Regularity of affine processes on general state spaces

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El e c t ro nic J

o f

Pr

ob a bi l i t y

Electron. J. Probab.18(2013), no. 43, 1–17.

ISSN:1083-6489 DOI:10.1214/EJP.v18-2043

Regularity of affine processes on general state spaces

Martin Keller-Ressel

^∗

Walter Schachermayer

^†

Josef Teichmann

^‡

Abstract

We consider a stochastically continuous, affine Markov process in the sense of Duffie, Filipovic and Schachermayer [9], with càdlàg paths, on a general state space D, i.e. an arbitrary Borel subset ofR^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 ofD =R^m>0×Rⁿ studied in Duffie 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 gener- alize the results of Keller-Ressel, Schachermayer and Teichmann [15] for the state spaceR^m>0×Rⁿand provide a new probabilistic approach to regularity.

Keywords:affine process; regularity; semimartingale; generalized Riccati equation.

AMS MSC 2010:60J25.

Submitted to EJP on May 22, 2012, final version accepted on February 8, 2013.

SupersedesarXiv:1105.0632v2.

1 Introduction

A time-homogeneous, stochastically continuous Markov processXon the state space D⊂R^dis called affine, if its transition kernelpt(x, dξ)has the following property: There exist functionsΦandψ, taking values inC^andC^drespectively, such that

Z

D

e^hξ,uip_t(x, dξ) = Φ(t, u) exp(hx, ψ(t, u)i) for allt∈R>0,x∈Danduin the setU =

u∈C^d: sup_x∈DRehu, xi<∞ .

The class of stochastic processes resulting from this definition is a rich class that includes Brownian motion, Lévy processes, squared Bessel processes, continuous-state branching processes with and without immigration [14], Ornstein-Uhlenbeck-type processes [17, Ch. 17], Wishart processes [1] and several models from mathematical finance, such as the affine term structure models of interest rates [10] and the affine stochastic volatility models [13] for stock prices.

∗TU Berlin, Germany. E-mail:[email protected]

†University of Vienna, Austria. E-mail:[email protected]

‡ETH Zurich, Switzerland. E-mail:[email protected]

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For a state space of the formD =R^m_>0×Rⁿ the class of affine processes has been originally defined and systematically studied in [9], under a regularity condition. In this context, regularity means that the time-derivatives

F(u) = ∂Φ(t, u)

∂t _t=0+

, R(u) = ∂ψ(t, u)

∂t _t=0+

exist for allu∈ Uand are continuous on subsetsUkofUthat exhaustU. Once regularity is established, the processX can be described completely in terms of the functionsF and R. The problem of showing that regularity of a stochastically continuous affine processX always holds true was originally considered for processes on the state space D=R^m_>0×Rⁿ and was proven – giving a positive answer – in [15], building on results from [8].

Already in [9] it has been remarked that affine processes can be considered on other state spacesD 6=R^m_>0×Rⁿ, where also no reduction to the ‘canonical’ case by embed- ding or linear transformation is possible. One such example is given by the Wishart process (ford≥2), which is an affine process taking values inS_d⁺, the cone of positive semidefinited×d-matrices. Recently, in [6] a full characterization of all affine processes with state spaceS⁺_d has been given and in [7] the even more general case whenDis an

‘irreducible symmetric cone’ in the sense of [11] is considered, which includes theS_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 ofD =R^m_>0×Rⁿ. Even though the affine processes onR^m_>0×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 spaceD. A simple example is given by the processX^(x,x

2)

t = (Bt+x,(Bt+x)²)t≥0, whereBis a standard Brownian motion. This process is an affine process that lives on the parabolaD=

(y, y²), y∈R ⊂R², and can be characterized by the functions Φ(t, u) = 1

√1−2tu₂exp

u²₁t 2(1−2tu2)

, ψ(t, u) = (u1, u2)/(1−2tu2).

It can even be extended into an affine process on the parabola’s epigraph (y, z) :z≥y², y∈R (see [9, Sec. 12.2]), but not into a process on the state space R^m_>0×Rⁿ, or on any symmetric cone. For more general results in this direction we refer to [18], where a classification of affine diffusion processes on polyhedral cones and state spaces which are level sets of quadratic functions (‘quadratic state spaces’) is provided. The authors of [18] 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 spaceD ⊂R^dunder the only assumptions thatDis a non-empty Borel set whose affine span isR^d and that the affine process has càdlàg paths. All existing regularity proofs, with the notable exception of [4] – which has been prepared in parallel to this article – use some particular properties of the state space: In the case ofS_d⁺ and the symmetric cones the fact that the setU has open interior, and in the caseR^m_>0×Rⁿ^{a de-} generacy argument that reduces the problem toR^m_>0, which is again a symmetric cone.

The existence of an non-empty interior ofU leads to a purely analytical proof based in

1A symmetric cone is a self-dual convex coneD, such that for any two pointsx, y∈Da linear automor- phismfofDexists, which mapsxintoy. It is called irreducible if it cannot be written as a non-trivial direct sum of two other symmetric cones.

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broad terms on the theory of differentiable transformation semigroups of Montgomery and Zippin ([16]); see [15]. For general state spaces D it is not true that U has non- empty interior, and the analytic technique ceases to work. An empty interior ofU also causes problems when applying the techniques from [9] and [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 in [4], where another probabilistic regularity proof for affine processes on general state spaces is given. Our proof is largely self-contained, while the approach in [4] uses the theory of full and complete function classes put forward in [3]. On the other hand, in [4] also the automatic right-continuity of the augmented natural filtration and the existence of a càdlàg modification of the affine process is shown, which results in slightly weaker assumptions than in this article.

