UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)
Affine Markov processes on a general state space
Veerman, E.
Publication date
2011
Link to publication
Citation for published version (APA):
Veerman, E. (2011). Affine Markov processes on a general state space. Uitgeverij BOXPress.
General rights
It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons).
Disclaimer/Complaints regulations
If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible.
Chapter
5
Admissible parameter sets
In this chapter we work out the admissibility conditions as given in Definition 4.3 for affine processes with an arbitrary polyhedral and with an arbitrary quadratic state space, which complements the characterization of admissible parameter sets for the state space Rm
+ × Rp−m and S p
+, as derived in [17] and [10]. First we derive some preliminary results in Section 5.1, where we translate the positive maximum-principle as discussed in Section 2.4 in explicit boundary conditions for general processes. Here we essentially consider state spaces where the boundary can be transformed locally into the boundary of the state space Rm
+× Rp−m. Our method to derive boundary conditions is an alternative to the analysis in [10], which is based on stochastic invariance results from [12]. In Section 5.2 we derive the admissibility conditions for a general polyhedral state space, see Theorem 5.12. For the proof we use results from convex analysis. Next we determine all possible quadratic state spaces in Section 5.3. We show that the parabolic state space and the Lorentz cone are the only possibilities for an affine process, for which we derive the required admissibility conditions in Sections 5.4 and 5.5, see Theorems 5.17 and 5.22.
We note that it suffices to characterize the admissible parameter sets up to an invertible affine transformation. The following proposition will be needed for this.
Proposition 5.1. Suppose (X, (Px)x∈E) is an affine jump-diffusion with state space E and parameters (b(x), c(x), γ(x), K(x, dz)). Let ` ∈ Rp and L ∈ Rp×p be non-singular. Write Y = ` + LX, eE = ` + LE, Qy = PL−1(y−`) for y ∈ eE.
Then (Y, (Qy)y∈ eE) is an affine jump-diffusion with state space eE and parameters 105
(eb(y),ec(y),eγ(y), eK(y, dz)) given by eb(y) = Lb(L−1(y − `)) e c(y) = Lc(L−1(y − `))L> e γ(y) = γ(L−1(y − `)) e K(y, dz) = K(L−1(y − `), L−1dz),
with respect to the truncation functionχ(z) = Lχ(Le −1z).
Proof. Let eA be given by (2.2) with (eb,ec,eγ, eK) instead of (b, c, γ, K) and with the truncation function χ instead of χ. We have to show that for all g ∈ Ce 2
c( eE) it holds that g(Yt) − Z t 0 e Ag(Ys)ds
is a Qy-martingale for all y ∈ eE. Let g ∈ Cc2( eE) be arbitrary. Define f ∈ Cc2(E) by f (x) = g(` + Lx) and let A given by (2.2) be the generator of X. Then it holds that
f (Xt) − Z t
0
Af (Xs)ds
is a Px-martingale for all x ∈ E. Note that ∇f (x) = L∇g(y) and ∇2f (x) = L>∇2g(y)L for y = Lx + `, x ∈ E. Using this, one easily verifies that Af (x) =
e
Ag(y) for y = Lx + `, x ∈ E. This gives the result.
Throughout this chapter the following notation regarding matrices and vectors is used. Let p, q ∈ N, P = {1, . . . , p}, Q = {1, . . . , q}, A ∈ Rp×q, I ⊂ P , J ⊂ Q. Write I = {i1, . . . , i#I}, J = {j1, . . . , j#J}, with i1≤ i2≤ . . . ≤ i#I and j1≤ j2≤ . . . ≤ j#J. Then AIJdenotes the (#I ×#J )-matrix with elements (AIJ)kl= Aikjl.
The same notation applies to matrix-valued functions φ, e.g. φIJ(x) stands for (φ(x))IJ. If #I = 1, say I = {i}, we write AiJ instead. If J = Q then we write AI instead. In particular, Ai denotes the i-th row of A. The j-th column is denoted by Aj and the transpose of A is denoted by A>. For a
1, . . . , ap ∈ R we write diag(a1, . . . , ap) for the p-dimensional diagonal matrix D with diagonal elements Dii = ai, i ∈ P . We also write diag(a) instead, where a denotes the vector with elements ai, sometimes explicitly denoted by a = vec(a1, . . . , ap). The unique positive semi-definite square root of a positive semi-definite matrix A is denoted by A1/2.
5.1
Preliminaries
In Proposition 2.16 we derived necessary boundary conditions for the positive maximum principle for general processes on general state spaces. In this section we elaborate on this and derive necessary and sufficient conditions for polyhedral and quadratic state spaces, which we apply in the next sections to the affine setting. The key steps are Propositions 5.2 and 5.3 below. In the first one we consider Rm
+ × Rp−m, while in the second we consider smooth transformations of this state space.
Proposition 5.2. Suppose E = Rm+ × Rp−m and let b ∈ Rp, c ∈ S p
+ and K be a measure on Rp\{0}. Let O ⊂ Rp be open and suppose x
0 ∈ O ∩ E. Then for all f ∈ C2(O ∩ E) such that f (x0) = supx∈O∩Ef (x) it holds that
(i) ∇f (x0)>c∇f (x0) = 0,
(ii) R |∇f (x0)>χ(z)|K(dz) < ∞,
(iii) ∇f (x0)>b −R ∇f (x0)>χ(z)K(dz) +12tr (∇2f (x0)c) ≤ 0, if and only if for all i = 1, . . . , m with x0,i= 0 it holds that
(1) cij= cji= 0, for all j = 1, . . . , p,
(2) R |χi(z)|K(dz) < ∞, (3) bi−R χi(z)K(dz) ≥ 0.
Proof. The “only if”-part follows from the proof of Proposition 2.16. We show the “if”-part. Note that if x0∈ E◦, then all first-order derivatives of f in x0 are zero and ∇2f (x
0) is negative semi-definite, whence the conditions are trivially satisfied. Therefore, suppose x0∈ ∂E. Define M = {i ≤ m : x0∈ ∂Ei}. By permuting the indices, we may assume without loss of generality that M = {1, . . . , q}, for some 1 ≤ q ≤ m. Write ∂if (x) = ∂xif (x). It holds that ∂if (x0) ≤ 0 for all i ∈ M and
∂jf (x0) = 0 for all j 6∈ M . The latter yields
ci∇f (x0) = p X j=1 cij∂jf (x0) = q X j=1 cij∂jf (x0) = 0,
for all i = 1, . . . , p, by assumption (1). This gives condition (i). Condition (ii) follows from assumption (2), as we have
∇f (x0)>χ(z) = q X
i=1
It remains to verify condition (iii). Suppose q < p. Write N = {q + 1, . . . , p} and decompose x ∈ E as x = (xM, xN), so that x0 = (0, x0,N). There exists an open set B ⊂ Rm−q+ × Rp−m containing x0,N and such that (0, xN) ∈ O ∩ E for all xN ∈ B. Define the function
g : B → R : xN 7→ f (0, xN).
Then g assumes a maximum at x0,N ∈ B. Therefore, the Hessian ∇2g(x0) is negative semi-definite. Note that by assumption (1) we have tr (∇2f (x0)c) = tr (∇2g(x
0,N)cN N), which is non-positive, since cN N is positive semi-definite. It follows that ∇f (x0)>b − Z ∇f (x0)>χ(z)K(dz) +12tr (∇2f (x0)c) = q X i=1 ∂if (x0) bi− Z χi(z)K(dz) +12tr (∇2g(x0)cN N) ≤ q X i=1 ∂if (x0) bi− Z χi(z)K(dz) ≤ 0,
in view of assumption (3) and the fact that ∂if (x0) ≤ 0 for i = 1, . . . , q. This yields condition (iii). If q = p, then c = 0 and the above argument simplifies.
Proposition 5.3. Let O ⊂ Rp
be open, E ⊂ Rparbitrary, x
0∈ O ∩E and suppose there exists a C2
-bijection φ : Rp → Rp
such that φ(O ∩ E) = B ∩ (Rm
+ × Rp−m) for some open set B ⊂ Rp
. Let b ∈ Rp, c ∈ Sp
+ and K be a measure on Rp\{0} satisfying
Z
(|z|2∧ 1)K(dz) < ∞. Then for all f ∈ C2(O ∩ E) such that f (x
0) = supx∈O∩Ef (x) the conditions (i)-(iii) of Proposition 5.2 are satisfied if and only if for all i = 1, . . . , m with φi(x0) = 0 it holds that
(1) ∇φi(x0)>c∇φj(x0) = 0, for all j = 1, . . . , p,
(2) R |∇φi(x0)>χ(z)|K(dz) < ∞,
Proof. Define eb ∈ Rp, e c ∈ Sp+and a measure eK on Rp\{0} by ebi= ∇φi(x0)>b +12tr (∇ 2φ i(x0)c) + Z χi(φ(x0+ z) − φ(x0)) − ∇φi(x0)>χ(z) K(dz) e cij = ∇φi(x0)>c∇φj(x0) e K(G) = Z 1G(φ(x0+ z) − φ(x0))K(dz), for G ∈ B(Rp\{0}).
