u = e un,ξ p tn x, dξ e u,ξ p t x, dξ e un,ξ ρξp tn x, dξ e u,ξ ρξp tn x, dξ + + e un,ξ 1 ρξp tn x, dξ e u,ξ 1 ρξp tn x, dξ + + e u,ξ p tn x, dξ e u,ξ p t x, dξ ɛ + kɛ + ɛ = ɛ2 + k. Since ɛ was arbitrary this shows the continuity of ft, u on R U k. Hence we conclude that t, u ft, u is continuous on R U k for each k N. Moreover ft, u = if and only if Φt, u = and f, u = e u,x for all u U. We conclude from the continuity of f that σu = inf {t : ft, u = } > for all u U and i follows. To obtain ii, note that for each x, we have just shown that the function t, u e u,ξ p t x, ξ maps Q k continuously into C \ {} for each k N. We claim that the mapping has a unique continuous logarithm 3, i.e. for each x there exists a unique function gx;.,., : Q C being continuous on Q k for k N, such that gx;, = and e u,ξ p t x, ξ = e gx;t,u. For each n N define the set K n = {t, u : u U n, u n, t [, σu 1/n]}. Clearly, the K n are compact subsets of Q n Q and exhaust Q as n. We show that every K n is contractible to. Let γ = tr, ur r [,1] be a continuous curve in K n. For each α [, 1] define γ α = αtr, ur r [,1]. Then γ α depends continuously on α, stays in K n for each α and satisfies γ 1 = γ and γ =, ur r [,1]. Thus any continuous curve in K n is homotopically equivalent to a continuous curve in {} U. Moreover, all continuous curves in {} U are contractible to, since U is a convex cone. We conclude that each K n is contractible to and in particular connected. Let H n : [, 1] K n K n be a corresponding contraction, and for some fixed x write f n t, u for the restriction of t, u e u,ξ p t x, ξ to K n. Since H n and f n are continuous and K n is compact, we have that lim t s f n H n t,. f n H n s,. =. Hence f n H n is a continuous curve in C b K n from f n to the constant function 1. By Bucchianico [2, Thm. 1.3] there exists a continuous logarithm g n C b K n that satisfies f n t, u = e gnt,u for all t, u K n. It follows that for arbitrary m n in N we have g m t, u = g n t, u + 2πi lt, u for all t, u K m, where lt, u is a continuous function from K m to Z satisfying l, =. But K m is connected, hence also the image of K m under l. We conclude that lt, u =, and that g m t, u = g n t, u for all t, u K m. Taking m = n this shows that g n is uniquely defined on each subset K n of Q. Taking m < n it shows that g n extends g m. Since the K n n N exhaust Q, it follows that there exists indeed, for each x, a unique function gx;. : Q C such that gx;, = and e u,ξ p t x, ξ = 3 We adapt a proof from Bucchianico [2, Thm.2.5] to our setting.

