Description of the driving process - Term structure modelling: L´evy setting

2.2 Term structure modelling: L´evy setting

2.2.1 Description of the driving process

Let T^∗ > 0 be a fixed time horizon. All processes considered in what follows are defined on a common probability space (Ω, FT^∗, P), endowed with a canonical filtration (Ft)0≤t≤T^∗, associated with a d–dimensional time-inhomogeneous L´evy process (L_t)_0≤t≤T∗. More specif-ically, L = (L₁, . . . , L_d) is a process with independent increments and absolutely continuous characteristics (shortly PIIAC), its law is characterised by the following characteristic func-tions:

E£ exp¡

iu^>Lt

¢¤ = exp



 Zt



iu^>ebs− 1 2u^>Csu

+ Z

R^d

¡exp¡ iu^>x¢

− 1 − iu^>h(x)¢

λ_s(dx)



 ds



 .

(2.2.54)

Here eb_t, u ∈ R^d, and C_t is a symmetric positive semidefinite d × d matrix such that

T^∗

¯¯

¯ebt

¯¯

¯ dt < ∞,

T^∗

kC_tk dt < ∞, (2.2.55)

where | · | denotes the Euclidean³ vector norm and and k · k is a spectral⁴ norm on the set of d × d matrices. More specifically, we have kC_tk = sup

|u|=1

|C_tu|, and kC_tk is equal to the square root from the maximal eigenvalue of C_t^>C_t.

The measure λ_t with t ∈ [0, T^∗] is a measure on R^d, which satisfies λ_t({0}) = 0 for all t ≥ 0 and integrates (|x|²∧ 1). Finally, h : R^d 7→ R^d is a truncation function, i.e. a bounded function with compact support that satisfies h(x) = x in the neighborhood of zero. First, we take h(x) = x1{|x|≤1}(x), which is a canonical one.

3Eucledian norm is also known as a L²-norm.

4Spectral norm is a matrix norm induced by the L²-norm.

According to Lemma 1 and 2 in Eberlein and Kluge (2004) L_t is an additive process in law. Therefore with respect to Theorem 9.1 in Sato (1999) its distribution is infinitely divisible with L´evy–Khintchine triplet ¡

B^L, C^L, ν^L¢

. Note that C^L and ν^L are independent of the truncation function h, while the value of B^L(h) varies with h according to the following equality:

B_t^L(h) − B_t^L(h⁰) =

According to Theorem 2.11.5 in Sato (1999), every additive process in law has a modifica-tion, which is c`adl`ag. We will always work with this modification of L.

The process L_thas the following canonical representation (see Theorem II.2.34 in Jacod and Shiryaev (1987)): is its deterministic compensator, W is a d–dimensional standard Brownian motion and c_s is a measurable version of the square-root of C_s.

We put an additional integrability assumption on measures λ_t, t ∈ [0, T^∗].

Assumption 2.2.1 There are constants M > 1 and ² > 0 such that

T^∗

Under this assumption L is a special semimartingale (see Lemma 7 in Eberlein and Kluge (2004)). Hence, ^T

R∗

|x|>1

|x|λs(dx) ds < ∞, and representation (2.2.54) holds for h(x) = x. More specifically,

E£

and the drift term in the triplet of the semimartingale characteristics is now given by

B_t^L:=

b_sds = Zt



eb_s+ Z

R^d

x1{|x|>1}(x)λ_s(dx)



 ds .

Henceforth, we will always work with the truncation function h(x) = x, therefore we omit it in the notation. Thus, canonical representation of L simplifies to

L_t = Zt

b_sds + Zt

c_sdW_s+ Zt

R^d

x(µ^L− ν^L)(ds, dx). (2.2.58)

Note that under (2.2.55) and (2.2.56) we automatically have ^T R∗

|bs| ds < ∞. Furthermore, if d = 1, i.e L is a one-dimensional PIIAC process, condition (2.2.56) guarantees that it is an exponentially special semimartingale, since ^T

R∗

x>1

exp (x) λ_s(dx) ds < ∞ (see Lemma 2.13 in Kallsen and Shiryaev (2002)).

Lemma 2.2.2 Assumption 2.2.1 holds if and only if there are positive constants M and ² such that E£

exp¡

u^>L_t¢¤

< ∞ for all t ∈ [0, T^∗] and for all u with |u| ≤ (1 + ²)M.

Proof. See the proof of Lemma 6 in Eberlein and Kluge (2004). ¤ Proposition 2.2.3 Let (Lt)_0≤t≤T∗be an R–valued PIIAC process given by equation (2.2.58).