Note that eb is well-defined, sinceR (|z|2∧ 1)K(dz) < ∞ and χ(z) = z in a neigh-borhood of 0. We claim that conditions (i)-(iii) of Proposition 5.2 hold for all f ∈ C2(O ∩ E) with f (x
0) = supx∈O∩Ef (x) if and only if (i) ∇g(y0)>ec∇g(y0) = 0,
(ii) R | Ppi=1∂ig(y0)>∇φi(x0)χ(z)|K(dz) < ∞,
(iii) ∇g(y0)>eb −R ∇g(y0)>χ(z) eK(dz) +12tr (∇2g(y0)ec) ≤ 0, holds for all g ∈ C2(φ(O ∩ E)) with g(y
0) = supy∈φ(O∩E)g(y), where we put y0= φ(x0). This is tedious, but straightforward to verify, using the identities
∂ ∂xi f (x) =X j ∂ ∂yj g(y) ∂ ∂xi φj(x) ∂2 ∂xi∂xj f (x) =X k,l ∂2 ∂yk∂yl g(y) ∂ ∂xi φk(x) ∂ ∂xj φl(x) + X k ∂ ∂yk g(y) ∂ 2 ∂xi∂xj φk(x),
where f ∈ C2(O ∩ E), g = f ◦ φ−1, x ∈ O ∩ E and y = φ(x). Note that condition (ii) is equivalent with
Z |∇g(y0)>χ(z)| eK(dz) < ∞, as Z χi(φ(x0+ z) − φ(x0)) − ∇φi(x0)>χ(z) K(dz) < ∞, for all i = 1, . . . , p. Since φ(O ∩ E) = B ∩ (Rm
+× Rp−m) for some open set B ⊂ Rp, the result follows from Proposition 5.2.
In the following, we let Ω be given by (2.8), A as in (2.2), we assume (2.17) and we let E ⊂ Rp be of the form E = X × Rp−m as in (3.2). We write Q = {2, . . . , m}. We derive two existence results from Proposition 5.3. In the first we give necessary and sufficient conditions for a parabolic state space, in the second for the Lorentz cone. We apply these in Section 5.4 respectively Section 5.5 to obtain the admissibility conditions for the corresponding affine processes.
Corollary 5.4. Suppose X = {x ∈ Rm: x1≥ m X i=2 x2i}.
Define Φ(x) = (1, −2xQ, 0) ∈ R × Rm−1× Rp−m for x ∈ E. Then for all x ∈ E there exists a solution Px of the martingale problem for (A, δx) in Ω if and only if for all x ∈ ∂E it holds that
(1) c(x)Φ(x) = 0,
(2) R |Φ(x)>χ(z)|K(x, dz) < ∞,
(3) Φ(x)>b(x) −R Φ(x)>χ(z)K(x, dz) − tr (cQQ(x)) ≥ 0.
Proof. This follows from Proposition 2.15 together with Proposition 5.3 with O = Rp, m = 1 and φ given by φ1(x) = x1−P
m i=2x
2
i, φi(x) = xifor i = 2, . . . , p. Corollary 5.5. Suppose E is the Lorentz cone given as
E = {x ∈ Rp: x1≥ |xQ|}.
Define Φ(x) = (2x1, −2xQ) ∈ R × Rp−1 for x ∈ E. Then for all x ∈ E there exists a solution Px of the martingale problem for (A, δx) in Ω if and only if for all x ∈ ∂E\{0} it holds that
(1) c(x)Φ(x) = 0, (2) R |Φ(x)>χ(z)|K(x, dz) < ∞, (3) Φ(x)>b(x) −R Φ(x)>χ(z)K(x, dz) + c 11(x) − tr (cQQ(x)) ≥ 0. and (1’) c(0) = 0, (2’) R |χ(z)|K(0, dz) < ∞, (3’) b(0) −R χ(z)K(0, dz) ∈ E.
Proof. We apply Proposition 2.15 together with Proposition 5.3. Clearly the con-ditions of Proposition 2.15 only depend on the values of f in a neighborhood of x0. Take an open set O ⊂ {x ∈ Rp : x1 > 0} with x0 ∈ O. Then there exists a C2
-bijection φ : Rp→ Rp such that for x ∈ O we have φ
1(x) = x21− Pp
i=2x 2 i and φi(x) = xi, for i = 2, . . . , p. Note that φ(O ∩ E) = B ∩ (R+× Rp−1), for some open
set B ⊂ Rp. Hence we can apply Proposition 5.3 to deduce that the conditions of Proposition 2.15 are satisfied for x0∈ ∂E\{0} if and only if conditions (1)-(3) hold. The conditions of Proposition 2.15 are obviously satisfied for x0 ∈ E◦. It remains to show they are equivalent with (1’)-(3’) in case x0= 0.
Assume the conditions of Proposition 2.15 hold for all f ∈ C2
c(E) such that f (0) = supx∈Ef (x) ≥ 0. Let y ∈ E be arbitrary and define f (x) = −y>x. Then f (x) ≤ 0 for all x ∈ E by self-duality of the Lorentz cone E, whence f (0) = supx∈Ef (x) ≥ 0. It holds that ∇f (0) = y. Since y ∈ E is arbitrary, the conditions of Proposition 2.15 yield
(i) ci(0)y = 0 for all i = 1, . . . , p, y ∈ E,
(ii) R |y>χ(z)|K(0, dz) < ∞, for all y ∈ E, (iii) y>b(0) −R y>χ(z)K(0, dz)) ∈ E.
By (i) we have ci(0)> ∈ E for all i. Taking y = ci(0)> gives ci(0) = 0 for all i, whence we have (1’). Likewise we see that (2’) holds. By the self-duality of E we infer (3’) from (iii).
Conversely, assume (1’)-(3’) holds. Let f ∈ Cc2(E) be such that f (0) = supx∈Ef (x) ≥ 0. Let y ∈ E be arbitrary. Define g : R+ → R : t 7→ f(ty). Then 0 ≥ ∂t+
t=0g(t) = ∇f (0)>y. Hence ∇f (0) ∈ −E, since E is a self-dual cone. This yields the result.
5.2
Polyhedral state space
In this section we derive the explicit form of the admissible parameters for an affine process with a general polyhedral state space. Throughout we let E ⊂ Rp be a (non-empty) polyhedron given by
E = q \ i=1 {x ∈ Rp: ζ i+ ηix ≥ 0}, with ζ ∈ Rq
, η ∈ Rq×p, for some q ≥ 1. We write u(x) = ζ + ηx and ∂E
i= E ∩ {x : ui(x) = 0}. We start with a general result on affine functions that are positive on E, using methods from convex analysis, see also [46].
Proposition 5.6. Suppose E ⊂ {x ∈ Rp : d(x) ≥ 0} for some affine function d(x) = a>x + b with a ∈ Rp
, b ∈ R. Then there exist c ≥ 0 and λ ∈ Rq+, such that d(x) = λ>u(x) + c, for all x ∈ Rp.
Proof. We give a proof by contradiction. Let
K = {(η>λ, ζ>λ + c) ∈ Rp× R : λ ∈ Rq+, c ≥ 0}.
Suppose (a, b) 6∈ K. Since K is a closed convex set, (a, b) is strictly separated from K by the Separating Hyperplane Theorem. Therefore, there exist y ∈ Rp and y0∈ R such that h(y, y0), (k, k0)i > h(y, y0), (a, b)i for all (k, k0) ∈ K, i.e.
k>y + k0y0> a>y + by0 for all (k, k0) ∈ K.
In other words, for all λi≥ 0 and c ≥ 0 we have X
i
λi(ηiy + ζiy0) + cy0> a>y + by0.
It easily follows that
a>y + by0< 0 (5.1)
ηiy + ζiy0≥ 0 (5.2)
y0≥ 0. (5.3)
Using this we construct x ∈ E for which d(x) < 0. Suppose y0> 0. Then we take x = y/y0. Indeed, ui(x) = (ηiy + ζiy0)/y0 ≥ 0, so x ∈ E. But d(x) = (a>y + by0)/y0< 0, which is a contradiction. Suppose y0= 0. Then we take an arbitrary x0∈ X and let xN = x0+ N y, with N ∈ N. Then ui(xN) = ui(x0) + N ηiy ≥ 0 for all N , so xN ∈ E, but d(xN) = d(x0) + N a>y < 0 for N big enough.