7 REGULARITY OF AFFINE PROCESSES ON GENERALSTATE SPACES 7 e gx;t,u. ue to 2.4 gx; t, u must be of the form φt, u + ψt, u, x, and since affinely spans R d also φt, u and ψt, u are jointly continuous on Q k for k N and uniquely determined on Q, whence we have shown ii. Next note that the rightmost term of 2.9 is uniformly bounded for all x. Thus also the middle term is, and we obtain that ψt, u U, as claimed in iii. Applying the Chapman-Kolmogorov equation to 2.4 and writing Φt, u = e φt,u yields that 2.1 exp φt + s, u + x, ψt + s, u = e ξ,u p t+s x, dξ = = p s x, dy e ξ,u p t y, dξ = e φt,u e y,ψt,u p s x, dy = = exp φt, u + φs, ψt, u + x, ψs, ψt, u for all x and for all u U such that t+s, u Q and s, ψt, u Q. Taking continuous logarithms on both sides iv follows. Remark 2.5. From now on φ and ψ shall always refer to the unique choice of functions described in Proposition Main Results 3.1. Presentation of the main result. We now introduce the important notion of regularity. efinition 3.1. An affine process X is called regular if the derivatives φt, u ψt, u 3.1 F u =, Ru = t t t=+ exist for all u U and are continuous on U k for each k N. t=+ Remark 3.2. Note that in comparison with the definition given in the introduction, we now define F u as the derivative at t = of t φt, u instead of t Φt, u. In light of Proposition 2.4 these definitions coincide, since φt, u is always defined for t small enough and satisfies Φt, u = e φt,u with φ, u =. Our main result is the following. Theorem 3.3. Let X be a càdlàg affine process on R d. Then X is regular. Before this result is proved in the subsequent sections, we illustrate why regularity is a crucial property. The following result has originally been established by uffie et al. [9] for affine processes on the state-space R n R m. Proposition 3.4. Let X be a regular affine process with state space. Then there exist R d -vectors b, β 1,..., β d ; d d-matrices a, α 1,..., α d ; real numbers c, γ 1,..., γ d and signed Borel measures m, µ 1,..., µ d on R d \ {}, such that for all u U the functions F u and Ru can be written as F u = 1 3.2a 2 u, au + b, u c + e ξ,u 1 hξ, u mdξ, 3.2b R i u = 1 2 R d \{} u, α i u + β i, u γ i + e ξ,u 1 hξ, u µ i dξ, R d \{}

8 8 MARTIN KELLER-RESSEL, WALTER SCHACHERMAYER, AN JOSEF TEICHMANN with truncation function hx = x1 { x 1}, and such that for all x the quantities 3.3a 3.3b 3.3c 3.3d Ax = a + x 1 α x d α d, Bx = b + x 1 β x d β d, Cx = c + x 1 γ x d γ d, νx, dξ = mdξ + x 1 µ 1 dξ + + x d µ d dξ have the following properties: Ax is positive semidefinite, Cx and ξ 2 1 νx, dξ <. R d \{} Moreover, for u U the functions φ and ψ satisfy the ordinary differential equations 3.4a φt, u = F ψt, u, t φ, u = 3.4b ψt, u = Rψt, u, t ψ, u = u. for all t [, σu. Remark 3.5. The differential equations 3.4 are called generalized Riccati equations, since they are classical Riccati differential equations, when mdξ = µ i dξ =. Moreover equations 3.2 and 3.3 imply that u F u + Ru, x is a function of Lévy-Khintchine form for each x. Note that the Proposition makes no claim on the uniqueness of the solutions of the generalized Riccati equations. In fact examples are known where uniqueness does not hold true cf. uffie et al. [9, Ex. 9.3]. Proof. The equations 3.4 follow immediately by differentiating the semi-flow equations 2.8. The form of F, R follows by the following argument: By 3.1 and the affine property 2.4 it holds for all x and u U that F u + x, Ru = lim t t { } e φt,u+ x,ψt,u u 1 = { } 1 = lim e ξ x,u p t x, dξ 1 = t t { 1 = lim e ξ x,u 1 p t x, dξ + p } tx, 1 = t t t { 1 } = lim e ξ,u p t x, 1 1 p t x, dξ + lim, t t t t x where we write p t x, dξ := p t x, dξ + x for the shifted transition kernel. Inserting u = into the above equation shows that lim t p t x, 1/t converges to F + x, R. Set c = F and γ = R and write F u = F u + c and Ru = Ru + γ, such that 3.6 exp F u + x, Ru { 1 } = lim exp e ξ,u 1 p t x, dξ. t t x For each t and x, the exponential on the right hand side is the Fourier- Laplace transform of a compound Poisson distribution with jump measure p t x, dξ and jump intensity 1 t cf. Sato [18, Ch. 4]. The Fourier-Laplace transforms converge