1. If L is a local martingale, then it is a martingale.

2. If exp (L) is a local martingale, then it is a uniformly integrable martingale.

Proof. Since (L_t)_0≤t≤T∗ is an additive process on R^d, the distribution of L_t, which we will denote by µ_t, is infinitely divisible for every t. According to Corollary 11.6 in Sato (1999) for every infinitely divisible distribution µ on R^d, there is a L´evy process Z such that the distribution of Z₁ is equal to µ. Moreover, L´evy process Z is unique up to identity in law.

Consequently, for a fixed t ∈ [0, T^∗] we have L_t = Z^d ₁, where the L´evy–Khintchine triplet (b^z, c^z, ν^z) of Z₁ is given by

B^z = Zt

b_sds, C^z = Zt

C_sds, ν^z(dx) = Zt

λ_s(dx) ds .

Since L_tis a local martingale, it is also a special semimartingale and with respect to II.8.24 in Jacod and Shiryaev (2003) we obtain

|x|>1

|x|λ_s(dx) ds < ∞ or equivalently Z

|x|>1

|x|ν^z(dx) < ∞.

Hence, Z₁ is a special semimartingale too, and by Corollary 25.8 in Sato (1999) (see also Example 25.12), its mean is finite. Evidently, for each t ∈ [0, T^∗] holds

E [|L_t|] = E [|Z₁|] < ∞.

According to results given in Chapter II.1 of Shiryaev (1999), a local martingale (L_t)_0≤t≤T∗

that satisfies

· sup

s≤t

|L_s|

< ∞ for t ≥ 0

is a martingale. With respect to Remark 25.19 in Sato (1999) and Theorem VII.5.1 in Doob (1953), for L^∗_t := sup_s≤t|L_t| holds

E [(L^∗_t)^α] ≤ 8E [|Lt|^α] < ∞ for t ∈ [0, T^∗] and α ≥ 1, and the first statement of the proposition follows.

It is necessary to note that there is another possibility to prove the first statement:

the idea of the proof comes from J. Jacod and is also based on the special semimartingale property of the process L (see Sidib´e (1979)).

To prove the second statement we will use the uniform integrability conditions for exponential local martingales introduced in Kallsen and Shiryaev (2002). Evidently, if exp (L) is a local martingale, it is also a special semimartingale and the PIIAC process (L_t)_0≤t≤T∗ is exponentially special by definition. With respect to Lemma 2.13 in Kallsen and Shiryaev (2002) this is equivalent to

Z∞

−∞

(exp (x) − 1 − x) λ_s(dx) ds < ∞. (2.2.59)

Furthermore, from the martingality of exp (L) it follows that the exponential compensator of L, which is given by its Laplace cumulant process eK_t^L, is equal to zero. More specifically,

Ke_t^L= Zt

b_sds +1 2

C_sds + Zt

Z∞

−∞

(exp (x) − 1 − x) λ_s(dx) ds = 0.

According to Corollary 3.10 in Kallsen and Shiryaev (2002) for exp (L) to be a uniformly integrable martingale it suffices that the following condition holds:

limδ→0 sup

t∈[0,T^∗]

δ ln µ

· exp

µ1 δ h

(1 − δ) eK_t^L− eK_t^L(1 − δ)i¶¸¶

= 0, (2.2.60)

where δ ∈ (0, 1) and

Ke_t^L(1 − δ) = (1 − δ) Zt

b_sds +(1 − δ)² 2

C_sds

+ Zt

Z∞

−∞

(exp ((1 − δ)x) − 1 − (1 − δ)x) λ_s(dx) ds .

Without loss of generality we can assume that δ ∈ [0, 1]. Since Lt has deterministic characteristics, the functions eK_t^L and eK_t^L(1 − δ) are deterministic, and condition (2.2.60) simplifies to

limδ→0 sup

t∈[0,T^∗]

− eK_t^L(1 − δ) i

= 0.

Observe that for each δ ∈ [0, 1] holds

| exp ((1 − δ)x) − 1 − (1 − δ)x| ≤ | exp (x) − 1 − x|, and in view of (2.2.59) we get

Z∞

−∞

(exp ((1 − δ)x) − 1 − (1 − δ)x) λs(dx) ds < ∞

for each δ ∈ [0, 1]. Hence, eK_t^L(1 − δ) is continuous in δ and

limδ→0

Ke_t^L(1 − δ) = lim

δ→0



(1 − δ) Zt

b_sds +(1 − δ)² 2

C_sds Zt

Z∞

−∞

(exp ((1 − δ)x) − 1 − (1 − δ)x) λ_s(dx) ds





= Zt

b_sds +1 2

C_sds

+ Zt

Z∞

−∞

δ→0lim[exp ((1 − δ)x) − 1 − (1 − δ)x] λs(dx) ds = eK_t^L= 0.