As a consequence of the above result, we obtain Proposition 5.7 and Proposi-tion 5.9 below, which we use later on to tackle the drift respectively the diffusion part of affine processes with a general polyhedral state space.
Proposition 5.7. Suppose ∅ 6= ∂Ei ⊂ {x : d(x) ≥ 0} for some i ≤ q, some affine function d(x) = a>x + b with a ∈ Rp
, b ∈ R. Then there exist c ≥ 0 and λ ∈ Rq with λj ≥ 0 for j 6= i, such that
d(x) = λ>u(x) + c.
Proof. Let u0 := −ui. Then ∂Ei = T q
j=0{uj ≥ 0} and d(x) ≥ 0 for x ∈ ∂Ei. Hence we can apply Proposition 5.6, which gives the existence of λj ≥ 0 with j = 0, . . . , q and c ≥ 0 such that
d(x) = q X j=0 λjuj(x) + c = q X j=1 f λjuj(x) + c,
Henceforth in this section, we assume that q is minimal in the sense that E is strictly contained in \ i≤q,i6=j {x ∈ Rp: ζi+ ηix ≥ 0}. for all j ≤ q.
Lemma 5.8. It holds that ∂Ei6= ∅ for all i ≤ q.
Proof. Fix i ≤ q. By minimality of q we can choose x ∈ Rp such that u
i(x) < 0 and uj(x) ≥ 0 for all j 6= i. Let y ∈ E. Then uj(y) ≥ 0 for all j. For t ∈ [0, 1] it holds that
uj(tx + (1 − t)y) = tuj(x) + (1 − t)uj(y),
which is non-negative for j 6= i. For t = ui(y)/(ui(y) − ui(x)) we have ui(tx + (1 − t)y) = 0, so tx + (1 − t)y ∈ ∂Ei.
Proposition 5.9. Suppose ∂Ei ⊂ {x : v(x) = 0} for some i ≤ q, some affine function v(x) = a>x + b with a ∈ Rp, b ∈ R. Then there exists λi ∈ R such that v(x) = λiui(x) for x ∈ E. If E◦6= ∅, then v(x) = λiui(x) for all x ∈ Rp.
Proof. We have ∂Ei ⊂ {v ≥ 0} and ∂Ei ⊂ {−v ≥ 0}. Applying Proposition 5.7 with d = v respectively d = −v, we derive that
v(x) = q X j=1 λjuj(x) + c1 −v(x) = q X j=1 µjuj(x) + c2,
for some λj, µj ∈ R with λj, µj≥ 0 for j 6= i and c1, c2≥ 0. Adding the equations in the above display gives
0 = q X
j=1
(λj+ µj)uj(x) + c1+ c2.
By Lemma 5.8 we can choose x ∈ ∂Ei and deduce that c1= c2= 0. So
−(λi+ µi)ui(x) = X
j6=i
(λj+ µj)uj(x).
By minimality of q we can choose x ∈ Rp such that ui(x) < 0 and uj(x) ≥ 0 for all j 6= i. This gives that c := λi+ µi≥ 0. If c > 0, then for x ∈ E we have
0 ≤ ui(x) = −c−1 X
j6=i
whence ui(x) = 0 for x ∈ E. This gives E = ∂Ei ⊂ {x : v(x) = 0}, so that v(x) = ui(x) = 0 for x ∈ E. If c = 0, thenPj6=i(λj+ µj)uj(x) = 0 for all x. This holds in particular for x ∈ E, i.e. for x such that uj(x) ≥ 0 for all j. Hence for x ∈ E we have λjuj(x) = µjuj(x) = 0 for all j 6= i, so
v(x) = q X
j=1
λjuj(x) + c1= λiui(x), (5.4)
for x ∈ E. If E◦6= ∅, then choosing x ∈ E◦gives u
j(x) > 0 for all j, which implies λj= 0 for all j 6= i. Then (5.4) holds for all x ∈ Rp.
Having proved the above general results, we now turn to affine processes on a polyhedron. Using Proposition 5.9 together with a necessary boundary condition on the diffusion matrix c(x) of an affine process, we are able to characterize the form of c(x) and further specify the state space E. The next proposition generalizes [23, Lemma 7.1] and considerably simplifies and improves upon the results given in the first part of the appendix in [18].
Proposition 5.10. Let c : Rp → Sp be affine and such that c(E) ⊂ Sp
+. Assume E◦6= ∅ and
∀i ≤ q, ∀x ∈ ∂Ei: ηic(x) = 0. (5.5) Then there exists a non-singular L ∈ Rp×p
and a vector ` ∈ Rp such that for all x ∈ Rp we have
Lc(L−1(x − `))L>= diag(xM, 0N) 0 0 Ψ(xM ∪N)
!
, (5.6)
for some index sets M = {1, . . . , m}, N = {m + 1, . . . , m + n} and affine function Ψ. In addition we have
LE + ` = Rm+ × C × R
p−m−n, (5.7)
for some convex polyhedron C =Tq−m i=1 {y ∈ R
n: e
ui(y) ≥ 0} ⊂ Rn+ with eui(y) = yi for i ≤ n.
Proof. We divide the proof into a couple of steps.
Step 1. There exists B ∈ Rq×p such that for i ≤ q it holds that
ηic(x) = Biui(x), (5.8) Biη>i > 0, if Bi 6= 0, (5.9) Biη>j = 0, for j 6= i. (5.10)
This is shown as follows. Fix i ≤ q. By (5.5) and Proposition 5.9 there exists Bi ∈ R1×p such that ηicj(x) = Bijui(x) for j ∈ P . By assumption there exists x0∈ E◦. It holds that ui(x0) > 0, so we can write
Bi= ui(x0)−1ηic(x0). (5.11)
By positive semi-definiteness of c(x0), it holds that Biη>i ≥ 0. We have Biηi>= 0 if and only if ηic(x0) = 0, i.e. Bi= 0. This yields (5.9). Moreover, if j 6= i, then by symmetry of c(x) it holds that
Biη>jui(x) = ηic(x)η>j = ηjc(x)η>i = Bjηi>uj(x),
for all x ∈ Rp. This implies Biη>j = 0, since q is minimal.
Step 2. Define M = {i : ηiB>i 6= 0} and permute indices such that M = {1, . . . , m}, with m = #M . Then ηM has full-rank, in view of
ηMB>M = diag(η1B1>, . . . , ηmB>m).
Now take N ⊂ {m + 1, . . . , q} such that ηM ∪N has full-rank. By permuting the indices we may assume N = {m + 1, . . . , m + n}, with n = #N .
Step 3. It holds that
c(x) = m X
i=1
(ηiBi>)−1Bi>Biui(x) + Φ(x),
for some affine function Φ : Rp → Sp satisfying ηΦ(x) = 0, for all x. This is an immediate consequence of Step 1. In addition, Φ can be written as a function of u(x), since it is positive semi-definite on E.
Step 4. The result follows by taking ` ∈ Rpwith `
M ∪N = ζM ∪N and L ∈ Rp×p non-singular with LM ∪N = ηM ∪N and the remaining rows orthogonal to the first m + n rows.
Proposition 5.7 together with a necessary boundary condition for the drift enables us to derive a kind of maximum principle, see the next proposition. Proposition 5.11. Suppose b : Rp→ Rpis affine and such that η
ib(x) ≥ 0 for all i ≤ q, x ∈ ∂Ei. Then for all f ∈ Cc2(E) and x0∈ E such that f has a maximum at x0∈ E, it holds that
∇f (x0)>b(x0) ≤ 0. Proof. Let Xtbe the unique solution to the ODE
It suffices to prove that Xt∈ E for all t ≥ 0, since in that case ∇f (x0)>b(x0) = ∂t+
t=0f (Xt) = lim
t↓0(f (Xt) − f (x0))/t ≤ 0. By Proposition 5.7 we have ηb(x) = a0+ au(x) for some a0 ∈ Rq
+, a ∈ Rq×q with aij ≥ 0 for i 6= j. Hence
du(Xt) = (a0+ au(Xt))dt, u(X0) = u(x0).
We claim that u(Xt) ∈ R q
+ for all t, which yields the result. From Proposition 5.2 we infer that the parameter set (ai, Ai, γi, Ki) with ai as above and Ai = 0, γi= 0, Ki
= 0, is admissible for the state space Rq+. By Theorem 4.4 there exists a corresponding affine jump-diffusion with state space Rq+. Necessarily this process equals u(Xt) in case the initial condition is u(x0). This proves the claim.