2 Definitions and Preliminaries

Let D be a non-empty Borel subset of the real Euclidian space R^d, equipped with the Borelσ-algebraD, and assume that the affine hull ofDis the full space R^d^{. To}D we add a pointδthat serves as a ‘cemetery state’, define

Db =D∪ {δ}, Db=σ(D,{δ}),

and equipDb with the Alexandrov topology, in which any open set with a compact com- plement inDis declared an open neighborhood ofδ.² Any continuous functionfdefined onDis tacitly extended toDb by settingf(δ) = 0.

Let(Ω,F,F)be a filtered space, on which a family(P^x)_x∈

Db of probability measures is defined, and assume thatFisP^x-complete for allx∈Db and thatFis right-continuous.

Finally letXbe a càdlàg process taking values inDb, whose transition kernel

pt(x, A) =P^x(Xt∈A), (t≥0, x∈D, Ab ∈D)b (2.1) is a normal time-homogeneous Markov kernel, for whichδis absorbing. That is,pt(x, .) satisfies the following:

(a) x7→pt(x, A)isDb-measurable for each(t, A)∈R>0×Db. (b) p0(x,{x}) = 1for allx∈Db,

(c) p_t(δ,{δ}) = 1for allt≥0

(d) pt(x,D) = 1b for all(t, x)∈R>0×Db, and (e) the Chapman-Kolmogorov equation

pt+s(x, dξ) = Z

pt(y, dξ)ps(x, dy) holds for eacht, s≥0and(x, dξ)∈Db×Db.

We equipR^dwith the canonical inner producth,i, and associate toDthe setU ⊆C^d defined by

U =

u∈C^d: sup

x∈D

Rehu, xi<∞

. (2.2)

2Note that the topology ofDbenters our assumptions in a subtle way: We require later thatXis càdlàg on Db, which is a property for which the topology matters.

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Note that the setU is the set of complex vectors usuch that the exponential function x7→e^hu,xiis bounded onD. It is easy to see thatUis a convex cone and always contains the set of purely imaginary vectorsiR^d. We will also need the sets

Uk=

u∈C^d: sup

x∈D

Rehu, xi ≤k

, k∈N, (2.3)

for which we note thatU =S

k∈NUk.

Definition 2.1(Affine Process). The processX is called affinewith state spaceD, if its transition kernelpt(x, dξ)satisfies the following:

(i) it is stochastically continuous, i.e.lim_s→tp_s(x, .) =p_t(x, .)weakly for allt≥0, x∈ D, and

(ii) its Fourier-Laplace transform depends on the initial state in the following way:

there existΦ :R>0× U →Cand a continuousψ:R>0× U →C^d, such that Z

D

e^hξ,uipt(x, dξ) = Φ(t, u) exp(hx, ψ(t, u)i) (2.4) for allt∈R>0,x∈Dandu∈ U.

Remark 2.2. Note that this definition does not specifyψ(t, u)in a unique way. However there is a natural unique choice forψ that will be discussed in Prop. 2.4 below. Also note that as long asΦ(t, u)is non-zero, there existsφ(t, u)such thatΦ(t, u) =e^φ(t,u)and (2.4)becomes

Z

D

e^hξ,uipt(x, dξ) = exp(φ(t, u) +hx, ψ(t, u)i). (2.5) This is the essentially the definition that was used in [9]; with this notation the Fourier- Laplace transform is the exponential of an affine functionof x. This is usually inter- preted 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 infinitesi- mal 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 [14]

also the ‘big-Φ’ notation is used to define a ‘continuous-state branching process with immigration’, which corresponds to an affine process onR>0in our terminology.

Remark 2.3. It has recently been shown in [5] (see also [4]), that any affine process on a general state spaceD has a càdlàg modification under every P^x, x ∈ D. More- over, when X is an affine process relative to an arbitrary filtration F⁰^{, then the} P^x^- augmentationF^x^ofF⁰is right-continuous, for anyx∈D. This implies that the assumptions that we make on the path properties ofX are in fact automatically satisfied after a suitable modification of the process.

Before we explore the first consequences of Definition 2.1, we introduce some additional notation. For anyu∈ U define

σ(u) := inf{t≥0 : Φ(t, u) = 0}, (2.6) and

Qk:={(t, u)∈R>0× Uk:t < σ(u)}, (2.7)

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fork∈N^{. We set}Q:=∪kQk. Finally onQletφbe a function such that Φ(t, u) =e^φ(t,u) ((t, u)∈ Q).

The uniqueness ofφwill be discussed. The functionsφandψhave the following properties:

Proposition 2.4. LetX be an affine process onD. Then (i) It holds thatσ(u)>0for anyu∈ U.

(ii) The functionsφandψare uniquely defined onQby requiring that they are jointly continuous onQ_k fork∈Nand satisfyφ(0,0) =ψ(0,0) = 0.

(iii) The functionψmapsQintoU.

(iv) The functionsφ andψ satisfy thesemi-flow property. For anyu∈ U andt, s ≥ 0 with(t+s, u)∈ Qand(s, ψ(t, u))∈ Qit holds that

φ(t+s, u) =φ(t, u) +φ(s, ψ(t, u)), φ(0, u) = 0

ψ(t+s, u) =ψ(s, ψ(t, u)), ψ(0, u) =u ^(2.8) Proof. Choose somex∈D, and for(t, u)∈R>0× U define the function

f(t, u) = Φ(t, u)e^hψ(t,u),xi= Z

D

e^hu,ξipt(x, dξ). (2.9) Fixk∈N^{and let}(t_n, u_n)_n∈_Nbe a sequence inR>0× Uk converging to(t, u)∈R>0× Uk. For any > 0 we can find a function ρ : D → [0,1] with compact support, such that R

D(1−ρ(ξ))pt(x, dξ)< . Moreover, there exists aN0∈N^{such that}

e^huⁿ^,ξi−e^hu,ξi

< , ∀n≥N₀, ξ∈suppρ.