9 REGULARITY OF AFFINE PROCESSES ON GENERALSTATE SPACES 9 pointwise for u U and in particular for all u ir d as t. By the assumption of regularity the pointwise limit is continuous at u = as function on ir d U k for each k N, which implies by Lévy s continuity theorem that the compound Poisson distributions converge weakly to a limiting probability distribution. Moreover, as the weak limit of compound Poisson distributions, the limiting distribution must be infinitely divisible. Let us denote the law of the limiting distribution, for given x, by Kx, dy. Since it is infinitely divisible, its characteristic exponent is of Lévy-Khintchine form, and we obtain the identity 3.7 F u + x, Ru = log e ξ,u Kx, dξ = R d = 1 2 uax, u + Bx, u R d e ξ,u 1 hξ, u νx, dξ, where for each x, Ax is a positive semi-definite d d-matrix, Bx R d, and νx, dξ a σ-finite Borel measure on R d \ {} and ξ 2 1 νx, dξ <. Note that in the step from 3.6 from 3.7 we have used that F u and Ru are continuous on every U k, k N, and hence that F u + x, Ru is the unique continuous logarithm of exp F u + x, Ru on each U k and for all x. Since 3.7 holds for all x, and contains at least d + 1 affinely independent points, we conclude that Ax, Bx and νx, dξ are of the form given in 3.3 and the decompositions in 3.2 follow. In general, the parameters a, α i, b, β i, c, γ i, m, µ i i {1,...,d} of F and R have to satisfy additional conditions, called admissibility conditions, that guarantee the existence of an affine Markov process X with state space and prescribed F and R. It is clear that such conditions depend strongly on the geometry of the state space, in particular of its boundary. Finding such necessary and sufficient conditions on the parameters for different types of state spaces has been the focus of several publications. For = R m Rn the admissibility conditions have been derived by uffie et al. [9], for = S + d, the cone of semi-definite matrices by Cuchiero et al. [6], and for cones that are symmetric and irreducible in the sense of Faraut and Korányi [11] by Cuchiero et al. [7]. Finally for affine diffusions m = µ i = on polyhedral cones and on quadratic state spaces the admissiblility conditions have been given by Spreij and Veerman [19]. The purpose of this article is not to derive these admissibility conditions for conrete specifications of, but merely to show that for any arbitrary state space there are parameters in terms of which admissibility conditions can be formulated Auxiliary Results. For the sake of simpler notation we define ϱt, u = ψt, u u. Note that we have ϱ, u = for all u U. The following Lemma is a purely analytical result that will be needed later. Lemma 3.6. Let K be a compact subset of U l for some l N and assume that φt, u ϱt, u 3.8 lim sup sup + =. t t t u K