We are now able to prove the main theorem of this section. We work out the admissibility conditions for general polyhedra, extending the conditions for the “canonical” polyhedron Rm
+ × Rp−m as given in [17, Theorem 2.4]. In view of Proposition 5.10, we assume that the state space E is of the form E = X × Rp−m, with X = Rn
+× C ⊂ Rm, for some n ≤ m and C ⊂ R m−n
+ a convex polyhedron as in Proposition 5.10. In addition, without loss of generality we assume that there does not exist an invertible affine transformation φ such that φ(C) = R+× C0 for some polyhedron C0 ⊂ Rm−n−1+ , i.e. n is maximal and C is minimal. We give the admissibility conditions up to an invariant transformation, i.e. an invertible affine transformation φ such that φ(E) is of the same form as E.
Theorem 5.12. Suppose E = X × Rp−m
is a polyhedron with X = Rn
+× C as given as above. The parameter set (ai, Ai, γi, Ki) is admissible if and only if the following properties hold, possibly after an invariant transformation.
(A) The killing part γ(x) = γ0+Pp i=1γ ix i can be written as γ(x) = λ0+ λ>u(x), for some λ0≥ 0, λ ∈ R q +.
(B) The jump part (|z|2∧ 1)K(x, dz) = K0(dz) +Pp i=1K i(dz)x i can be written as (|z|2∧ 1)K(x, dz) = µ0(dz) + q X i=1 µi(dz)ui(x),
for some finite positive measures µi with supp µi ⊂ K for some convex cone K ⊂ E such that E + K ⊂ E. In addition, R |ηiχ(z)|(|z|2∧ 1)−1µj(dz) < ∞ for all i = 1, . . . , q, j = 0, . . . , q with j 6= i.
(C) The diffusion part c(x) = A0+Pp i=1A ix i is of the form c(x) = diag(xI1, 0I2) 0 0 Ψ(xI) ! ,
for some index sets I1 = {1, . . . , m1}, I2 = {m1+ 1, . . . , m}, with m1 ≤ n and an affine function Ψ : Rm→ Sp−m such that Ψ(X ) ⊂ Sp−m
+ . (D) The drift part b(x) = a0+Pp
i=1a ix i satisfies ηb(x) =ea0+eau(x), with ea0∈ Rq and e a ∈ Rq×q such that e aji ≥R ηiχ(z)(|z|2∧ 1)−1µj(dz) for all i = 1, . . . , q, j = 0, . . . , q with j 6= i.
Proof. We first prove the “only if”-part using Proposition 2.16. Since γ(x) ≥ 0 and K(x, dz) is a positive measure for x ∈ E, Proposition 5.6 yields (A) and
Z G (|z|2∧ 1)K(x, dz) = µ0(G) + q X i=1 µi(G)ui(x),
for some µi(G) ≥ 0, for all G ∈ B(Rp\{0}). The coefficients µi(G) are the unique non-negative numbers with the above property. Therefore, µi(∅) = 0 and σ-additivity of G 7→ K(x, G) carries over to G 7→ µi(G), so that µiare finite positive measures. Their support is contained in some convex cone K ⊂ E with E +K ⊂ E, in view of Lemma 4.7. Proposition 2.16 (ii) gives
Z |ηiχ(z)|(|z|2∧ 1)−1µ0(dz) + q X j=1 Z |ηiχ(z)|(|z|2∧ 1)−1µj(dz)uj(x) < ∞,
for all i = 1, . . . , q, x ∈ ∂Ei, which proves (B). Necessity of (C) follows from Proposition 2.16 (i) and Proposition 5.10 together with Proposition 5.1. Finally, Proposition 2.16 (iii) together with Proposition 5.7 yields (D), since for all i it holds that d(x) := ηib(x) + Z ηiχ(z)(|z|2∧ 1)−1µ0(dz) + X j6=i Z ηiχ(z)(|z|2∧ 1)−1µj(dz)uj(x)
is a (well-defined) affine function satisfying d(x) ≥ 0 for all x ∈ ∂Ei. We now show the “if”-part. Let f ∈ C2
c(E) assume a non-negative maximum at x0∈ E. Property (a) of Definition 4.3 is obvious. Note that ∇f (x0)> is in the linear span of L := {ηi: x0∈ ∂Ei}, since
for all v orthogonal to L. Hence (b) follows from (B). It remains to show (c). By the same argument as in the proof of Proposition 5.2 we have tr (∇2f (x
0)c(x0)) ≤ 0. To show ∇f (x0)>b(x0) −R ∇f (x0)>χ(z)K(x0, dz) ≤ 0 we argue as follows. Since supp K(x, dz) ⊂ K ⊂ E for some cone K, we have ηiz ≥ 0 for all i and z ∈ supp K(x, dz) (as ui(z) = ζi+ ηz ≥ 0 for all z ∈ E). Let ε > 0 and write
bε(x) = b(x) − Z
{|z|>ε}
χ(z)K(x, dz).
Then for all i = 1, . . . , q it holds that
ηibε(x) = ηib(x) − Z {|z|>ε} ηiχ(z)K(x, dz) ≥ ηib(x) − Z ηiχ(z)K(x, dz),
for ε > 0 small enough, since χ(z) = z ∈ E in a neighborhood of 0. Hence by (D) we have ηibε(x) ≥ 0 for all i and x ∈ ∂Ei. Proposition 5.11 yields
∇f (x0)>bε(x0) ≤ 0. The left-hand side equals
∇f (x0)>b(x0) − Z
{|z|>ε}
∇f (x0)>χ(z)K(x0, dz),
so dominated convergence gives the result when we let ε tend to zero (recall that we already showed thatR |∇f (x0)>χ(z)|K(x0, dz) < ∞).
Remark 5.13. An important difference between the admissibility conditions for the canonical state space Rm
+× Rp−m and a general polyhedron is the form of the diffusion matrix. In case X = Rm+, it is easy to see that Ψ(X ) ⊂ S
p
+ if and only if Ψ(xI) = B0+Pmi=1Bixi with Bi ∈ S+p−m. In general such a characterization is not possible. More precise, for Ψ(X ) ⊂ S+p it is not necessary that Ψ(xI) can be written as B0+Pm
i=1B iu
i(x) for some Bi ∈ Sp−m+ . For example, let X = T3 i=1{x ∈ R 2: u i(x) ≥ 0} ⊂ R2 with u1(x) = x1, u2(x) = x2, u3(x) = x1+ x2−32 and take Ψ(x) = x1+ 1 2 1 1 x2+12 ! . Then Ψ(X ) ⊂ S2
+, but one verifies that Ψ(x) is not of the form B0+ P3 i=1B iu i(x) for Bi∈ S2 +.
5.3
Characterizing all quadratic state spaces
In this section we consider quadratic spaces as defined below and determine all possible quadratic state spaces for an affine process. As we will see, the only possibilities are the parabolic state space and the Lorentz cone. Our analysis extends the classification of [27] for the two-dimensional case to higher dimensions. Let E ⊂ Rp be closed and convex with non-empty interior. We say E is a quadratic state space if there exists a quadratic function Φ : Rp→ R given byΦ(x) = x>M x + v>x + w, (5.12)
for some symmetric non-zero M ∈ Rp×p
, a vector v ∈ Rp
and a constant w ∈ R, such that ∂E ⊂ {x : Φ(x) = 0} and E◦ ⊂ {x : Φ(x) 6= 0}, or equivalently, E◦ is a connected component of {x : Φ(x) 6= 0} (i.e. a maximal connected subset of {x : Φ(x) 6= 0}). By the convexity of E, there are only three possibilities for Φ, as we show in the following proposition.
Proposition 5.14. Let E ⊂ Rp be convex and assume E◦ is a non-empty con-nected component of {Φ 6= 0}, with Φ given by (5.12). Then, up to an invertible affine transformation, Φ(x) is of one of the following forms:
(i) Φ(x) = x1− q X i=2 x2i, (ii) Φ(x) = q X i=1 x2i + d, (iii) Φ(x) = x21− q X i=2 x2i + d, for some d ∈ R, q ≤ p.