By stochastic continuity ofpt(x, dξ)we can findN1≥N0such that Z

D

(1−ρ(ξ))pt_n(x, dξ)< , ∀n≥N1, and also

Z

D

e^hu,ξipt_n(x, dξ)− Z

D

e^hu,ξipt(x, dξ)

< , ∀n≥N1, Forn≥N₁, we now have

|f(tn, un)−f(t, u)|= Z

D

e^huⁿ^,ξipt_n(x, dξ)− Z

D

e^hu,ξipt(x, dξ)

≤

≤ Z

D

e^huⁿ^,ξiρ(ξ)pt_n(x, dξ)− Z

D

e^hu,ξiρ(ξ)pt_n(x, dξ)

+ +

Z

D

e^huⁿ^,ξi(1−ρ(ξ))ptn(x, dξ)− Z

D

e^hu,ξi(1−ρ(ξ))ptn(x, dξ)

+ +

Z

D

e^hu,ξip_t_n(x, dξ)− Z

D

e^hu,ξip_t(x, dξ)

≤

≤+k+=(2 +k).

Sincewas arbitrary this shows the continuity off(t, u)onR>0×Uk. Hence we conclude that(t, u)7→f(t, u)is continuous onR>0×Ukfor eachk∈N^{. Moreover}f(t, u) = 0if and

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only ifΦ(t, u) = 0andf(0, u) =e^hu,xi6= 0for allu∈ U. We conclude from the continuity off thatσ(u) = inf{t≥0 :f(t, u) = 0}>0for allu∈ U and (i) follows.

To obtain (ii), note that for eachx∈D, we have just shown that the function(t, u)7→

R

De^hu,ξipt(x, ξ)maps Qk continuously into C\ {0} for each k ∈ N. We claim that the mapping has a unique continuous logarithm³, i.e. for eachx∈D there exists a unique functiong(x;., .,) :Q →Cbeing continuous onQkfork∈N, such thatg(x; 0,0) = 0and R

De^hu,ξip_t(x, ξ) =e^g(x;t,u). For eachn∈Ndefine the set

K_n={(t, u) :u∈ U_n,kuk ≤n, t∈[0, σ(u)−1/n]}.

Clearly, the Kn are compact subsets ofQn ⊂ Q and exhaustQ as n → ∞. We show that everyKn is contractible to 0. Let γ = (t(r), u(r))_r∈[0,1] be a continuous curve in Kn. For eachα ∈ [0,1]define γα = (αt(r), u(r))_r∈[0,1]. Then γα depends continuously on α, stays in K_n for each α and satisfiesγ₁ = γ and γ₀ = (0, u(r))_r∈[0,1]. Thus any continuous curve inK_n is homotopically equivalent to a continuous curve in {0} × U. Moreover, all continuous curves in{0} × U are contractible to 0, since U is a convex cone. We conclude that eachKn is contractible to 0and in particular connected. Let Hn : [0,1]×Kn → Kn be a corresponding contraction, and for some fixed x ∈ D writefn(t, u)for the restriction of(t, u) 7→R

De^hu,ξipt(x, ξ)toKn. SinceHn andfn are continuous andK_n is compact, we have thatlim_t→skfn(H_n(t, .))−f_n(H_n(s, .))k_∞ = 0. Hencef_n◦H_n is a continuous curve inC_b(K_n)fromf_n to the constant function1. By [2, Thm. 1.3] there exists a continuous logarithmgn ∈ Cb(Kn)that satisfiesfn(t, u) = e^gⁿ^(t,u)for all(t, u)∈Kn. It follows that for arbitrarym≤ninN^{we have}

gm(t, u) =gn(t, u) + 2πi l(t, u) for all (t, u)∈Km,

wherel(t, u) is a continuous function from K_m to Z ^satisfyingl(0,0) = 0. But K_m is connected, hence also the image ofK_munderl. We conclude thatl(t, u) = 0, and that gm(t, u) = gn(t, u) for all (t, u) ∈ Km. Taking m = n this shows that gn is uniquely defined on each subset Kn of Q. Taking m < nit shows that gn extends gm. Since the(Kn)_n∈N exhaust Q, it follows that there exists indeed, for eachx ∈ D, a unique functiong(x;.) :Q →C^{such that}g(x; 0,0) = 0andR

De^hu,ξipt(x, ξ) =e^g(x;t,u). Since the affine span ofD isR^d we may assume without loss of generality that0∈Dand obtain thatφ(t, u) =g(0;t, u)is the unique choice ofφ(t, u)withφ(0,0) = 0. Moreover we know from (2.9) that

g(x;t, u) =φ(t, u) +hψ(t, u), xi+ 2πi l(x;t, u),

for allx∈D,(t, u)∈ Qk and wherelis anZ-valued function that satisfiesl(x; 0,0) = 0 for allx∈D. By definition 2.1ψ(t, u)is continuous onR>0× U and hence alsol(x;t, u) is, for each fixedx ∈ D, a continous function on Qk. Connected sets are mapped to connected sets by continuous functions and we conclude that in fact l(x;t, u) ≡ 0 on D× Qk, which completes the proof of (ii).

Next note that the rightmost term of (2.9) is uniformly bounded for allx∈D. Thus also the middle term is, and we obtain thatψ(t, u)∈ U, as claimed in (iii). Applying the Chapman-Kolmogorov equation to (2.4) and writingΦ(t, u) =e^φ(t,u)yields that

exp (φ(t+s, u) +hx, ψ(t+s, u)i) = Z

D

e^hξ,uipt+s(x, dξ) =

= Z

D

ps(x, dy) Z

D

e^hξ,uipt(y, dξ) =e^φ(t,u) Z

D

e^hy,ψ(t,u)ips(x, dy) =

= exp (φ(t, u) +φ(s, ψ(t, u)) +hx, ψ(s, ψ(t, u)))i) (2.10)

3We adapt a proof from [2, Thm.2.5] to our setting.

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for allx ∈ D and for allu ∈ U such that (t+s, u) ∈ Q and (s, ψ(t, u)) ∈ Q. Taking (continuous) logarithms on both sides (iv) follows.