10 1 MARTIN KELLER-RESSEL, WALTER SCHACHERMAYER, AN JOSEF TEICHMANN Then there is x, ε >, η >, z C with z = 1, a sequence t k k=1 of positive real numbers, a sequence M k k=1 of integers satisfying 3.9 lim k t k =, lim M k =, k lim M kt k =, k and a sequence of complex vectors u k k= in K such that u k u and 3.1 φt k, u k + x, ϱt k, u k η ϱt k, u k. Moreover, for all ξ R d satisfying x ξ < ε, 3.11 M k φt k, u k + ξ, ϱt k, u k = z + e k,ξ, where the complex numbers e k,ξ describing the deviation from z satisfy e k,ξ < 1 2 and lim k sup {ξ: x ξ <ε} e k,ξ =. Remark 3.7. The essence of the above Lemma is that the behavior of φt, u and ϱt, u as t approaches can be crystallized along the sequences t k and M k. Equation 3.9 then states that t k = o 1 M k, and 3.11 asserts that the asymptotic equivalence φt k, u k + ξ, ϱt k, u k 1 M k, holds uniformly for all ξ in an ε-ball around x. Proof. We first show all assertions of the Lemma for a sequence M k k N of positive but not necessarily integer numbers. In the last step of the proof we show that it is possible to switch from M k k N to the integer sequence M k k N. By Assumption 3.8 we can find a sequence t k k= and a sequence u k k= with u k K, such that φt k, u k + ϱt k, u k t k. Passing to a subsequence, and using the compactness of K, we may assume that u k converges to some point u K. For more concise notation, we write from now on φ k = φt k, u k and ϱ k = ϱt k, u k. Note that φ k and ϱ k, by joint continuity of φ and ϱ on U l, and the fact that φ, u = and ϱ, u =. Let us now show 3.1. By assumption, contains d + 1 affinely independent vectors x, x 1,..., x d. Assume for a contradiction that φ k + ϱ k, x j 3.12 lim k ϱ k for all x j, j {,..., d}. Since the vectors x j affinely span R d, the vectors {x 1 x,..., x d x } are linearly independent, and we can find some numbers α j,k C, such that 3.13 ϱ k / ϱ k = j=1 d α j,k x j x, j=1 for all k N. Moreover, since ϱ k / ϱ k is bounded also the α j,k are bounded by a constant. By direct calculation we obtain d φk + ϱ k, x j 3.14 α j,k φ k + ϱ k, x = ϱ k/ ϱ k, ϱ k = 1, ϱ k ϱ k ϱ k

11 REGULARITY OF AFFINE PROCESSES ON GENERALSTATE SPACES 11 for all k N. On the other hand, 3.12 implies that the left hand side of 3.14 converges to as k, which is a contradiction. We conclude that there exists x for which φ k + ϱ k, x 3.15 η ϱ k for some η > after possibly passing to subsequences, whence 3.1 follows. To show 3.11, set M k = φ k + x, ϱ k 1. Passing once more to a subsequence, and using the compactness of the complex unit circle, we can find some α [, 2π such that arg φ k + x, ϱ k α. Now φ k + ξ, ϱ k = φ k + x, ϱ k + ξ x, ϱ k = 1 Mk e iα + e 1 k + ξ x, ϱ k where e 1 k as k. Multiplying by M k and setting z = e iα we obtain M k φ k + ξ, ϱ k = z + e 1 k + e 2 k,ξ by 3.1 we can where we can estimate e 2 k,ξ M k ε ϱ k. Since M k ϱ k 1 η make e 2 k,ξ arbitrarily small by choosing a small enough ε. Setting e k,ξ = e 1 k + e2 k,ξ we obtain Finally, for each k N let M k be the nearest integer greater than M k. It is clear that after possibly removing a finite number of terms from all sequences, the assertion of the Lemma is not affected from switching from M k to M k. Lemma 3.8. Let X = X t t be an affine process starting at X and let u U, >, ε >. efine n 3.16 Ln,, u = exp u, X n X φ, u + ϱ, u, Xj 1 and the stopping time N = inf {n N : X n X > ε} Then n Ln N,, u is a F n n N -martingale under every measure P x, x. Proof. It is obvious that each Ln,, u is F n -measurable. The definition of the stopping time N guarantees the integrability of Ln N,, u. We show the martingale property by combining the affine property of X with the tower law for conditional expectations. Write n S n = φ, u + ϱ, u, Xj 1, j=1 and note that S n is F n 1 -measurable. On {n N } we have that E x [ Ln,, u F n 1 ] = E x [ exp u, X n X F n 1 ] e S n = j=1 = exp φ, u + ψ, u, X n 1 u, X S n = = exp u, X n 1 X Sn 1 = Ln 1,, u, showing that n Ln N,, u is indeed a F n n N -martingale under every P x, x. We combine the two preceding Lemmas to show the following.