Proof. Since M is symmetric, it is diagonalizable by an orthogonal matrix. By further scaling one can take the diagonal elements equal to −1, 0 or 1. Using the equality
x>x + v>x = (x>+12v>)(x +12v) −14v>v,
we deduce that, up to an invertible affine transformation, the quadratic function Φ is of the form Φ(x) = x1− X i∈Q x2i +X i∈Q0 x2i, (5.13)
or Φ(x) = x21− X i∈Q x2i + X i∈Q0 x2i + d, (5.14)
for some disjoint sets Q, Q0⊂ {2, . . . , p} and d ∈ R. If Φ is of the form (5.13), then E is of the form E = {x ∈ Rp: x1≥ X i∈Q x2i −X i∈Q0 x2i},
possibly after replacing x1 by −x1 and interchanging Q and Q0. Convexity of E yields that the Hessian ofP
i∈Qx 2 i −
P
i∈Q0x2i is positive semi-definite, which
implies that Q0 = ∅. Permuting coordinates gives Q = {2, . . . , q} with q = #Q + 1. Now assume Φ is of the form (5.14). We have to show that either Q0 = ∅ or #Q ≤ 1. Assume Q0∪ Q 6= ∅. Define the function f by f (xQ∪Q0) =P
i∈Qx 2 i − P
i∈Q0x2i−d. One verifies that there are two possible forms E can assume (possibly
after replacing x1 by −x1), namely
E = {x ∈ Rp: x1≥ f (xQ∪Q0)1/2, xQ∪Q0∈ K}
or
E = {x ∈ Rp: |x1| ≤ f (xQ∪Q0)1/2, xQ∪Q0 ∈ K},
where K is convex and with K◦a non-empty connected component of {f > 0}. By convexity of E, we have in the first case that the Hessian of f (xQ∪Q0)1/2is positive
semi-definite, while in the second case the Hessian is negative semi-definite. Now suppose Q06= ∅. We show that in that case #Q ≤ 1. Let xQ∪Q0 ∈ K◦. For i ∈ Q0
we have ∂2 ∂x2
i
f (xQ∪Q0)1/2= −f (xQ∪Q0)−1/2− x2if (xQ∪Q0)−3/2 < 0,
wherefore necessarily the Hessian is negative semi-definite. For i ∈ Q we have ∂2
∂x2 i
f (xQ∪Q0)1/2= f (xQ∪Q0)−1/2− x2if (xQ∪Q0)−3/2, (5.15)
which therefore also has to be negative. Now if #Q > 1, then there exists i, j ∈ Q with i 6= j. For such i, j it holds that
∂2 ∂xi∂xj f (xQ∪Q0)1/2= −f (xQ∪Q0)−3/2xixj, whence det ∂2 ∂x2 i ∂2 ∂xi∂xj ∂2 ∂xj∂xi ∂2 ∂x2 j f (xQ∪Q0)−1/2 = = f (xQ∪Q0)−1/2 f (xQ∪Q0)−1/2− (x2i + x2j)f (xQ∪Q0)−3/2 ,
which is negative as (5.15) is negative for i ∈ Q. This contradicts the negative semi-definiteness of the Hessian of f (xQ∪Q0)1/2. Thus it holds that #Q ≤ 1, as we
needed to show.
Next we eliminate possibility (ii) for all d and (iii) for d ≤ 0, by exploiting the necessary boundary condition for the diffusion matrix c(x). For this we need the following lemma.
Lemma 5.15. Let Ψ : Rp→ Sp: x 7→ A0+Pp k=1A
kx
k with Ak∈ Sp and assume x>Ψ(x) = 0 for all x ∈ Rp
. Then Ψ(x) = 0 for all x ∈ Rp. Proof. We show that Ak
= 0 for k ≥ 0. For all j = 1, . . . , p and x ∈ Rp it holds that 0 = x>Ψj(x) = x>(A0+ p X k=1 Akxk) j = p X i=1 xiA0ij+ p X k=1 p X i=1 xixkAkij = p X i=1 xiA0ij+ X 1≤i<k≤p xixk(Akij+ A i kj) + p X i=1 x2iAiij.
Hence for all i, j = 1, . . . , p we have A0
ij = Aiij = 0 and Akij = −Aikj for k 6= i. Since Ak is symmetric, the latter gives Akij = −Aikj = −Aijk. So if we permute the indices i, j, k by the cycle (i 7→ j, j 7→ k, k 7→ i), then Ak
ij gets a minus sign. Permuting the indices repeatedly we obtain
Akij= −Aijk= Ajki= −Akij,
which implies Ak
ij = 0 for all i, j and k 6= i. Hence Ak = 0 for all k, as we have already shown that Aiij= 0 for all i, j.
Proposition 5.16. Suppose (X, (Px)x∈E) is an affine jump-diffusion with param-eters (b(x), c(x), γ(x), K(x, dz)) given by (4.3), with E of the form (3.2) and X◦ a connected component of {Φ 6= 0}, for some quadratic function Φ given by (5.12). In addition, suppose cII(x) 6= 0 for some x ∈ E. Then, up to an invertible affine transformation, Φ is of the form
Φ(x) = x1− m X i=2 x2i or Φ(x) = x21− m X i=2 x2i.
Proof. By Corollary 4.22 we may assume without loss of generality that X = E, i.e. m = p. There exists an invertible affine transformation such that Φ is of one of the three forms as stated in Proposition 5.14. We show that Φ cannot be of the second form for all d ∈ R and not of the third form unless d = 0. Let x ∈ ∂E be arbitrary. Recall that U given by (3.3) is assumed to have non-empty interior. Therefore, all the components xi for i ≤ p are present in the expression for Φ(x), so that q = p. In addition, Φ(x) =Pp
i=1x 2
i + d with d ≥ 0 is impossible, as this would imply X = Rp.
Now suppose Φ(x) = Pp
i=1x2i + d with d < 0. Then E = {Φ ≤ 0}, whence Φ(x) = supy∈EΦ(y) = 0 for all x ∈ ∂E = {Φ = 0}. Define Ψ(x) : Rp → Sp : x 7→ A0+Pp
k=1Akxk, so that Ψ|E = c. By the admissibility conditions as given in Definition 4.3 it follows that
x>Ψi(x) = 0, for all i and all x ∈ Rp such that Φ(x) = 0.
Regarding Φ(x) as a univariate polynomial in x1, we see that it has distinct roots ±(−d−Pp
j=2x 2
j)1/2, for x close enough to zero. Therefore, these are also the roots of x>Ψi(x). Since the latter has maximal degree 2, it follows that
x>Ψi(x) = PiΦ(x), for all i and x ∈ Rp,
for some constant Pi. Note that the right-hand side of the above display has a constant term Pid. Since the left-hand side only contains multiples of x, this yields Pi = 0 for all i, so that x>Ψ(x) = 0 for all x ∈ Rp. Lemma 5.15 yields that Ψ(x) = 0 for all x, which contradicts the assumption that c(x) 6= 0 for some x ∈ E.
Likewise we show that Φ(x) = x2 1−
Pp
i=2x2i + d with d 6= 0 is impossible. Indeed, suppose Φ is of this form. Write Q = {2, . . . , p} and define the function f by f (xQ) =Pi∈Qx2i − d. Then convexity of E yields
E = {x ∈ Rp: x1≥ f (xQ), xQ ∈ K},
where K is convex and with K◦ a non-empty connected component of {f > 0}. It holds that E ⊂ {Φ ≥ 0}, so −Φ assumes a non-negative maximum at the boundary. The admissibility conditions yield
(−x1, xQ)>Ψi(x) = 0, for all i and all x ∈ E such that Φ(x) = 0,
where Ψ is defined as before. If x ∈ E and Φ(x) = 0, then x1 = f (xQ)1/2. Hence f (xQ)1/2 is a root of x>Ψi(x), regarded as a univariate polynomial in x1.
Since the degree does not exceed 2, necessarily also −f (xQ)1/2 is a root, so that x>Ψi(x) = PiΦ(x) for some constant Pi, for all x ∈ Rp. The rest of the argument is verbatim the previous paragraph.
From Proposition 5.16 it follows that in order to characterize all affine jump-diffusions with a quadratic boundary ∂E ⊂ {x : Φ(x) = 0}, there are two cases to consider, namely Φ(x) = x1−P
m
i=2x2i and Φ(x) = x21− Pm
i=2x2i, for some m ≤ p. In the next two sections we work out the corresponding admissibility conditions.
5.4
Parabolic state space
In this section we state the admissibility conditions for the parabolic state space E = X × Rp−m with X given by X = {x ∈ Rm: x1≥ m X i=2 x2i}.
For characterizing the diffusion matrix, we introduce the following notation. For x ∈ Rp we write y = (x
2, . . . , xm) and we define affine matrix-valued functions ζ and η by ζ(x) = 4x1 2y > 2y I ! , η(x) = 0 0 . . . 0
T12(y) T13(y) . . . Tm−2,m−1(y) !
,
with Tij : Rm−1→ Rm−1 for 1 ≤ i < j < m given by Tij(y)i= yj, Tij(y)j= −yi, Tij(y)k= 0 for k 6= i, j. For example, for m = 4 we have
η(x) = 0 0 0 y2 y3 0 −y1 0 y3 0 −y1 −y2 .
As in the previous section, we give the admissible parameters up to an invariant transformation, i.e. an invertible affine transformation that leaves the state space E unaltered. In addition, we assume that the diffusion part for the first m com-ponents does not vanish on the whole of E. The following theorem extends [27, Proposition 2] from the 2-dimensional continuous case to higher dimensions includ-ing jumps and killinclud-ing. As in Chapter 4 we write I = {1, . . . , m}, J = {m+1, . . . , p} and we write Q = I\{1}.