Remark 2.5. From now onφandψshall always refer to the unique choice of functions described in Proposition 2.4.

3 Main Results

3.1 Presentation of the main result

We now introduce the important notion ofregularity.

Definition 3.1. An affine processX is called regularif the derivatives F(u) = ∂φ(t, u)

∂t _t=0+

, R(u) = ∂ψ(t, u)

∂t _t=0+

(3.1) exist for allu∈ U and are continuous onUkfor eachk∈N^.

Remark 3.2. Note that in comparison with the definition given in the introduction, we now defineF(u)as the derivative att = 0oft7→φ(t, u)instead oft 7→Φ(t, u). In light of Proposition 2.4 these definitions coincide, sinceφ(t, u)is always defined fort small enough and satisfiesΦ(t, u) =e^φ(t,u)withφ(0, u) = 0.

Our main result is the following.

Theorem 3.3. LetX be a càdlàg affine process onD⊂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 in [9] for affine processes on the state-spaceRⁿ×R^m_>0.

Proposition 3.4. Let X be a regular affine process with state space D. Then there existR^d^-vectorsb, β¹, . . . , β^d;d×d-matricesa, α¹, . . . , α^d; real numbersc, γ¹, . . . , γ^d and signed Borel measuresm, µ¹, . . . , µ^d onR^d\ {0}, such that for allu ∈ U the functions F(u)andR(u)can be written as

F(u) = 1

2hu, aui+hb, ui −c+ Z

R^d\{0}

e^hξ,ui−1− hh(ξ), ui

m(dξ), (3.2a) R_i(u) = 1

2

u, αⁱu +

βⁱ, u

−γⁱ+ Z

R^d\{0}

µⁱ(dξ), (3.2b) with truncation functionh(x) =x1_{kxk≤1}, and such that for allx∈Dthe quantities

A(x) =a+x1α¹+· · ·+xdα^d, (3.3a) B(x) =b+x1β¹+· · ·+xdβ^d, (3.3b) C(x) =c+x1γ¹+· · ·+xdγ^d, (3.3c) ν(x, dξ) =m(dξ) +x1µ¹(dξ) +· · ·+xdµ^d(dξ) (3.3d) have the following properties: A(x) is positive semidefinite, C(x) ≤ 0 and R

R^d\{0}

kξk²∧1

ν(x, dξ)<∞.

Moreover, foru∈ U the functionsφandψsatisfy the ordinary differential equations

∂

∂tφ(t, u) =F(ψ(t, u)), φ(0, u) = 0 (3.4a)

∂

∂tψ(t, u) =R(ψ(t, u)), ψ(0, u) =u. (3.4b) for allt∈[0, σ(u)).

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Remark 3.5. The differential equations(3.4)are called generalized Riccati equations, since they are classical Riccati differential equations, whenm(dξ)=µⁱ(dξ) = 0. More- over equations (3.2) and (3.3)imply that u 7→ F(u) +hR(u), xi is a function of Lévy- Khintchine form for each x ∈ D. 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. [9, Ex. 9.3]).

Proof. The equations (3.4) follow immediately by differentiating the semi-flow equations (2.8). The form ofF, Rfollows by the following argument: By (3.1) and the affine property (2.4) it holds for allx∈Dandu∈ U that

F(u) +hx, R(u)i= lim

t↓0

1 t

n

eφ(t,u)+hx,ψ(t,u)−ui−1o

=

= lim

t→0

1 t

Z

D

e^hξ−x,uip_t(x, dξ)−1

=

= lim

t→0

1 t

Z

D

e^hξ−x,ui−1

pt(x, dξ) +pt(x, D)−1 t

=

= lim

t→0

1 t

Z

D−x

e^hξ,ui−1

pet(x, dξ)

+ lim

t→0

pt(x, D)−1

t , (3.5)

where we writepet(x, dξ) :=pt(x, dξ+x)for the shifted transition kernel. Insertingu= 0 into the above equation shows thatlimt↓0(pt(x, D)−1)/tconverges toF(0) +hx, R(0)i. Setc=−F(0)andγ=−R(0)and writeFe(u) =F(u) +candR(u) =e R(u) +γ, such that

exp

Fe(u) +D

x,R(u)e E

= lim

t↓0exp 1

t Z

D−x

e^hξ,ui−1

pe_t(x, dξ)

. (3.6)

For eacht≥0andx∈D, the exponential on the right hand side is the Fourier-Laplace transform of a compound Poisson distribution with jump measure pet(x, dξ)and jump intensity ¹_t (cf. [17, Ch. 4]). The Fourier-Laplace transforms converge pointwise for u∈ U – and in particular for all u∈ iR^d ^{– as}t → 0. By the assumption of regularity the pointwise limit is continuous at u = 0 as function on iR^d ⊂ Uk 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 givenx∈D, byK(x, dy). Since it is infinitely divisible, its characteristic exponent is of Lévy-Khintchine form, and we obtain the identity

F(u) +e D

x,R(u)e E

= log Z

R^d

e^hξ,uiK(x, dξ) =

=−1

2huA(x), ui+hB(x), ui − Z

R^d

ν(x, dξ), (3.7) where for each x ∈ D, A(x) is a positive semi-definite d×d-matrix, B(x) ∈ R^d^{, and} ν(x, dξ)aσ-finite Borel measure onR^d\ {0}and R

kξk²∧1

ν(x, dξ)< ∞. Note that in the step from (3.6) from (3.7) we have used thatFe(u) and R(u)e are continuous on everyUk, k∈N, and hence thatFe(u) +D

x,R(u)e E

is the unique continuous logarithm of exp

Fe(u) +D

x,R(u)e E

on eachUk and for all x∈ D. Since (3.7) holds for allx∈ D, andDcontains at leastd+ 1affinely independent points, we conclude thatA(x),B(x) andν(x, dξ)are of the form given in (3.3) and the decompositions in (3.2) follow.