12 12 MARTIN KELLER-RESSEL, WALTER SCHACHERMAYER, AN JOSEF TEICHMANN Proposition 3.9. Let X be a càdlàg affine process. Then the associated functions φt, u and ϱt, u = ψt, u u satisfy φt, u ϱt, u 3.17 lim sup sup + < t u K t t for each compact subset K of U l and each l N. Proof. We argue by contradiction: Fix l N and assume that 3.17 fails to hold true. Then by Lemma 3.6 there exist ε > and sequences u k u in K, t k and M k such that t k M k and equations 3.1, 3.11 hold. efine the F ntk n N -stopping times N k = inf {n N : X ntk X > ε}. Then setting = t k in Lemma 3.8 yields that 3.18 n Ln N k, t k, u k = = exp n N k u k, X n Nk t k X j=1 φtk, u k + ϱt k, u k, X j 1tk is a F ntk n N -martingale. It follows in particular that E [LM k N k, t k, u k ] = 1 for all k N. By 3.11, we have the uniform bound M k N k LM k N k, t k, u k C exp φtk, u k + ϱt k, u k, X j 1tk 3.19 C exp3/2, j=1 where C = exp Re u, X. Let δ > and x. Since X is càdlàg we can find a T > such that P sup x t [,T ] X t X > ε < δ. For k large enough t k M k T and hence PM k > N k < δ. We conclude that P lim x M K N k k M k = 1 1 δ, M and since δ was arbitrary lim k N k k M k = 1 holds P x -a.s. for any x. Together with 3.11 and 3.19 we obtain by dominated convergence that [ ] 3.2 lim k Ex [LM k N k, T k, u k ] = E x lim LM k N k, T k, u k = k [ ] = E x lim exp M k N k φt k, u k + ϱt k, u k, x = e z. k where z = 1. But E x [LM k N k, T k, u k ] = 1 by its martingale property, which is the desired contradiction Affine processes are regular. In this section we prove the main result, Theorem 3.3. Lemma 3.1. Let a sequence t k u be assigned to each u U. Then each of these sequences has a subsequence Su := s k u k N such that the limits 3.21 F S u := lim s k u φs k u, u ϱs k u, u, R S u := lim s k u s k u s k u are well-defined and finite. Moreover the subsequences Su can be chosen such that the numbers F S u and R S u are bounded on each compact subset K of U l for each l N.

13 REGULARITY OF AFFINE PROCESSES ON GENERALSTATE SPACES 13 Proof. Let the sequences t k u be given, but assume that the assertion of the Lemma does not hold true. Then either t k u for some u U has no subsequence for which the limits in 3.21 exist, or the limits F u and Ru exist for each u U, but at least one of them is not bounded in some compact K U l for some l N. Consider the first case. By the Bolzano-Weierstrass theorem an R d -valued sequence that contains no convergent subsequence must be unbounded, and we conclude that lim sup t k u φtk u, u t k u + ϱt ku, u t k u =, in contradiction to Proposition 3.9. Consider now the second assertion. Fix l N. For each u U l there is a sequence s k u such that 3.21 holds, but F S u or R S u is not bounded in K U l, i.e. there exists a sequence u n u in K for which F u n + Ru n. Fix some η >. Then for each k N there exists an N k N such that φs Nk u k, u k s Nk u k F u k η/2 We conclude that φsk, u lim sup sup s k u K s k and + ϱs k, u s k ϱs Nk u k, u k s Nk u k Ru k η/2. lim sup F u k + Ru k η =, k again in contradiction to Prop Having shown Lemma 3.1, only a small step remains to show regularity. Comparing with efinition 3.1 we see that two ingredients are missing: First we have to show that the limits F u and Ru do not depend on the choice of subsequence, i.e. they are proper limits and hence the proper derivatives of φ and ψ at t =, and second we have to show that F and R are continuous on U l for each l N. Proof of Theorem 3.3. Our first step is to show that the derivatives F u and Ru in 3.1 exist. By Lemma 3.1 we already know that they exist as limits along a sequence Su which depends on the point u U and has been chosen as a particular subsequence of a given sequence t k u k N. We show now that the limit is in fact independent of the choice of Su and even of the original sequence t k u k N, and hence that F u and Ru are proper derivatives in the sense of 3.1. To this end, fix some u U, and let Su be an arbitrary other sequence s k u, such that 3.22 FS u := lim s k u φ s k u, u, RS u := lim s k u s k u ϱ s k u, u. s k u We want to show that F S u = F S u and R S u = R S u. Assume for a contradiction that this were not the case. Then we can find x and r > such that the convex set {F S u + R S u, ξ : ξ x r} and its counterpart involving S are disjoint, i.e { } { } F S u + R S u, ξ : ξ x r FS u + RS u, ξ : ξ x r =.