Theorem 5.17. Suppose E = X × Rp−m with X parabolic. The parameter set (ai, Ai, γi, Ki) is admissible with A0
II + Pp
i=1AiIIxi 6= 0 for some x ∈ E, if and only if the following properties hold, possibly after an invariant transformation. (A) The killing parameters satisfy γ0≥ 0, γ1≥ 0, γi= 0 for i > m and
4γ0γ1≥ m X
i=2 (γi)2,
(B) For the jump parameters it holds that K0(G) ≥ 0, K1(G) ≥ 0, Ki(G) = 0 for i > m and 4K0(G)K1(G) ≥ m X i=2 (Ki(G))2,
for all G ∈ B(Rp\{0}). In addition, supp |Ki
| ⊂ R+ × {0} × Rp−m and R
{|z1|≤1}|z1||z|
−2|Ki|(dz) < ∞, for all i. (C) The diffusion part c(x) = A0+Pp
i=1A ix i is of the form c(x) = ζ(x) η(x)M M>η(x)> B(x) ! , (5.16)
for some M ∈ Rp−m× Rp−m, with B(x) affine and
B(x) ≥ M>η(x)>η(x)M, for all x ∈ E. (5.17)
(D) Let α0 ∈ Rp
and α ∈ Rp×p be given by αi1 = ai1−R χ1(z)(|z|2∧ 1)−1Ki(dz) and αi
j= aij for i = 0, . . . , p, j 6= 1. Then these drift parameters satisfy
αIJ = 0 αQ1 = 0 αQQ=12α11I − 12diag(d) α1i= 2α0i, for q + 2 ≤ i ≤ m α01≥ m − 1 + q X i=1 1 4d −1 i (α1,i+1− 2α0i+1) 2,
for some vector d ∈ Rm−1 with d
i > 0 for i ≤ q and di = 0 for i > q, for some q ≤ m − 1.
Proof. Define Φ(x) = (1, −2y, 0) and let b(x), c(x), γ(x), K(x, dz) be given by (4.3). In view of Corollary 5.4 and positivity of c, γ and K, we have to verify that the above parameter restrictions are equivalent with the following properties:
(i) c(x) ≥ 0 for all x ∈ E, cII(x0) 6= 0 for some x0∈ E, and c(x)Φ(x) = 0 for all x ∈ Rp such that x
1= y>y.
(ii) K(x, dz) ≥ 0, supp K(x, dz) ⊂ E − x for x ∈ E andR |Φ(x)>χ(z)|K(x, dz) < ∞ for all x ∈ Rp such that x
1= y>y.
(iii) γ(x) ≥ 0 for all x ∈ E.
(iv) Φ(x)>b(x) −R Φ(x)>χ(z)K(x, dz) − tr (cQQ(x)) ≥ 0 for all x ∈ Rp such that x1= y>y.
We treat each item separately in the next subsections.
5.4.1
Admissibility for the diffusion parameters
In this section we show the equivalence between (C) and (i) (up to an invariant transformation). We first prove that (C) implies (i). Suppose c(x) is of the form (5.16) and (5.17) holds. To show that c(x) ≥ 0 for all x ∈ E, it suffices to find a square root σ(x) for c(x), i.e. a matrix σ(x) such that σ(x)σ(x)> = c(x) for all x ∈ E. Write ξ(x) = 2 p x1− y>y 2y> 0 I ! ,
for x ∈ E. Note that ξ(x)η(x) = η(x). Since B(x) − M>η(x)η(x)>M> is positive semi-definite, it admits a square root, so that we can define
σ(x) = ξ(x) 0
M>η(x)> (B(x) − M>η(x)η(x)>M>)1/2 !
.
One easily verifies that σ(x)σ(x)> = c(x) for all x ∈ E, whence c(x) is positive semi-definite. It is also clear that c(x)(1, −2y, 0) = 0 whenever x1 = y>y. Thus we have shown that (C) implies (i).
Next we show the other direction, for which we need the following two lemmas. Lemma 5.18. Consider the linear space
L =na : Rp→ Rmaffine |1 −2y>a(x) = 0 for all x with x
1= y>y o
. (5.18) Then a basis for L is formed by the columns of ζ and η.
Proof. Clearly these columns are linearly independent elements of L. To prove that they span L we use a dimension argument. Let Aff(Rp, Rm) denote the space
of affine functions from Rp to Rm
and let Quadr(Rp
, R)/(x1− y>y) be the space of quadratic functions from Rp
to R, modulo x1−y>y (that is, φ and ψ are equivalent if φ(x) − ψ(x) = k(x1− y>y) for some constant k). Consider the linear operator
L : Aff(Rp, Rm) → Quadr(Rp, R)/(x1− y>y) : a(x) 7→
1 −2y>a(x), and note that L = ker L. By the dimension theorem for linear operators, we have
dim Aff(Rp, Rm) = dim ker L + dim im L.
It holds that dim Aff(Rp, Rm) = pm + m. Since x1≡ y>y, a basis for im L is given by {1, x2, . . . , xp} ∪ {xixj : 2 ≤ i ≤ m, 1 ≤ j ≤ p}, whence dim im L = p + (m − 1)p − m − 1 2 ! = pm − m − 1 2 ! . It follows that dim ker L = m +1
2(m − 1)(m − 2), which is the number of columns in ζ and η. Thus the columns span the kernel of L.
Lemma 5.19. Let L be defined by (5.18) and suppose M : Rp→ Smis affine. If the columns of M are in L, then M = cζ for some c ∈ R.
Proof. By Lemma 5.18 there exist matrices A and B such that M (x) = ζ(x)A + η(x)B.
Write T (y) = (Tij(y))1≤i<j<m and B =
B1 e B
. Then the above display reads
M (x) = 4x1A11+ 2y >A
Q1 4x1A1Q+ 2y>AQQ 2yA11+ AQ1+ T (y)B1 2yA1Q+ AQQ+ T (y) eB
! .
Since M (x) is symmetric it immediately follows that A1Q = 0 and AQ1 = 0. We have to show that N (x) := M (x) − A11ζ(x) is zero. Note that N is symmetric and that N (x) = 0 2y >C T (y)B1 C + T (y) eB ! ,
with C = AQQ−A11I. This yields C = C>and T (y)B1= 2Cy. Since y>T (y) = 0, the latter implies y>Cy = 0 for all y ∈ Rp−1, whence C = 0, as C is symmetric. Thus AQQ= A11I and it remains to show that eB = 0.
It holds that T (y) eB is symmetric and y>T (y) eB = 0. Lemma 5.15 yields T (y) eB = 0, whence eB = 0 by linear independence of the columns of T (y), as we needed to prove.
Proposition 5.20. Suppose c : Rp→ Sp
+ is affine and such that c(x)(1, −2y, 0) = 0 whenever x1= y>y. Then necessarily c is of the form
c(x) = kζ(x) N (x) N (x)> B(x) !
, (5.19)
for some k ≥ 0, with N (x) = ζ(x)M1+ η(x)M2 for some matrices M1, M2 and B : Rp→ S(p−m) affine. Moreover, if k = 1 and M
1= 0, then it holds that
B(x) − M2>η(x)>η(x)M2≥ 0, for all x ∈ E.
Proof. The first part follows from Lemma 5.18 and Lemma 5.19. It remains to show (5.17). Suppose k = 1 and M1 = 0. By positive semi-definiteness of c, we have
0 ≤v> w>c(x) v w
!
= v>ζ(x)v + 2v>η(x)M2w + w>B(x)w,
for all v ∈ Rm, w ∈ Rp−m, x ∈ E. Fix w ∈ Rp−m, let x ∈ E be arbitrary and take v = −η(x)M2w. Noting that ζ(x)η(x) = η(x) for all x ∈ Rp, the above display then reads
w>B(x)w − w>M2>η(x)>η(x)M2w ≥ 0,
which gives the result.
In view of the above proposition, (i) implies that c(x) is of the form (5.19). It remains to find an invariant transformation such that we can take k = 1 and M1= 0, in other words, we have to find a non-singular matrix L ∈ Rp×p, a vector ` ∈ Rp such that
Lc(L−1x − `)L>= ζ(x) η(x)M2 M2>η(x)> B(x)
!
, and LE + ` = E, (5.20)
see Proposition 5.1. Since by assumption cII(x0) 6= 0 for some x0 ∈ E, it holds that k > 0. Define matrices
L1= k−1 0 0 k−1/2I 0 0 I , L2= I 0 −M> 1 I ! .