Using the same arguments, we obtain lim

δ→1

Ke_t^L(1 − δ) = 0.

Thus, eK_t^L(1−δ) is a continuous function on a compact set [0, T^∗]×[0, 1], and with respect to the Heine-Cantor Theorem, it is uniformly continuous on [0, T^∗] × [0, 1]. In particular, for each ε > 0 there exists 0 < γ < 1 such that for all δ < γ and for all t ∈ [0, T^∗] holds

¯¯

¯ eK_t^L(1 − δ) − eK_t^L(1)

¯¯

¯ < ε. Since eK_t^L(1) = 0, we obtain sup

t∈[0,T^∗]

Ke_t^L(1 − δ) ≤ ε. Hence, limδ→0 sup

t∈[0,T^∗]

Ke_t^L(1 − δ) = 0, which completes the proof. ¤

Proposition 2.2.4 Let L be the R^d–valued PIIAC process given by equation (2.2.58) and f : [0, T^∗] → R^d be a continuous deterministic function of time such that |f (t)| < M for all t ∈ [0, T^∗], where M is the constant introduced in Assumption 2.2.1. Then

1. The function f is integrable with respect to L, i.e f ∈ L(L).

2. The process f · L is a special semimartingale.

Proof. According to Theorem III.6.30 in Jacod and Shiryaev (2003), f is integrable with respect to L if and only if the following condition is satisfied:

f (s)^>b_sds + Zt

f (s)^>C_sf (s) ds

+ Zt

R^d

f (s)^>x1{|x|≤1, |f (s)^>x|≤1} − f (s)^>h(x) i

λ_s(dx) ds < ∞

(2.2.61)

for all t ∈ [0, T^∗]. First, observe that ¯

¯f(s)^>b_s¯

¯ < M|b_s| and f (s)^>C_sf (s) < M²kC_sk for all s ∈ [0, T^∗], therefore, in view of (2.2.55), the first two integrals on the left-hand side of inequality (2.2.61) are finite.

Let us now consider the last integral on the left-hand side of (2.2.61). Since L is a special semimartingale, we have h(x) = x, and one can also easily check that

f (s)^>x = f (s)^>x1{|x|≤1}+ f (s)^>x1{|x|>1}

= f (s)^>x1{^|x|≤1,|^{f (s)}^>^x|^≤1} + f (s)^>x1{^|x|≤1,|^{f (s)}^>^x|^>1} + f (s)^>x1{|x|>1}. Thus, the last integral on the left hand side of (2.2.61) can be represented as

R^d

f (s)^>x1{|x|≤1, |f (s)^>x|≤1} − f (s)^>h(x) i

λs(dx) ds =

= Zt

R^d

f (s)^>x1{|x|≤1, |f (s)^>x|≤1} − f (s)^>x i

λs(dx) ds

= − Zt

R^d

f (s)^>x1{|x|>1}+ f (s)^>x1{|x|≤1, |f (s)^>x|>1} i

λs(dx) ds .

Due to the fact that |f (s)| < M for all s ∈ [0, T^∗] Zt

|x|>1

f (s)^>xλ_s(dx) ds < M Zt

|x|>1

|x|λ_s(dx) ds < ∞

holds for all t ∈ [0, T^∗]. Note that, since L is a semimartingale, we have it follows that |x| > 1/M > 0. Altogether, we obtain

Thus, all the integrals on the left hand side of (2.2.61) are well defined and the first statement of the lemma follows.

Observe that the process µ _t

is a special semimartingale if and only if the following condition holds:

T^∗

is a PIIAC process, it is a semimartingale, and we have

T^∗

Hence, to prove the second statement of the Lemma, it remains to show that

T^∗

Note that

T^∗

|f (s)^>x|>1

¯¯f(s)^>x¯

¯ λ_s(dx) ds < M

T^∗

|f (s)^>x|>1

|x| λ_s(dx) ds

≤ M







T^∗

|f (s)>x|>1,

|x|>1

|x| λ_s(dx) ds +

T^∗

|f (s)>x|>1,

|x|≤1

|x| λ_s(dx) ds





 . (2.2.63)

Since L is a special semimartingale, the first integral on the right-hand side of (2.2.63) is finite, and one can easily check that the second integral is finite too, using the same argument as in the proof of statement 1 (see equation (2.2.62)). ¤ Proposition 2.2.5 Let L and f be defined as in Proposition 2.2.4 above. Suppose that Assumption 2.2.1 holds. Then

1. The process f · L is exponentially special.

2. The exponential compensator of f · L is given by

Ke^L(f ) = Zt

f (s)^>b_sds +1 2

f (s)^>C_sf (s) ds

+ Zt

R^d

¡exp¡

f (s)^>x¢

− 1 − f (s)^>x¢

λ_s(dx) ds,

where eK^L(f ) stands for the Laplace cumulant process of L in f .