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In general, the parameters(a, αⁱ, b, βⁱ, c, γⁱ, m, µⁱ)i∈{1,...,d}ofF andRhave to satisfy additional conditions, calledadmissibility conditions, that guarantee the existence of an affine Markov processX with state spaceDand prescribedF andR. It is clear that such conditions depend strongly on the geometry of the state space D, 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. ForD= R^m_>0×Rⁿ the admissibility conditions have been derived in [9], forD =S_d⁺, the cone of semi-definite matrices in [6], and for conesDthat are symmetric and irreducible in the sense of [11] in [7]. Finally for affine diffusions (m=µⁱ = 0) on polyhedral cones and on quadratic state spaces the admissiblility conditions have been given in [18].

The purpose of this article is not to derive these admissibility conditions for conrete specifications ofD, but merely to show that for any arbitrary state spaceD there are parameters in terms of which admissibility conditions can be formulated.

3.2 Auxiliary Results

For the sake of simpler notation we define

%(t, u) =ψ(t, u)−u.

Note that we have%(0, u) = 0for allu∈ U. The following Lemma is a purely analytical result that will be needed later.

Lemma 3.6. LetKbe a compact subset ofU_lfor somel∈Nand assume that lim sup

t→0

sup

u∈K

|φ(t, u)|

t +k%(t, u)k t

=∞. (3.8)

Then there isx∈D,ε >0,η >0,z∈C^with|z|= 1, a sequence(t_k)^∞_k=1of positive real numbers, a sequence(M_k)^∞_k=1of integers satisfying

lim

k→∞t_k= 0, lim

k→∞M_k =∞, lim

k→∞M_kt_k = 0, (3.9) and a sequence of complex vectors(uk)^∞_k=0inKsuch thatuk→u0and

|φ(tk, uk) +hx, %(tk, uk)i | ≥ηk%(tk, uk)k. (3.10) Moreover, for allξ∈R^d^satisfyingkx−ξk< ε,

Mk(φ(tk, uk) +hξ, %(tk, uk)i) =z+ek,ξ, (3.11) where the complex numbersek,ξ describing the deviation fromz satisfy|ek,ξ|< ¹₂ and limk→∞sup{ξ:kx−ξk<ε}|ek,ξ|= 0.

Remark 3.7. The essence of the above Lemma is that the behavior ofφ(t, u)and%(t, u) ast approaches0 can be crystallized along the sequences tk and Mk. Equation (3.9) then states thattk=o

1 Mk

, and (3.11)asserts that the asymptotic equivalence

|φ(tk, uk) +hξ, %(tk, uk)i| ∼ 1 Mk

, holds uniformly for allξin anε-ball aroundx.

Proof. We first show all assertions of the Lemma for a sequence(fMk)_k∈Nof positive but not necessarily integer numbers. In the last step of the proof we show that it is possible to switch from(fMk)k∈Nto the integer sequence(Mk)k∈N.

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By Assumption (3.8) we can find a sequence (t_k)^∞_k=0 ↓ 0 and a sequence (u_k)^∞_k=0 with u_k ∈K, such that

|φ(tk, uk)|+k%(tk, uk)k tk

→ ∞.

Passing to a subsequence, and using the compactness of K, we may assume thatu_k converges to some point u0 ∈ K. For more concise notation, we write from now on φk =φ(tk, uk)and%k =%(tk, uk). Note thatφk →0and%k →0, by joint continuity ofφ and%onUl, and the fact thatφ(0, u) = 0and%(0, u) = 0.

Let us now show (3.10). By assumption,D containsd+ 1affinely independent vectors x₀, x₁, . . . , x_d. Assume for a contradiction that

k→∞lim

|φk+h%k, xji|

k%kk →0 (3.12)

for allx_j,j∈ {0, . . . , d}. Since the vectorsx_jaffinely spanR^d, the vectors{x1−x₀, . . . , x_d−x₀} are linearly independent, and we can find some numbersα_j,k∈C, such that

%k/k%kk=

d

X

j=1

αj,k(xj−x0), (3.13)

for all k ∈ N. Moreover, since %k/k%kk is bounded also the |αj,k| are bounded by a constant. By direct calculation we obtain

d

X

j=1

αj,k

φk+h%k, xji

k%_kk −φk+h%k, x0i k%_kk

=h%k/k%kk, %ki

k%_kk = 1, (3.14) for allk ∈ N. On the other hand, (3.12) implies that the left hand side of (3.14) converges to0ask → ∞, which is a contradiction. We conclude that there existsx^∗ ∈D for which

|φk+h%k, x^∗i|

k%kk ≥η (3.15)

for someη >0after possibly passing to subsequences, whence (3.10) follows.

To show (3.11), setMfk=|φk+hx^∗, %ki|⁻¹. Passing once more to a subsequence, and using the compactness of the complex unit circle, we can find someα∈[0,2π)such that arg (φk+hx^∗, %ki)→α. Now

φ_k+hξ, %_ki= (φ_k+hx^∗, %_ki) +hξ−x^∗, %_ki= 1 Mfk

(e^iα+e⁽¹⁾_k ) +hξ−x^∗, %_ki

wheree⁽¹⁾_k →0ask→ ∞. Multiplying byMfkand settingz=e^iαwe obtain Mfk(φk+hξ, %ki) =z+e⁽¹⁾_k +e⁽²⁾_k,ξ

where we can estimate|e⁽²⁾_k,ξ| ≤ Mfkεk%kk. SinceMfkk%kk ≤ ¹_η by (3.10) we can make e⁽²⁾_k,ξarbitrarily small by choosing a small enoughε. Settinge_k,ξ =e⁽¹⁾_k +e⁽²⁾_k,ξ we obtain (3.11). Finally, for eachk∈N^letM_k be the nearest integer greater thanMf_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 fromMfktoMk.