The invariant transformation x 7→ L1x gives (5.19) with k = 1, while x 7→ L2x gives (5.19) with M1= 0, so that the composition x 7→ L2L1x is the transformation we are looking for. The reader verifies that (5.20) indeed holds for L = L2L1, ` = 0. Thus we have shown the equivalence between (C) and (i), up to an invariant transformation.
5.4.2
Admissibility for the killing parameters
Here we show the equivalence between (A) and (iii). Suppose (A) holds. If γ1= 0, then (A) yields γi = 0 for all i = 1, . . . , p, so that γ(x) = γ0 ≥ 0 for all x. If γ1> 0, then for x ∈ E we can write
γ(x) = γ0+ m X i=1 γixi≥ γ0+ m X i=2 (γ1x2i + γixi) ≥ γ0− m X i=2 (γi)2 4γ1 ,
which is non-negative by (A). Conversily, suppose (iii) holds. Since 0 ∈ E and since we can choose x1 arbitrarily large, we necessarily have γ0≥ 0 and γ1≥ 0. If γ1= 0, then necessarily γi= 0 for all i = 1, . . . , p, as we can choose x
i arbitrarily large. Hence we have (A). If γ1> 0, then we define x ∈ E by
x1= m X i=2 x2i and xi= − γi 2γ1, for i > 1. It holds that 0 ≤ γ(x) = γ0− m X i=2 (γi)2 4γ1 ,
which yields (A).
5.4.3
Admissibility for the jump parameters
The first part of the equivalence between (B) and (ii) follows by similar arguments as in the previous paragraph. It remains to show the equivalence
supp |Ki| ⊂ R
+× {0} × Rp−m, ∀i, ⇐⇒ supp K(x, dz) ⊂ E − x, ∀x ∈ E and the equivalence
R
|z1|≤1|z1||z|
−2|Ki|(dz) < ∞, ∀i, ⇐⇒R |Φ(x)>χ(z)|K(x, dz) < ∞, ∀x ∈ ∂E. By Lemma 4.7 it holds that supp K(x, dz) ⊂ E − x for all x ∈ E if and only if supp K(x, dz) ⊂ C for all x ∈ E, for some closed convex cone C ⊂ E with E + C ⊂ E. The maximal cone that is contained in E is R+× {0} × Rp−m. This proves the first equivalence. The second equivalence is straightforward.
5.4.4
Admissibility for the drift parameters
Assume that (C) holds. We conclude the proof of Theorem 5.17 by showing the equivalence between (D) and (iv), up to an invariant transformation that leaves the diffusion matrix unaltered. We have tr (cQQ(x)) = m − 1 for all x, so the boundary condition (iv) for the drift reads
y>(α11I − 2αQQ)y + (α1Q− 2α0Q >
)y + α01− m + 1 ≥ 0, for all y ∈ Rm−1. For this it is necessary that M := α11I − 2αQQis positive semi-definite. Moreover, if y is in the kernel of M , then y should also be in the kernel of α1Q− 2α0Q
> . We can diagonalize M by an orthogonal matrix O, so D = OM O> is diagonal with positive diagonal elements di for i ≤ q and di = 0 for i > q, for some q ≤ m − 1. Applying the orthogonal transformation y 7→ Oy, the above condition becomes
q X i=1 diyi2+ q X i=1
(α1,i+1− 2α0i+1)yi+ α01− m + 1 ≥ 0, for all y,
in view of Lemma 5.21 below. We can write the left-hand side as q X i=1 di(yi+12d−1i (α1,i+1− 2α0i+1)) 2−1 4 q X i=1 d−1i (α1,i+1− 2α0i+1) 2+ α0 1− m + 1, which is non-negative for all y if and only if
−1 4 q X i=1 d−1i (α1,i+1− 2α0i+1)2+ α01− m + 1 ≥ 0.
This yields the result.
Lemma 5.21. Let α0 and α be given as in Theorem 5.17 and c(x) be given by (5.16). Let O ∈ R(m−1)×(m−1) be orthogonal. Then the invariant transformation given by y 7→ Oy and xi7→ xi for i 6∈ Q, leaves c(x) unaltered and transforms α0 and α into U α0 and U αU>, with U ∈ Rp×p given by
U = 1 0 0 0 O 0 0 0 I .
Proof. This is a consequence of Proposition 5.1 and the fact that χ1(z) only de-pends on z1.
5.5
The Lorentz cone
In this section we work out the admissibility conditions for the Lorentz cone E given by
E = {x ∈ Rp: x1≥ |(x2, . . . , xp)|}.
As in the previous section, we introduce some notation for characterizing the dif-fusion matrix. We write y = (x2, . . . , xp) and let ζ(x) be given by
ζ(x) = x1 y > y x1I
! ,
and η and T be given as in the previous section. In addition, we define affine functions ρ(i) : Rp→ Sp for i = 1, . . . , p − 1 by
ρ(i)i+1: x 7→ x1 y> , ρ(i)11: x 7→ yi,
ρ(i)jj : x 7→ −yi, for j 6= 1, i + 1, ρ(i)jk: x 7→ 0, if j, k 6= i + 1 and j 6= k.
For example, for p = 4 we have
ρ(1)(x) = y1 x1 0 0 x1 y1 y2 y3 0 y2 −y1 0 0 y3 0 −y1 , ρ(2)(x) = y2 0 x1 0 0 −y2 y1 0 x1 y1 y2 y3 0 0 y3 −y2 , ρ(3)(x) = y3 0 0 x1 0 −y3 0 y1 0 0 −y3 y2 x1 y1 y2 y3 .
Similar as in Theorem 5.17 we assume the diffusion part does not vanish on the whole of E. Moreover, we assume an integrability condition for the jump-measures Ki, in order to give a more precise characterization of the drift. As in the previous section we write Q = {2, . . . , p}.
Theorem 5.22. Let E be the Lorentz cone and let (ai, Ai, γi, Ki) be a parameter set such that A0+Pp
i=1A ix
i6= 0 for some x ∈ E and R
{|z|≤1}|zi||z|−2|Kj|(dz) < ∞ for all i, j ≤ p. Then (ai, Ai, γi, Ki) is admissible if and only if the following properties hold, possibly after an invariant transformation:
(A) The killing parameters satisfy γ0≥ 0 and (γ1, . . . , γp) ∈ E.
(B) For the jump parameters it holds for all G ∈ B(Rp\{0}) that K0(G) ≥ 0 and (K1(G), . . . , Kp(G)) ∈ E.
In addition, supp |Ki| ⊂ E and we haveR
{|z|≤1}|z|
−1K0(dz) < ∞.
(C) The diffusion part c(x) = A0+Pp i=1A ix i is of the form c(x) = v1ζ(x) + p−1 X i=1 vi+1ρ(i)(x), for some v ∈ E. (D) Let α0∈ Rp and α ∈ Rp×pbe given by αj i = a j i−R χi(z)(|z| ∧ 1)−2Kj(dz) for all i = 1, . . . , p, j = 0, . . . , p. Then these drift parameters satisfy
α11I − αQQ= diag(d) for some d ∈ Rp−1+ (5.21) p−1
X
i=1
(diyi2+ (α1,i+1− αi+1,1)yi) ≥ 0 for all y ∈ Rp−1 with |y| = 1 (5.22)
α0− (p − 2)v ∈ E, (5.23)
where v is given as in (C).
Proof. Define Φ(x) = (2x1, −2y) and let b(x), c(x), γ(x), K(x, dz) be given by (4.3). In view of Corollary 5.5, we have to verify that the above parameter re-strictions are equivalent with the following properties:
(i) c(x) ≥ 0 for all x ∈ E, c(0) = 0, c(x0) 6= 0 for some x0∈ E, and c(x)Φ(x) = 0 for all x ∈ Rpsuch that x
1= |y|.
(ii) K(x, dz) ≥ 0, supp K(x, dz) ⊂ E − x for all x ∈ E,R |Φ(x)>χ(z)|K(x, dz) < ∞ for all x ∈ Rp such that x
1= |y|, andR χ(z)K(0, dz) < ∞. (iii) γ(x) ≥ 0 for all x ∈ E.
(iv) Φ(x)>b(x) −R Φ(x)>χ(z)K(x, dz) + c
11(x) − tr (cQQ(x)) ≥ 0 for all x ∈ Rp such that x1= y>y, and b(0) −R χ(z)K(0, dz) ∈ E.
Remark 5.23. Condition (5.22) can be worked out using a Lagrange multiplier, as follows. Let w ∈ Rp−1 be given by w
i= a1,i+1− ai+1,1for i = 1, . . . , p − 1 and write f (y) = p−1 X i=1 (diyi2+ wiyi).