To prove the first statement of the proposition we need two auxiliary results.

x ∈ R^d\ {0} : ∠(x, uⁱ) < δ, for fixed uⁱ ∈ R^d\ {0}ª

where ∠(a, b) denotes an angle between the vectors a and b, and i = 1, . . . , n such that [n

i=1

K_i = R^d\ {0}.

Proof. Let us denote by S¹ a sphere with radius 1, i.e S¹ := {x ∈ R^d: |x| = 1}. Obviously, a family of open cones covers R^d\ {0} if and only if its intersection with S¹ covers the whole S¹.

x ∈ R^d : ∠(x, u) < δ,ª

indexed by u ∈ R^d \ {0}. Evidently, S¹ ⊂ [

u∈R^d\{0}

K(u). Since S¹ is compact, there exists a

finite subfamily of open cones {K_i with i = 1, . . . , n}, which satisfies S¹ ⊂ [n i=1

K_i and the

statement of the lemma follows. ¤

Lemma 2.2.7 Let L be an R^d–valued PIIAC process satisfying condition (2.2.56). Then Zt

|x|>1

exp (M|x|) λ_s(dx) ds < ∞,

where M is a constant introduced in Assumption 2.2.1.

Proof. Let us fix a vector u ∈ R^d such that |u| = (1 + ²)M with ² > 0. Consider an open cone K_u given by

K_u :=

x ∈ R^d: ∠(x, u) < arccos µ 1

1 + ²

¶¾

for a fixed u ∈ R^d\ {0}. By definition cos (∠(x, u)) = x^>u/(|x||u|). Observe that x^>u ≥ |x||u| inf

x∈Ku

cos (∠(x, u)) ≥ |x|M(1 + ²)

(1 + ²) ≥ |x|M. (2.2.64) Hence, for each x ∈ K_u holds x^>u ≥ |x|M.

According to Lemma 2.2.6 for δ = arccos (1/(1 + ²)) > 0 there exists a finite family of open cones

K_i :=©

x ∈ R^d\ {0} : ∠(x, uⁱ) < δ, for a fixed uⁱ ∈ R^d\ {0}ª ,

where i = 1, . . . , n, such that [n i=1

K_i = R^d \ {0}. Without loss of generality we can set

|ui| = (1 + ²)M for each i = 1, . . . , n. Then making use of (2.2.64), we obtain Zt

|x|>1

exp (M|x|) λs(dx) ds = Xn

i=1

|x|>1, x∈Ki

exp (M|x|) λs(dx) ds

≤ Xn

i=1

|x|>1, x∈Ki

exp¡ x^>ui

¢λs(dx) ds < ∞

and the lemma follows. ¤

Now we are in position to prove Proposition 2.2.5.

Proof. With Lemma 2.13 in Kallsen and Shiryaev (2002) the process Zt

f (s)^>dL_s is exponentially special if and only if

Let us consider the first integral on the right hand side of equation (2.2.65). Note that, since f is bounded in absolute value by M for all t ∈ [0, T^∗], we have f (t)^>x ≤ |f (t)||x| < M |x|

for all t ∈ [0, T^∗]. Furthermore, sinceR^t

f (s)^>dL_sis a semimartingale, with II.2.13 in Jacod and Shiryaev (1987) holds

≤ e^M Zt

R^d

³¯¯f(s)^>x¯

¯²∧ 1´

λ_s(dx) ds < ∞.

Now we return to equation (2.2.65) and consider the second integral on its right hand side.

Using Lemma 2.2.7, we obtain Zt

|x|>1, f (s)>x>1

exp¡

f (s)^>x¢

λs(dx) ds <

|x|>1, f (s)>x>1

exp (M|x|) λs(dx) ds

≤ Zt

|x|>1

exp (M|x|) λs(dx) ds < ∞.

Thus,

R^d

1{^{f (s)}^>^x>1}(x) exp

¡f (s)^>x¢

λ_s(dx) ds < ∞

and we are done with the proof of the first statement.

According to Proposition 2.2.4, f is integrable with respect to L, and we have just proved that

µR_t

f (s)^>dL_s

0≤t≤T^∗

is exponentially special. The second statement follows now from Theorems 2.18 and 2.19 in Kallsen and Shiryaev (2002). ¤

In document Time-inhomogeneous Lévy Processes in Cross-Currency Market Models (Page 68-78)