14 14 MARTIN KELLER-RESSEL, WALTER SCHACHERMAYER, AN JOSEF TEICHMANN For the next part of the proof, we set τ = inf {t : X t X r}, and introduce the following notation: a u t := F S u + R S u, X t, A u t := t a u s ds, G u t := expa u t, Yt u := exp u, X t X with ã u t, Ã u t and G u t the corresponding counterparts for F S and R S. We show that 3.24 L u t τ = Y t τ u t τ G u = exp u, X t τ X F S u + R S u, X s ds t τ is a martingale under every P x, x. This reduces to showing that ] E x [exp u, X h τ X h τ since then by the Markov property of X F S u + R S u, X s ds = 1, 3.25 ] u, E [exp x t+h τ Xt+h τ X t τ F S u + R S u, X s ds t τ F t = ] u, E [exp x t+h τ Xt+h τ X t τ F S u + R S u, X s ds 1 τ t F t +1 τ t = = E Xt [exp u, X h τ X t τ h τ F S u + R S u, X s ds ] 1 τ t +1 τ t = 1 holds true. Now, use the sequence Su = s n u n N to define a sequence of Riemannian sums approximating the above integral. efine M k = h/s k and N k = inf {n N : X nsk X > r}. First we show that s k N k τ almost surely under every P x. Fix ω Ω such that t X t ω is a càdlàg function. Let Ñkω be a sequence in N such that s k Ñ k ω τω. It follows from the right-continuity of t X t ω that for large enough k it holds that X sk Ñ k X > r and hence that eventually Ñkω N k ω. On the other hand X sk N k X > r for all k N, which implies that N k ωs k τω. Hence, for large enough k N it holds that s k Ñ k ω s k N k ω τω. We also know that s k Ñ k ω τω as k, such that we conclude that s k N k τ P x -almost surely, as claimed. By Riemann approximation and the fact that X is càdlàg it then holds that M k N k j=1 FS u + h τ R S u, X j 1sk sk F S u + R S u, X s ds P x -almost-surely as k for all x. From Lemma 3.1 we know that φs k, u = F S us k + os k and φs k, u = R S us k + os k. Moreover M k N k os k since M k s k. Thus we have

15 REGULARITY OF AFFINE PROCESSES ON GENERALSTATE SPACES 15 that LM k N k, s k, u = = exp M k N k u, X Mk N k s k X M k N k j=1 j=1 = exp u, X Mk N k s k X φtk, u + ϱt k, u, X j 1sk = FS u + R S u, X j 1sk sk + M k N k os k exp u, X h τ X h τ F S u + R S u, X s ds as k almost surely with respect to all P x, x. But by Lemma 3.8 and optional stopping, E x [LM k N k, s k, u] = 1, such that by dominated convergence cf we conclude that [ ] E exp u, X h τ X h τ F S u + R S u, X s ds = 1, and hence that t L u t τ is a martingale. Summing up we have established that Yt τ u = L u t τ G u t τ, where L u t τ is a martingale and hence a semimartingale. Clearly, the process G u t τ is predictable and of finite variation and hence a semimartingale too. We conclude that also the product Yt τ u = exp u, Xt τ x x is a semimartingale. It follows from Jacod and Shiryaev [12, Thm. I.4.49] that Mt τ u = Yt τ u t τ L u s dg u s is a local martingale. We can rewrite Mt u as M u t = Y u t τ t L u s G u s da u s = Y u t t Ys da u u s = Yt u t Y u s a u s ds. Hence Yt τ u = Mt τ u + t τ Y u s a u s ds is the decomposition of the semi-martingale Yt τ u into a local martingale and a finite variation part. But t τ Y u s a u s ds is even predictable, such that Y