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Lemma 3.8. LetX = (X_t)_t≥0be an affine process starting atX₀and letu∈ U,∆>0, ε >0. Define

L(n,∆, u) = exp



hu, Xn∆−X₀i −

n

X

j=1

φ(∆, u) +

%(∆, u), X_(j−1)∆



 (3.16)

and the stopping timeN∆= inf{n∈N:kXn∆−X0k> ε} Thenn7→L(n∧N∆,∆, u)is a(Fn∆)_n∈N-martingale under every measureP^x^,x∈D.

Proof. It is obvious that eachL(n,∆, u)isFn∆-measurable. The definition of the stopping timeN_∆guarantees the integrability ofL(n∧N_∆,∆, u). We show the martingale property by combining the affine property ofX with the tower law for conditional ex- pectations. Write

Sn=

n

X

j=1

φ(∆, u) +

%(∆, u), X_(j−1)∆

,

and note thatS_nisF_(n−1)∆-measurable. On{n≤N_∆}we have that E^x

L(n,∆, u)| F_(n−1)∆

=E^x

exp (hu, Xn∆−X0i)| F_(n−1)∆

e^−Sⁿ=

= exp φ(∆, u) +

ψ(∆, u), X_(n−1)∆

− hu, X0i −Sn

=

= exp

u, X_(n−1)∆−X₀

−S_n−1

=L(n−1,∆, u),

showing thatn7→L(n∧N∆,∆, u)is indeed a(Fn∆)_n∈N-martingale under everyP^x, x∈ D.

We combine the two preceding Lemmas to show the following.

Proposition 3.9. Let X be a càdlàg affine process. Then the associated functions φ(t, u)and%(t, u) =ψ(t, u)−usatisfy

lim sup

t↓0

sup

u∈K

|φ(t, u)|

t +k%(t, u)k t

<∞ (3.17)

for each compact subsetKofUland eachl∈N^.

Proof. We argue by contradiction: Fixl ∈ Nand assume that (3.17) fails to hold true.

Then by Lemma 3.6 there existε >0and sequencesu_k →u₀inK,t_k ↓0 andM_k ↑ ∞ such thatt_kM_k → 0 and equations (3.10), (3.11) hold. Define the(F_nt_k)_n∈_N-stopping timesNk = inf{n∈N:kXnt_k−X0k> ε}. Then setting∆ =tkin Lemma 3.8 yields that

n7→L(n∧Nk, tk, uk) =

= exp





u_k, X_(n∧N_k_)t_k−X₀

−

n∧Nk

X

j=1

φ(t_k, u_k) +

%(t_k, u_k), X_(j−1)t_k



 (3.18) is a(Fnt_k)n∈N-martingale. It follows in particular thatE[L(Mk∧Nk, tk, uk)] = 1for all k∈N. By (3.11), we have the uniform bound

|L(M_k∧N_k, t_k, u_k)| ≤Cexp





M_k∧Nk

X

j=1

φ(t_k, u_k) +

%(t_k, u_k), X_(j−1)t_k



≤

≤Cexp(3/2), (3.19)

whereC = exp (−Rehu, X0i). Letδ > 0 and x ∈ D. SinceX is càdlàg we can find a T > 0such that P^x

sup_t∈[0,T_]kXt−X₀k> ε

< δ. Fork large enought_kM_k ≤ T and

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henceP(Mk > Nk)< δ. We conclude thatP^x

limk→∞MK∧Nk

M_k = 1

≥1−δ, and sinceδ was arbitrarylim_k→∞^M^k_M^∧N^k

k = 1holdsP^x-a.s. for anyx∈D. Together with (3.11) and (3.19) we obtain by dominated convergence that

k→∞lim E^x[L(Mk∧Nk, Tk, uk)] =E^x

k→∞lim L(Mk∧Nk, Tk, uk)

=

=E^x

lim

k→∞exp ((M_k∧N_k) (φ(t_k, u_k) +h%(tk, u_k), xi))

=e^−z. (3.20) where|z| = 1. ButE^x[L(Mk∧Nk, Tk, uk)] = 1by its martingale property, which is the desired contradiction.

3.3 Affine processes are regular

In this section we prove the main result, Theorem 3.3.

Lemma 3.10. Let a sequencetk(u)↓0be assigned to eachu∈ U. Then each of these sequences has a subsequenceS(u) := (sk(u))_k∈N such that the limits

F_S(u) := lim

s_k(u)↓0

φ(sk(u), u)

sk(u) , R_S(u) := lim

s_k(u)↓0

%(sk(u), u)

sk(u) ^(3.21)

are well-defined and finite. Moreover the subsequencesS(u)can be chosen such that the numbersF_S(u) and R_S(u)are bounded on each compact subset K of Ul for each l∈N^.

Proof. Let the sequencestk(u)↓0be given, but assume that the assertion of the Lemma does not hold true. Then eithertk(u)for someu∈ Uhas no subsequence for which the limits in (3.21) exist, or the limitsF(u)andR(u)exist for eachu∈ U, but at least one of them is not bounded in some compactK⊂Ulfor somel∈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

tk(u)↓0

|φ(t_k(u), u)|

tk(u) +k%(t_k(u), u)k tk(u)

=∞,

in contradiction to Proposition 3.9. Consider now the second assertion. Fixl ∈N^{. For} eachu∈ Ul there is a sequencesk(u)such that (3.21) holds, butF_S(u)orR_S(u)is not bounded in K ⊂ Ul, i.e. there exists a sequence un → u0 in K for which |F(un)|+ kR(un)k → ∞. Fix someη >0. Then for eachk∈Nthere exists anNk∈N^{such that}

φ(s_N_k(u_k), u_k) sN_k(uk)

≥ |F(u_k)| −η/2 and

%(s_N_k(u_k), u_k) sN_k(uk)

≥ kR(uk)k −η/2.

We conclude that lim sup

s_k↓0

sup

u∈K

|φ(sk, u)|

s_k +k%(sk, u)k s_k

≥lim sup

k→∞

|F(u_k)|+kR(u_k)k −η=∞, again in contradiction to Prop. 3.9.

Having shown Lemma 3.10, only a small step remains to show regularity. Comparing with Definition 3.1 we see that two ingredients are missing: First we have to show that the limits F(u) and R(u) do not depend on the choice of subsequence, i.e. they are proper limits and hence the proper derivatives ofφandψatt= 0, and second we have to show thatF andRare continuous onUlfor eachl∈N^.

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Proof of Theorem 3.3. Our first step is to show that the derivatives F(u) and R(u) in (3.1) exist. By Lemma 3.10 we already know that they exist as limits along a sequence S(u)which depends on the pointu∈ Uand has been chosen as a particular subsequence of a given sequence(tk(u))_k∈N. We show now that the limit is in fact independent of the choice ofS(u) and even of the original sequence(tk(u))k∈N, and hence that F(u)and R(u)are proper derivatives in the sense of (3.1). To this end, fix someu∈ U, and let eS(u)be an arbitrary other sequencees_k(u)↓0, such that

Fe_S(u) := lim

es_k(u)↓0

φ(esk(u), u)

es_k(u) , Re_S(u) := lim

es_k(u)↓0

%(esk(u), u)

es_k(u) . (3.22) We want to show thatF_S(u) = Fe_S(u)and R_S(u) = Re_S(u). Assume for a contradiction that this were not the case. Then we can findx∈Dandr >0such that the convex set {F_S(u) +hR_S(u), ξi:kξ−xk ≤r}and its counterpart involvingeSare disjoint, i.e.

nF_S(u) +hR_S(u), ξi:kξ−xk ≤ro

∩n

Fe_S(u) +D

Re_S(u), ξE

:kξ−xk ≤ro

= ∅. (3.23) For the next part of the proof, we setτ = inf{t≥0 :kXt−X0k ≥r}, and introduce the following notation:

a^u_t := F_S(u) +hR_S(u), X_ti, A^u_t :=

Z t 0

aû_s−ds, Gû_t := exp(Aû_t), Y_tû:= exp(hu, X_t−X₀i

witheaû_t,Aeû_t andGeû_t the corresponding counterparts forFe_SandRe_S. We show that Lû_t∧τ = Y_t∧τû

G^u_t∧τ = exp

hu, Xt∧τ−X0i − Z t∧τ

0

(F_S(u) +hR_S(u), Xs−i)ds

(3.24) is a martingale under everyP^x, x∈D. This reduces to showing that

E^x

"

exp hu, Xh∧τ−X₀i − Z h∧τ

0

(F_S(u) +hR_S(u), X_s−i)ds

!#

= 1, since then by the Markov property ofX

E^x

"

exp

u, X_(t+h)∧τ−X_t∧τ

−

Z (t+h)∧τ t∧τ

!

F_t

#

=

E^x

"

exp

u, X_(t+h)∧τ−X_t∧τ

−

Z (t+h)∧τ t∧τ

! 1_τ≥t

Ft

#

+ 1_τ≤t=

=E^X^t

"

exp hu, X_h∧τ−X0i − Z h∧τ

0

!#

1_τ≥t+ 1_τ≤t= 1 (3.25) holds true. Now, use the sequence S(u) = (sn(u))n∈N ↓ 0 to define a sequence of Riemannian sums approximating the above integral. Define M_k = bh/skc and N_k = inf{n∈N:kX_ns_k−X₀k> r}. First we show thats_kN_k →τ almost surely under every P^x^{. Fix}ω ∈Ωsuch thatt →X_t(ω)is a càdlàg function. LetNe_k(ω)be a sequence inN such thatskNek(ω)↓τ(ω). It follows from the right-continuity oft7→Xt(ω)that for large enoughkit holds that

X_s

kNek−X₀

> rand hence that eventuallyNe_k(ω)≥N_k(ω). On

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the other hand kXskNk−X₀k > r for all k ∈ N, which implies that N_k(ω)s_k ≥ τ(ω). Hence, for large enoughk∈Nit holds that

s_kNe_k(ω)≥s_kN_k(ω)≥τ(ω).

We also know thatskNek(ω) → τ(ω)ask → ∞, such that we conclude that skNk → τ P^x-almost surely, as claimed. By Riemann approximation and the fact thatX is càdlàg it then holds that

M_k∧Nk

X

j=1

F_S(u) +

R_S(u), X_(j−1)s_k sk→

Z h∧τ 0

(F_S(u) +hR_S(u), X_s−i)ds P^x-almost-surely ask→ ∞for allx∈D.

From Lemma 3.10 we know thatφ(s_k, u) =F_S(u)s_k+o(s_k)andφ(s_k, u) =R_S(u)s_k+ o(sk). Moreover(Mk∧Nk)o(sk)→0sinceMksk →0. Thus we have that

L(Mk∧Nk, sk, u) =

= exp





u, X_(M_k_∧N_k_)s_k−X0

−

M_k∧Nk

X

j=1

φ(tk, u) +

%(tk, u), X_(j−1)s_k



=

= exp





u, X_(M_k_∧N_k_)s_k−X0

−

Mk∧Nk

X

j=1

F_S(u) +

R_S(u), X_(j−1)s_k

sk+ (Mk∧Nk)o(sk)



→

→exp hu, X_h∧τ−X0i − Z h∧τ

0

! ,

ask→ ∞almost surely with respect to allP^x, x∈D. But by Lemma 3.8 and optional stopping, E^x[L(Mk∧Nk, sk, u)] = 1, such that by dominated convergence (cf. (3.19)) we conclude that

E

"

exp hu, X_h∧τ−X0i − Z h∧τ

0

!#

= 1,

and hence thatt7→Lû_t∧τ is a martingale. Summing up we have established thatY_t∧τû = Lû_t∧τGû_t∧τ, whereLû_t∧τ is a martingale and hence a semimartingale. Clearly, the process Gû_t∧τ is predictable and of finite variation and hence a semimartingale too. We conclude that also the productY_t∧τû = exp (hu, X_t∧τ^x −xi)is a semimartingale. It follows from [12, Thm. I.4.49] thatM_t∧τû =Y_t∧τû −Rt∧τ

0 Lû_s−dGû_s is a local martingale. We can rewriteM_tû as

M_t^u=Y_t∧τ^u − Z t

0

Lû_s−Gû_s−dAû_s =Y_tû− Z t

0

Y_s−û dAû_s =Y_tû− Z t

0

Y_s−^u a^u_s−ds.

HenceY_t∧τ^u =M_t∧τ^u +Rt∧τ

0 Y_s−ûaû_s−dsis the decomposition of the semi-martingaleY_t∧τû into a local martingale and a finite variation part. ButRt∧τ

0 Y_s−û aû_s−dsis even predictable, such thatYûis a special semi-martingale, and the decomposition is unique. The same derivation goes through withAû replaced byAeû and by the uniqueness of the special semi-martingale decomposition we conclude that

Z t∧τ 0

Y_s−^ua^u_s−ds= Z t∧τ

0

Y_s−^uea^u_s−ds,

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up to aP^x-nullset. Taking derivatives we see thatY_t−ûaû_t− =Y_t−ûeaû_t− on{t≤τ}. As long ast≤τ it holds thatY_t−û 6= 0, and dividing byY_t−û, we see thataû_t−=eaû_t−, that is

F_S(u) +hR_S(u), X_t−i=Fe_S(u) +D

Re_S(u), X_t−E

on{t≤τ},

P^x-a.s, in contradiction to (3.23). We conclude that the limitsF_SandR_Sare independent from the sequenceS, and hence thatF(u)and R(u)exist as proper derivatives in the sense of (3.1).

It remains to show thatF(u)andR(u)are continuous onUlfor eachl∈N^{. Fix}l∈N and suppose for a contradiction that there exists a sequenceuk → u0 inUl such that F(u_k) → F^∗ and R(u_k) → R^∗, such that either F(u₀) 6= F^∗ orR(u₀) 6= R^∗. SinceD affinely spansR^dthis means that there isx∈Dwith

F(u0) +hR(u0), xi 6=F^∗+hR^∗, xi. Using the fact thatE^x[L^u_t∧τ^k ] = 1for allk∈N^{we obtain}

1

t(exp(hφ(t, u₀) +ψ(t, u₀), xi)−1) = lim

k→∞

1 tE^xh

e^hu⁰^,X^t^−X⁰ⁱ−L^u_t∧τ^k i

=

= lim

k→∞

1 tE^x

e^hu⁰^,X^t^−X⁰ⁱ

1−exp(−

Z t∧τ 0

(F(u_k) +hR(uk), X_s−i)ds

=

=E^x 1

te^hu⁰^,X^t^−X⁰ⁱ

1−exp(−

Z t∧τ 0

(F^∗+hR^∗, X_s−i)ds

. (3.26) for allt ≤σ(0)(see (2.6) for the definition ofσ(.)) by dominated convergence. Writing C=|F^∗|+kR^∗kεand using the elementary inequality|1−e^z| ≤ |z|e^|z|we can bound

1

te^hu⁰^,X^t^−X⁰ⁱ

1−exp(−

Z t∧τ 0

(F^∗+hR^∗, Xs−i)ds

≤Ce^2l+Ct

and therefore apply again dominated convergence to the right hand side of (3.26) as t→0. Taking the limit on both sides, we obtain

F(u0) +hR(u0), xi=F^∗+hR^∗, xi leading to the desired contradiction.

We conclude with a corollary that gives conditions for an affine process to be a D- valued semimartingale, up to its explosion time. Let τ_n = inf{t≥0 :kX_t−X₀k> n}

and define the explosion timeτ_exp as the pointwise limitτ_exp = lim_n→∞τn. Note that τexpis predictable.

Corollary 3.11. Let X be a càdlàg affine process and suppose that the killing terms vanish, i.e.c = 0 andγ = 0. Then under everyP^x, x∈ Dthe process X is a D-valued semi-martingale on[0, τexp)with absolutely continuous semimartingale characteristics

A_t= Z t

0

A(X_s−)ds Bt=

Z t 0

B(X_s−)ds K([0, t], dξ) =

Z t 0

ν(Xs−, dξ)ds.

whereA(.), B(.)andν(., dξ)are given by (3.3).

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Proof. In the proof of Theorem 3.3 we have shown that t 7→ Lû_t∧τ, withLû_t defined in (3.24) andτ = inf{t≥0 :kX_t−X₀k> r}, is a martingale under everyP^x, x∈Dand for everyu∈ U. Sincer >0 was arbitrary, alsoLû_t∧τ_n is a martingale for everyn∈N^{. By} dominated convergence and using thatF(0) +hR(0), xi=c+hγ, xi= 0for allx∈Dwe obtain

P^x X_t∧τ_exp6=δ

= lim

n→∞P(X_t∧τ_n6=δ) =E L⁰_t∧τ

n

= 1.

HenceXtand Xt− stayP^x-almost surely inD ⊂R^d ^fort∈ [0, τexp). Moreovert 7→L^u_t is a local martingale on[0, τ_exp)for allu∈ U. Thus [12, Cor. II.2.48b] can be applied to the local martingaleL^u_t withu∈iR^dand the assertion follows.

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Acknowledgments. The first and third author gratefully acknowledge the support by the ETH foundation. The second author gratefully acknowledges financial support from

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the Austrian Science Fund (FWF) under grant P19456, from the European Research Council (ERC) under grant FA506041 and from the Vienna Science and Technology Fund (WWTF) under grant MA09-003. Furthermore this work was financially supported by the Christian Doppler Research Association (CDG).

The authors would like to thank Enno Veerman and Maurizio Barbato for comments on an earlier draft