Then f (y) ≥ 0 for all |y| = 1 if and only if the critical values of g(y, λ) := f (y) + λ(y>y − 1)
are non-negative. The critical points (y, λ) of g satisfy
2(di+ λ)yi+ wi= 0, for all i, (5.24)
y>y = 1, (5.25)
so that the critical values of g are equal to g(y, λ) =
p−1 X
i=1
(di+ λ)yi+12wi yi+12w>y − λ = 12w>y − λ.
Therefore, condition (5.22) holds if and only if for all (y, λ) ∈ Rp satisfying (5.24) and (5.25) we have
1 2w
>y − λ ≥ 0. (5.26)
As an example we work this out for the case di = d1 for all i. Without loss of generality we may assume wi6= 0 for all i. Then (5.24) gives y = −12(d1+ λ)−1w. Substituting this expression in (5.25) yields λ = −d1± 12|w| and y = ∓w/|w|. From this one infers that (5.26) holds if and only if d1≥ |w|.
Remark 5.24. There is a close connection between affine processes on the Lorentz cone and matrix-valued affine processes living on the cone of positive semi-definite matrices as treated in [10]. Indeed, let X be a 3-dimensional affine process on the Lorentz cone. Then we can construct a matrix-valued affine process on S3
+ by X1− X2 X3
X3 X1+ X2 !
, see also [26, Example 1].
5.5.1
Admissibility for the diffusion parameters
The equivalence between (C) and (i) (up to an invariant transformation) is a consequence of Proposition 5.29 below, together with its proof. We first prove the following two lemmas, which are similar to Lemma 5.18 and Lemma 5.19.
Lemma 5.25. Consider the linear space L =na : Rp
→ Rp affine |
x1 −y>
a(x) = 0 for all x with x21= y>yo.
(5.27) Then a basis for L is formed by the columns of ζ and η.
Proof. Similar to the proof of Lemma 5.18. Lemma 5.26. Consider the linear space
M =M : Rp→ Sp affine | Mi ∈ L for all x, i
(5.28) with L defined by (5.27). Then a basis for M is given by
B = {ζ, ρ(1), . . . , ρ(p − 1)}.
Proof. Clearly the elements of B are linearly independent elements of M. It re-mains to show that they span M. Let M ∈ M be arbitrary. By Lemma 5.25 there exist matrices A and B such that
M (x) = ζ(x)A + η(x)B. Write T (y) = (Tij(y))1≤i<j<p as in Section 5.4 and let B =
B1
e
B. Then the above display reads
M (x) = x1A11+ y >A Q1 x1A1Q+ y>AQQ yA11+ x1AQ1+ T (y)B1 yA1Q+ x1AQQ+ T (y) eB ! . Symmetry of M (x) yields A1Q= A>Q1, AQQ= A>QQ, yA11+ T (y)B1= A>QQy,
yA1Q+ T (y) eB = (yA1Q+ T (y) eB)>.
Since y>T (y) = 0, the second equation together with the third gives
0 = y>T (y)B1= y>(AQQ− A11I)y,
which implies AQQ−A11I = 0, as AQQ−A11I is symmetric and thus diagonalizable by an orthogonal matrix. Define
N = M − A11ζ − X
i∈Q
Then N ∈ M and N is of the form N (x) = 0 0 0 P k∈QCkyk ! ,
for some matrices Ck∈ S+p−1. By Lemma 5.15 it follows that N = 0. For the proof of Proposition 5.29 we use the following lemmas. Lemma 5.27. For all x ∈ E it holds that ζ(x) ≥ 0.
Proof. Let v = (v1, vQ) ∈ Rp. Then we can write
v>ζ(x)v = x1(v21+ |vQ|2) + 2v1y>vQ. If x ∈ E, then x1≥ |y|, whence Cauchy-Schwarz gives
x1v>v + 2v1y>vQ≥ |y|(v12+ |vQ|2) + 2v1y>vQ≥ |y|(v21+ |vQ|2) − 2|v1||y||vQ| = |y|(|v1| − |vQ|)2≥ 0.
This proves the result.
Lemma 5.28. For all x ∈ E it holds that ζ(x) + ρ(1)(x) ≥ 0.
Proof. Let v = (v1, v2, w) ∈ Rp, write x = (x1, y) and y = (y1, yR). Then we can write v>ρ(1)(x)v = y1(v12+ v 2 2) + 2x1v1v2+ 2v2y>Rw − y1|w|2. Consequently, v>(ζ(x) + ρ(1)(x))v = x1(v21+ v 2 2+ |w| 2) + 2y 1v1v2+ 2v1yR>w + y1(v21+ v 2 2) + 2x1v1v2+ 2v2yR>w − y1|w|2 = (x1+ y1)(v1+ v2)2+ (x1− y1)|w|2+ 2(v1+ v2)yR>w. If x ∈ E, then x1≥ |y|, whence Cauchy-Schwarz gives
v>(ζ(x) + ρ(1)(x))v ≥ (|y| + y1)(v1+ v2)2+ (|y| − y1)|w|2− 2|v1+ v2||yR||w| =|y| + y1)1/2|v1+ v2| − (|y| − y1)1/2|w|
2 ≥ 0. This proves the result.
Proposition 5.29. Let z ∈ Rp. We have k(x, z) := z
1ζ(x)+P p−1
i=1zi+1ρ(i)(x) ≥ 0 for all x ∈ E if and only if z ∈ E.
Proof. The “only if”-part follows from the observation that k(x, z)11 = x>z, together with the self-duality of E. For the “if”-part it suffices to show that k(x, z) ≥ 0 for all x ∈ E and z ∈ Rp with z
1= 1, |zQ| = 1, since ζ(x) ≥ 0 for all x ∈ E by Lemma 5.27. Write c(x) = ζ(x) + ρ(1)(x). Then c(x) ≥ 0 for all x ∈ E by Lemma 5.28. Let |zQ| = 1 be arbitrary and let L ∈ Rp×pbe of the form
L = 1 0 0 O
! ,
with O orthogonal and O1= zQ. Then Lx ∈ E for all x ∈ E, so that L>c(Lx)L ≥ 0 for all x ∈ E. We conclude the proof by showing that L>c(Lx)L = ζ(x) + ρ(x). It holds that (L>c(Lx)L)11= ζ11(x) + p−1 X i=1 zi+1ρ(i)11(Lx) = x1+ zQ>Oy = x1+ y1.
In addition, we have L>c(Lx)L ∈ M, where M is given by (5.28). Therefore, by Lemma 5.26 we can write L>c(Lx)L as a linear combination of ζ(x) and ρ(1)(x), . . . , ρ(p − m)(x). By the above display we necessarily have L>c(Lx)L = ζ(x) + ρ(x) for all x, as we needed to show.
5.5.2
Admissibility for the jump and killing parameters
The facts that 0 ∈ E and that E is a self-dual cone immediately yield the equiv-alence of (A) and (iii) and part of the equivequiv-alence between (B) and (ii). The remaining part follows from Lemma 4.7.
5.5.3
Admissibility for the drift parameters
Assume that (C) holds. We conclude the proof of Theorem 5.22 by showing the equivalence between (D) and (iv). We have
c11(x) − tr (cQQ(x)) = v1x1− v>Qy − (p − 1)v1x1− p X i=2 (3 − p)vixi ! = (2 − p)(v1x1− vQ>y), so the boundary condition (iv) for the drift reads
x1(b1(x) − Z χ1(z)K(x, dz)) − y(bQ(x) − Z χQ(z)K(x, dz)) + c11(x) − tr cQQ(x) = x1(α11x1+ α1Qy + α01) − y(αQ1x1+ αQQy + α0Q) + (2 − p)(v1x1− v>Qy) = y>(α11I − αQQ)y + x1(α1Q− α>Q1)y + (α 0− (p − 2)v)>(x 1, −y) ≥ 0,
for all x1 = |y|, and in addition α0 ∈ E. For the above display it is sufficient and necessary that both
y>(α11I − αQQ)y + (α1Q− α>Q1)y ≥ 0, for all y ∈ Rp−m with |y| = 1, (5.29) and
(α0− (p − 2)v)>
(|y|, y) ≥ 0, for all y ∈ Rp−m. (5.30) Assertion (5.30) holds if and only if α0− (p − 2)v ∈ E. Indeed, sufficiency follows from self-duality of E, while necessity follows by taking y = −(α0
Q− (p − 2)vQ). Note that since v ∈ E, we also have α0∈ E if α0− (p − 2)v ∈ E.
For assertion (5.29) it is necessary that M := α11I − αQQ is positive semi-definite. In that case M is diagonalizable by an orthogonal matrix O, i.e. OM O> = diag(d) with d ∈ Rp−1+ . Applying the invariant transformation x1 7→ x1, y 7→ Oy we obtain the condition
p−1 X
i=1
diyi2+ (α1Q− α>Q1)y ≥ 0, for all y ∈ Rp−1 with |y| = 1,