u is a special semi-martingale, and the decomposition is unique. The same derivation goes through with A u replaced by Ãu and by the uniqueness of the special semi-martingale decomposition we conclude that t τ Y u s a u s ds = t τ Y u s ã u s ds, up to a P x -nullset. Taking derivatives we see that Yt a u u t = Yt ã u u t on {t τ}. As long as t τ it holds that Yt u, and dividing by Yt, u we see that a u t = ã u t, that is F S u + R S u, X t = F S u + RS u, X t on {t τ}, P x -a.s, in contradiction to We conclude that the limits F S and R S are independent from the sequence S, and hence that F u and Ru exist as proper derivatives in the sense of 3.1. It remains to show that F u and Ru are continuous on U l for each l N. Fix l N and suppose for a contradiction that there exists a sequence u k u,

16 16 MARTIN KELLER-RESSEL, WALTER SCHACHERMAYER, AN JOSEF TEICHMANN in U l such that F u k F and Ru k R, such that either F u F or Ru R. Since affinely spans R d this means that there is x with F u + Ru, x F + R, x. Using the fact that E x [L u k t τ ] = 1 for all k N we obtain t exp φt, u + ψt, u, x 1 = lim 1 [e t Ex u,xt X 1 exp = lim k k t τ = E x [ 1 t e u,xt X 1 exp 1 [ t Ex e u,xt X L u k t τ ] F u k + Ru k, X s ds t τ ] = = ] F + R, X s ds. for all t σ see 2.6 for the definition of σ. by dominated convergence. Writing C = F + R ε and using the elementary inequality 1 e z z e z we can bound 1 t e u,xt X 1 exp t τ F + R, X s ds Ce 2l+Ct and therefore apply again dominated convergence to the right hand side of 3.26 as t. Taking the limit on both sides, we obtain leading to the desired contradiction. F u + Ru, x = F + R, x We conclude with a corollary that gives conditions for an affine process to be a - valued semimartingale, up to its explosion time. Let τ n = inf {t : X t X > n} and define the explosion time τ exp as the pointwise limit τ exp = lim n τ n. Note that τ exp is predictable. Corollary Let X be a càdlàg affine process and suppose that the killing terms vanish, i.e. c = and γ =. Then under every P x, x the process X is a -valued semi-martingale on [, τ exp with absolutely continuous semimartingale characteristics A t = B t = K[, t], dξ = t t t where A., B. and ν., dξ are given by 3.3. AX s ds BX s ds νx s, dξds. Proof. In the proof of Theorem 3.3 we have shown that t L u t τ, with L u t defined in 3.24 and τ = inf {t : X t X > r}, is a martingale under every P x, x and for every u U. Since r > was arbitrary, also L u t τ n is a martingale for every n N. By dominated convergence and using that F + R, x = c + γ, x = for all x we obtain P x X t τexp δ = lim n P X t τ n δ = E [ L t τ n ] = 1.

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18 18 MARTIN KELLER-RESSEL, WALTER SCHACHERMAYER, AN JOSEF TEICHMANN [19] P. Spreij and E. Veerman. Affine diffusions with non-canonical state space. Stochastic Analysis and Applications, 34:65 641, 212. arxiv: epartment of Mathematics, TU Berlin, Germany address: Faculty of Mathematics, University of Vienna, Austria address: epartment of Mathematics, ETH Zurich, Switzerland address: