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Affine Markov processes on a general state space
Veerman, E.
Publication date
2011
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Citation for published version (APA):
Veerman, E. (2011). Affine Markov processes on a general state space. Uitgeverij BOXPress.
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Chapter
1
Semimartingales
Since a large part of this thesis heavily relies on methods from stochastic calculus for semimartingales, we recall some notation, definitions and results. For a com-plete overview the reader is referred to [33]; see also [35] for a didactic note. This chapter is not only for the sake of making this thesis self-contained. We also adapt and restate known results into a form tailored to our needs in subsequent chap-ters. In addition, some results for semimartingales we want to apply, are scattered through the literature, or are well-known, but not easy to find. These are also included here. The chapter starts with a summary of notation and concepts in the theory of semimartingales, including stochastic integration with respect to a semimartingale and with respect to a random measure, as well as square and angle bracket processes. In Section 1.2 we recall the definition of the characteristics of a semimartingale and state Itˆo’s formula in terms of these, together with some corollaries. Next we consider stochastic exponentials and measure changes in Sec-tion 1.3 and we prove a version of Girsanov’s Theorem in order to derive the form of the “inverse” of a stochastic exponential. The latter is stated in Proposition 1.12, a result which is surprisingly hard to find in the literature. It is applied in Chap-ter 2 to obtain a sufficient condition for the martingale property of a stochastic exponential.
1.1
Definitions
Let (Ω, F , (Ft)t≥0, P) be a filtered probability space with a right-continuous
filtra-tion (Ft)t≥0. An (Ft)-adapted, c`adl`ag stochastic process X with values in R is
called a semimartingale if it admits a decomposition X = X0+ M + A, where M
is a local martingale and A is a c`adl`ag adapted process with locally finite varia-tion, with M0= A0= 0. By convention, every local martingale is assumed to be
c`adl`ag. We call the semimartingale special if A can be chosen to be predictable (i.e. measurable with respect to the σ-algebra that is generated by all left-continuous adapted processes, viewed as mappings on Ω × R+). Every local martingale M can
be decomposed uniquely as M = M0+ Mc+ Md, where Mc is a continuous local
martingale and Md is a purely discontinuous local martingale (i.e. MdN is a local
martingale for all continuous local martingales N ), with Mc
0 = M0d = 0. There
exists a unique continuous local martingale denoted by Xc, such that Xc = Mc
for every decomposition X = X0+ M + A as above. We call Xc the continuous
martingale part of X.
Every semimartingale has a decomposition such that M is a local L2-martingale.
Under this decomposition, the stochastic integral of a predictable locally bounded process H with respect to the semimartingale X, denoted by H · X, is defined by H · X = H · M + H · A, where H · M is the stochastic integral in the sense of [33, I.4.40] and H · A is defined pathwise as the Lebesgue-Stieltjes integral R·
0Ht(ω)dAt(ω).
The quadratic covariation or square bracket process of two semimartingales X and Y is defined by
[X, Y ] = XY − X0Y0− X−· Y − Y−· X.
By [33, I.4.50.a and I.4.55.b], a local martingale M is purely discontinuous if and only if [M, N ] = 0 for all continuous local martingales N . In case [X, Y ] has locally integrable variation (i.e. there exists stopping times Tn↑ ∞ such that the variation
on the stopped interval [0, Tn] has finite expectation), it has a predictable
compen-sator called the predictable quadratic covariation or angle bracket process, which is denoted by hX, Y i and defined as the predictable process of locally bounded variation such that [X, Y ] − hX, Y i is a local martingale. We write hXi = hX, Xi. If M is a local L2-martingale, then hM i is the predictable compensator of M2, i.e.
M2− hM i is a local martingale.
In the sequel we consider p-dimensional semimartingales. We call a p-dimensi-onal process X = (X1, . . . , Xp) a semimartingale in Rpif all its components Xiare
semimartingales in R. For a predictable locally bounded process H in Rpwe define
H ·X :=Pp
i=1Hi·Xi. The square (resp. angle) brackets between a semimartingale
X in Rp
and a semimartingale Y in Rqis defined as the matrix-valued process with
In addition to stochastic integrals with respect to semimartingales, we also need stochastic integration with respect to random measures. Therefore, let us introduce the random measure associated to the jumps of X, which is denoted by µX and defined as µX([0, t] × B) = P
s≤t1{∆Xs∈B\{0}}, for t ≥ 0, B ∈ B(R
p).
There exists a predictable random measure νX, called the compensator of µX,
with the property that W ∗ µX− W ∗ νX is a local martingale for all predictable
W : Ω × R+× Rp → R where |W | ∗ µX is locally integrable. Here, the processes
W ∗ µX and W ∗ νX are defined pathwise as integrals
W ∗ µXt (ω) = Z Z [0,t]×Rp W (ω, s, z)µX(ω; ds, dz) (1.1) W ∗ νtX(ω) = Z Z [0,t]×Rp W (ω, s, z)νX(ω; ds, dz). (1.2)
Let Gloc(µX) denote the set of all predictable W : Ω × R+× Rp → R with the
property that (P
s≤tXes2)1/2 is locally integrable, where we write
e Xs= Z W (s, z)(µX− νX)({s} × dz) =Z Z {s}×Rp W (r, z)µX(dr, dz) − Z Z {s}×Rp W (r, z)νX(dr, dz).
Then for W ∈ Gloc(µX) the stochastic integral W ∗ (µX− νX) is defined as the
unique purely discontinuous local martingale starting at 0 with jumps equal to eXt.
1.2
Itˆ
o’s formula and characteristics
The cornerstone of stochastic calculus is Itˆo’s formula, which takes the following form in the present setting.
Theorem 1.1 (Itˆo’s formula). If f ∈ C2
(Rp
) and X is a semimartingale in Rp,
then f (X) is a semimartingale and
f (X) = f (X0) + ∇f (X−) · X +12tr Z · 0 ∇2f (X t−)dhXcit + (f (Xt−+ z) − f (Xt−) − ∇f (Xt−)>z) ∗ µX,
with all terms well-defined.
Here, the Lebesgue-Stieltjes integral R0·∇2f (X
t−)dhXcitis defined as the
ma-trix M with components
Mij= p X k=1 Z · 0 ∂i∂kf (Xt−)dhXkc, Xjcit.
Our aim is to state Itˆo’s formula in terms of the characteristics of X, which are defined as follows. Fix a truncation function χ : Rp → Rp, i.e. χ is bounded
and χ(z) = z in a neighborhood of 0 (for example χ(z) = z1{|z|≤1}). For every semimartingale X in Rp there exists a triplet (B, C, νX) of characteristics relative
to the truncation function χ. This triplet consists of a predictable process B in Rp of locally bounded variation, a continuous process C in S+p of locally bounded
finite variation, namely C = hXci (i.e. the quadratic variation of the continuous
martingale part of X), and a predictable random measure νX
on R+× Rp, namely
the compensator of the random measure µX associated to the jumps of X. It
holds that χi ∈ Gloc(µX) for all i and the semimartingale X can be decomposed
according to its characteristics as
X = X0+ B + Xc+ χ ∗ (µX− νX) + (z − χ(z)) ∗ µX. (1.3)
Here, χ ∗ (µX− νX) is p-dimensional and should be read componentwise. In case
X is a special semimartingale we have z ∈ Gloc(µX) and (after modifying B) X
admits the decomposition
X = X0+ B + Xc+ z ∗ (µX− νX),
in other words, in (1.3) one can replace χ(z) with z, see [33, II.2.38]. For special semimartingales we therefore often use the improper truncation function χ(z) = z. Throughout and henceforth we restrict ourselves to semimartingales where the characteristics (B, C, νX) are absolutely continuous with respect to the Lebesgue
measure, in the sense that
Bt= Z t 0 bsds, Ct= Z t 0 csds, νX(ω; [0, t], A) = Z t 0 Kω,s(A)ds,
for some adapted processes b in Rp, c in Sp
+ and a predictable transition kernel
Kω,t(dz) from Ω × R+ into Rp\{0} satisfying
Z
(|z|2∧ 1)Kω,t(dz) < ∞, for all ω ∈ Ω, t ∈ R+. (1.4)
We call (b, c, K) the differential characteristics of X. Note that in this case (B, C, νX) is a “good” version of the characteristics in the sense of [33, II.2.9].
We call X a time-homogeneous jump-diffusion if the differential characteristics take the form
bt(ω) = b(Xt(ω)),
ct(ω) = c(Xt(ω)),
for some measurable functions b : Rp
→ Rp
, c : Rp → Sp
+ and a transition kernel
K from Rp
into Rp\{0}. In that case, X can be written as the solution to an
SDE with jumps, see [33, III.2.26]. For continuous diffusions, this comes down to the existence of a Brownian motion W on some probability space together with a process Y that solves the SDE
dYt= b(Yt)dt + c(Yt)1/2dWt,
such that the law of X is equal to the law of Y .
We are now able to state a version of Itˆo’s formula in terms of the differential characteristics of X.
Theorem 1.2. Suppose that X is a semimartingale with differential characteris-tics (b, c, K) and f (X) is a special semimartingale for some f ∈ C2
(Rp). Then it
holds that f (X) = f (X0) + Mc+ Md+ A, with Mc= ∇f (X−) · Xc the continuous
martingale part of f (X), Md= W ∗ (µX− νX) a purely discontinuous local
mar-tingale, where W (t, z) = f (Xt−+ z) − f (Xt−), and A a continuous process given
by At= Z t 0 ∇f (Xs)>bs+12tr (∇2f (Xs)cs) + Z (f (Xs+ z) − f (Xs) − ∇f (Xs)>χ(z))Ks(dz) ds.
Proof. By Itˆo’s formula and [33, II.1.30] we can write
f (X) − f (X0) =
Z ·
0
∇f (Xt)>bt+21tr (∇2f (Xt)c(Xt)) dt + V ∗ µX
+ ∇f (X−) · Xc+ ∇f (X−)>χ ∗ (µX− νX),
where V (t, z) = f (Xt−+ z) − f (Xt−) − ∇f (Xt−)>χ(z). The last two terms on the
right are local martingales, while the first term on the right has locally integrable variation, as it is continuous. Since f (X) is a special semimartingale, it follows from [33, I.4.23] that V ∗ µX has locally integrable variation, whence V ∈ G
loc(µX)
and V ∗ µX = V ∗ (µX− νX) + V ∗ νX by [33, II.1.28]. This yields the assertion,
as W = V + ∇f (X−)>χ.
A sufficient criterion for f (X) being a special semimartingale is stated in the next proposition.
Proposition 1.3. Let f ∈ C2
(Rp) and X be a semimartingale. If |f (z)|1
{|z|>1}∗
Proof. By Theorem 1.1, f (X) is a semimartingale. It is a special semimartingale if and only if Yt:= sups≤t|f (Xs) − f (X0)| is locally integrable in view of [33, I.4.23].
Let Sn ↑ ∞ be a sequence of stopping times such that E|f(z)|1{|z|>1}∗ νSXn< ∞.
Define
Tn= inf{t ≥ 0 : |Xt−| ≥ n or |Xt| ≥ n} ∧ Sn.
Then it holds that
EYTn≤ 2n + E|f(∆XTn)| ≤ 2n + sup |z|≤1 |f (z)| + E(|f(z)|1{|z|>1}∗ µXSn) = 2n + sup |z|≤1|f (z)| + E(|f(z)|1{|z|>1} ∗ νX Sn) < ∞,
which gives the result.
The following theorem gives an important characterization for X being a semi-martingale with differential characteristics (b, c, K). One direction is a consequence of Theorem 1.2. For a full proof we refer to [33, II.2.42].
Theorem 1.4. Let us be given predictable processes b in Rp, c in Sp
+ and a
pre-dictable transition kernel Kω,t(dz) from Ω × R+ into Rp satisfying (1.4). Then a
c`adl`ag adapted process X is a semimartingale and it admits the differential char-acteristics (b, c, K) if and only if
Mtf = f (Xt) − f (X0)− Z t 0 ∇f (Xs)>bs+12tr (∇2f (Xs)cs) + Z (f (Xs+ z) − f (Xs) − ∇f (Xs)>χ(z))Ks(dz) ds (1.5)
is a local martingale for all f ∈ C2 b(R
p), which in turn holds if and only if Mf is
a local martingale for all f of the form f (x) = exp(u>x) with u ∈ iRp.
As a corollary we determine the differential characteristics of a semimartingale under stopping.
Proposition 1.5. Suppose X is a semimartingale with differential characteristics (b, c, K) and T is a stopping time. Then Y := XT is a semimartingale with differential characteristics (b1[0,T ], c1[0,T ], K(dz)1[0,T ]). Moreover, Yc = (Xc)T
and for W ∈ Gloc(µX) it holds that W 1[0,T ]∈ Gloc(µX), W ∈ Gloc(µY) and
(W ∗ (µX− νX))T = W 1
Proof. The first assertion is an immediate consequence of Theorem 1.4. Suppose W ∈ Gloc(µX). Then obviously also W ∈ Gloc(µY), whence the stochastic integral
W ∗(µY−νY) exists. It is the unique purely discontinuous local martingale starting
at 0 with jumps equal to
W (t, ∆Yt)1{∆Yt6=0}= W (t, ∆Xt)1{∆Xt6=0}1[0,T ](t).
By [33, II.1.30.a], we have (W ∗ (µX− νX))T = W 1
[0,T ]∗ (µX− νX), so it is a
purely discontinuous local martingale with the same jumps as W ∗ (µY − νY). It
follows that (W ∗ (µX− νX))T = W ∗ (µY − νY). This yields
Y = Y0+ BT + (Xc)T + χ(z) ∗ (µY − νY) + (z − χ(z)) ∗ µY,
which implies that (Xc)T = Yc by the uniqueness of the continuous martingale
part.
In the next proposition, inspired by [37, Problem 5.3.15], we establish finite second moments for a semimartingale under a growth condition. We write kZkt=
sups≤t|Zs|, where Z is a stochastic process.
Proposition 1.6. Let X be a special semimartingale with differential character-istics (b, c, K) relative to the (improper) truncation function χ(z) = z. Assume E|X0|2< ∞ and
|bt|2+ tr ct+
Z |z|2K
t(dz) ≤ C(1 + kXk2t), for some C > 0, all t ≥ 0, P-a.s.
(1.6)
Then for all t ≥ 0
EkXk2t ≤ 4(E|X0|2+ Ct(8 + t)) exp(4Ct(8 + t))
holds. In addition, Xc and z ∗ (µX− νX) are proper martingales.
Proof. Define stopping times Tn= inf{t ≥ 0 : |Xt| ≥ n or |Xt−| ≥ n}. By
Propo-sition 1.5, Y := XTn is a special semimartingale with differential characteristics
(b1[0,Tn], c1[0,Tn], K1[0,Tn]) relative to χ. It holds that
1 4kY k 2 t ≤ |X0|2+ k Z · 0 bs1[0,Tn](s)dsk 2 t+ kY c k2t+ kz ∗ (µ Y − νY)k2t. The Cauchy-Schwarz inequality gives
sup u≤t Z u 0 bs1[0,Tn](s)ds 2 ≤ t Z t 0 |bs|21[0,Tn](s)ds ≤ Ct Z t 0 (1 + kY k2s)ds,
while Doob’s inequality [33, I.1.43] yields EkYck2t ≤ 4E|Y c t| 2 = 4Etr hYci∞= 4E Z t 0 tr cs1[0,Tn]ds ≤ 4C Z t 0 (1 + EkY k 2 s)ds and Ekz ∗ (µY − νY)k2t ≤ 4E|z ∗ (µ Y − νY) t|2= 4Etr hz ∗ (µY − νY)it= 4E|z|2∗ νYt = E Z t 0 Z |z|21[0,Tn](s)Ks(dz)ds ≤ 4C Z t 0 (1 + EkY k2s)ds It follows that EkY k2t ≤ 4|X0|2+ 4Ct(8 + t) + 4C(8 + t) Z t 0 EkY k2sds. Since
EkXTnk2t ≤ E|X0|2+ n2+ E|∆Xt∧Tn|
2 ≤ E|X0|2+ n2+ E(|z|2∗ µXt∧Tn) = E|X0|2+ n2+ E(|z|2∗ νt∧TX n) ≤ E|X0|2+ n2+ CE Z t∧Tn 0 (1 + kXk2s)ds < ∞,
the integral form of the Gronwall-Bellman inequality yields
EkXTnk2t ≤ 4(E|X0|2+ Ct(8 + t)) exp(4Ct(8 + t)).
Let n → ∞, then the left-hand side converges by the Monotone Convergence Theorem to EkXk2
t, which is bounded by the right-hand side. This yields the first
assertion of the lemma. The second assertion is an immediate consequence in view of [33, I.4.50 and II.1.33.a], since both
E[Xic]t= EhXicit= E Z t 0 cii,sds and Ehzi∗ (µX− νX)it= E(z2i ∗ ν X t ) = E Z t 0 Z zi2Ks(dz)ds
are finite due to the growth-condition (1.6) and the derived moment inequality for kXk2
1.3
Girsanov’s Theorem
We finish this chapter by discussing locally absolutely continuous measure changes. Recall that a probability measure Q is said to be locally absolutely continuous with respect to P with density process Z if Q|Ft is absolutely continuous with respect
to P|Ft for all t ≥ 0 and Z is the density process defined as
Zt= dQ|Ft
dP|Ft
,
which is a P-martingale. We consider the case that the density process is given by a stochastic exponential . The stochastic exponential Z = E (X) of a semimartingale X is defined as the unique solution to the stochastic differential equation Z = 1 + Z−· X, which exists and is equal to
E(X)t= exp(Xt−12hXcit)
Y
s≤t
(1 + ∆Xs) exp(−∆Xs).
In case ∆Xt> −1, the above expression can be simplified to
E(X)t= exp(Xt−12hXcit+ (log(1 + z) − z) ∗ µXt ). (1.7)
We need the following version of the “classical” Girsanov’s Theorem.
Theorem 1.7 (Girsanov). Suppose that Q is locally absolutely continuous with respect to P with density process Z. If M is a P-local martingale with bounded jumps, then
M − 1 Z−
· hM, Zi
is well-defined Q-a.s. and is a Q-local martingale. Consequently, if Z is of the form Z = E (Y ) for some P-local martingale Y , then M − hM, Y i is a Q-local martingale.
Proof. The first assertion follows from [33, III.3.11 and III.3.14]. For the second assertion, [33, III.3.14] yields that both [M, Z] and [M, Y ] have locally integrable variation. Therefore, their compensators hM, Zi respectively hM, Y i exist. It holds that [M, Z] = [M, Z−· Y ] = Z−· [M, Y ] by [33, I.4.54]. The compensator of the
latter process equals Z−· hM, Y i by [33, I.3.18]. Plugging in hM, Zi = Z−· hM, Y i
in the above display gives the result.
In the following, L2
loc(Xc) denotes the class of predictable processes H with
stochastic integral H·Xcis defined as in [33, III.4.5]. In order to clarify under which
measure the stochastic integral is taken, we also write H ·PXc and L2
loc(Xc, P), as
well as W ∗P(µX− νX) and G
loc(µX, P). Likewise we write hXiP, [X]P, etc. We
are interested in how the differential characteristics of a semimartingale translate under a change of measure where the density process Z takes the typical form
Z = E (H · Xc+ W ∗ (µX− νX)), (1.8)
for some H ∈ L2 loc(X
c), W ∈ G
loc(µX). For this, we have to calculate the angle
brackets, for which we use the following lemma.
Lemma 1.8. Let X be a semimartingale with differential characteristics (b, c, K). Suppose Y = H · Xc+ W ∗ (µX− νX), eY = eH · Xc+ fW ∗ (µX− νX) for some
H, eH ∈ L2 loc(X
c) and some bounded W, fW ∈ G
loc(µX). Then it holds that
hY, eY it=
Z t
0
Hs>csHesds + W fW ∗ νtX.
Proof. By [33, I.4.52 and III.4.5.c] it holds that
[Y, eY ]t= hYc, eYcit+ X s≤t ∆Ys∆ eYs = hH · Xc, eH · Xcit+ X s≤t W (s, ∆Xs)fW (s, ∆Xs)1{∆Xs6=0} = Z t 0 Hs>csHesds + W fW ∗ µXt .
The predictable compensator hY, eY it exists by [33, III.3.14] and is equal to the
processRt
0H >
s csHesds + W fW ∗ νtX, since νX is the compensator of µX.
The next proposition describes the behavior of the differential characteristics under an absolutely continuous change of measure. The same proposition is also stated in [35, Proposition 4], but without proof. However, a proof can be found inside the proof of [6, Theorem 2.4]. For completeness we include it here.
Proposition 1.9. Let X be a semimartingale with differential characteristics (b, c, K). Suppose that Q is locally absolutely continuous with respect to P with density process
for some H ∈ L2 loc(X
c
, P), W ∈ Gloc(µX, P) with W ≥ −1. Then X is a
semi-martingale under Q with differential characteristics (eb,ec, eK) given by ebt= bt+ ctHt+ Z W (t, z)χ(z)Kt(dz) e ct= ct e Kt(dz) = (1 + W (t, z))Kt(dz). (1.9)
Proof. First we observe that P-a.s. (whence Q-a.s.) it holds that ebtis well-defined,
Rt
0ebsds exists and eK satisfies (1.4). Indeed, W ∈ Gloc(µ X
, P) implies that (W21{|W |≤1}+ |W |1{|W |>1}) ∗ νtX
is locally P-integrable by [33, II.1.33.c]. In particular it is finite for all t ≥ 0, P-a.s. Since also χ ∈ Gloc(µX, P), one can derive the desired properties with the aid of
Cauchy-Schwarz, which is left to the reader.
Now let Mf be given by (1.5) and similarly let fMf be given by the same
expression with b and K replaced by eb and eK. Note that fMf is related to Mf by
f Mtf = Mtf− Z t 0 ∇f (Xs)>csHs+ Z (f (Xs+ z) − f (Xs))W (s, z)Ks(dz) ds.
In view of Theorem 1.4, we have to show that fMf
is a Q-local martingale for all f ∈ C2
b(Rp). Since f (X) is bounded, it is a special semimartingale, whence by
Theorem 1.2 we can write
Mf = ∇f (X−) · Xc+ (f (Xt−+ z) − f (Xt−)) ∗ (µX− νX).
It holds that Mf
is a P-local martingale by Theorem 1.4. Moreover, |∆Mtf| =
|∆f (Xt)| ≤ kf k∞, so Theorem 1.7 yields that
Mf− hMf, H · Xc+ W ∗ (µX− νX)i
is a Q-local martingale. From Lemma 1.8 it follows that the angle brackets in the above display are equal to
Z t 0 ∇f (Xs)>csHs+ Z (f (Xs+ z) − f (Xs))W (s, z)Ks(dz) ds,
which yields the result.
Proposition 1.10. Let X be a semimartingale with differential characteristics (b, c, K). Suppose H ∈ L2loc(Xc, P) and W ∈ Gloc(µX, P) with W > −1 are such
that
E exp(12
Z ∞
0
Then Z = E (H · Xc+ W ∗ (µX− νX)) is a uniformly integrable martingale.
Proof. This follows from [39, Theorem IV.3].
We now consider locally equivalent measure changes. In Proposition 1.12 below we show that if the density process Z is a strictly positive stochastic exponential of the form (1.8), then Z−1 is also a stochastic exponential of the form (1.8), and we determine its explicit expression. This result extends a result in [37], which is limited to continuous semimartingales. To handle the jumps, we need the following lemma.
Lemma 1.11. Consider the situation of Proposition 1.9 and suppose Z > 0. If V ∈ Gloc(µX, P) and V is bounded, then V ∈ Gloc(µX, Q) and
V ∗Q(µX−
e
νX) = V ∗P(µX− νX) − V W ∗ νX, (1.10)
where the stochastic integration on the left- resp. right-hand side is taken under Q resp. P, andνe
X(dt, dz) = eK
t(dz)dt denotes the compensator of µX under Q.
Proof. Since V ∈ Gloc(µX, P) and V is bounded, it holds that V2∗ νX is locally
integrable, by [33, II.1.33.c]. This yields that V2∗
e
νX = V2(W + 1) ∗ νX is locally
integrable. Indeed, we have
V2|W | ∗ νX ≤ V2∗ νX+ kV k2
∞|W |1{|W |>1}∗ νX.
The second term on the right is locally integrable by [33, II.1.33.c], since W ∈ Gloc(µX, P). To show (1.10), it suffices to show that the right-hand side is a
purely discontinuous Q-local martingale, as it has the same jumps as the left-hand side. It is a Q-local martingale by Theorem 1.7 and Lemma 1.8. To prove that Y := V ∗P(µX− νX) − V W ∗ νX is purely discontinuous, we have to show that
[Y, N ]Q = 0 for all continuous Q-local martingales N . By [33, III.3.13], N is a continuous semimartingale under P and the square brackets are the same under P and Q. Under P, the continuous martingale part Yc of Y is 0, as V ∗P(µX− νX)
is purely discontinuous and V W ∗ νX is of locally bounded variation. Since in
addition ∆Nt= 0, it follows from [33, I.4.52] that
[Y, N ]P = hYc, N iP = 0,
Proposition 1.12. Consider the situation of Proposition 1.9. Suppose W > −1. Then it holds that −H ∈ L2
loc( eXc, Q) and −W/(W + 1) ∈ Gloc(µX, Q). Moreover,
Z > 0 and Z−1= E (−H ·Q e Xc− W/(W + 1) ∗Q(µX− e νX)), (1.11) where eXc
denotes the continuous martingale part of X under Q andeν
X(dt, dz) =
e
Kt(dz)dt the compensator of µX under Q.
Proof. Write V = −W/(W + 1). Then V > −1 and one verifies that
(1 −√1 + V )2∗νeX= (1 −√1 + W )2∗ νX.
Note that the latter process is continuous (as νX(dt, dz) = Kt(dz)dt). Therefore,
it is locally integrable under Q if and only if it is locally integrable under P, so W ∈ Gloc(µX, P) if and only if V ∈ Gloc(µX, Q) by [33, II.1.33.d]. Likewise we
have −H ∈ L2
loc( eXc, P). That Z > 0 holds, follows from [33, I.4.61.c], since we
have that ∆(W ∗ (µX− νX))
t= W (t, ∆Xt)1{∆Xt6=0} > −1.
It remains to show (1.11). By using (1.7) we infer that the right-hand side equals exp(−H ·Q e Xc− W/(W + 1) ∗Q(µX− e νX) −1 2 Z · 0 Ht>ctHtdt + (W − log(1 + W )) ∗ µX− W2/(W + 1) ∗ µX),
and the left-hand side equals
exp(−H ·PXc− W ∗P(µX− νX) +1 2
Z ·
0
Ht>ctHtdt + (W − log(1 + W )) ∗ µX).
Therefore, it suffices to show
H ·Q e Xc= H ·PXc− Z · 0 Ht>ctHtdt, (1.12) W/(W + 1) ∗Q(µX− e νX) = W ∗P(µX− νX) − W2/(W + 1) ∗ µX. (1.13)
We start with the latter. It holds that |W |1{|W |>1}∗ νX is locally integrable by
[33, II.1.33.c], whence [33, II.1.28] enables us to split the integrals to obtain
W 1{|W |>1}∗P(µX− νX) − W2/(W + 1)1{|W |>1}∗ µX
= W/(W + 1)1{|W |>1}∗ µX− W/(W + 1)1{|W |>1}∗eν
X
= W/(W + 1)1{|W |>1}∗Q(µX−eν
We now show that W 1{|W |≤1}∗P(µX− νX) − W 2 W + 11{|W |≤1}∗ µ X= W W + 11{|W |≤1}∗ Q(µX− e νX).
Applying Lemma 1.11 with V = W 1{|W |≤1} yields
W 1{|W |≤1}∗Q(µX−eν X) = W 1 {|W |≤1}∗P(µX− νX) − W2 W + 11{|W |≤1}∗eν X. (1.14)
Since W21{|W |≤1}∗ νX is locally integrable by [33, II.1.33.c], it follows from [33,
II.1.28] that W2/(W + 1)1 {|W |≤1}∈ Gloc(µX, Q) and W2 W + 11{|W |≤1}∗ Q(µX− e νX) = W 2 W + 11{|W |≤1}∗ µ X− W 2 W + 11{|W |≤1}∗eν X.
Subtracting this from (1.14) gives the result. Thus we have proved (1.12). To show (1.13) we note that Theorem 1.7 and Lemma 1.8 yield that
Xc− hXc, M i P = X c− Z · 0 ctHtdt
is a continuous Q-local martingale. Moreover, from Lemma 1.11 it follows that χ ∗Q(µX−
e
νX) = χ ∗P(µX− νX) − W χ ∗ νX,
whence X can be decomposed as
X = X0+ Z · 0 e btdt + Xc− Z · 0 ctHtdt + χ ∗Q(µX−eν X) + (z − χ(z)) ∗ µX.
By the uniqueness of the decomposition of a local martingale in its continuous and purely discontinuous part, it follows that eXc = Xc−R·
0ctHtdt. Now by [33,
III.4.5.b], H ·Q
e
Xc is characterized as the unique continuous Q-local martingale, null at 0, such that
hH ·Q
e
Xc, Y iQ= H · h eXc, Y iQ,
for all local L2(P)-martingales Y . From [33, I.4.52 and III.3.13] one infers that h eXc, Y iQ= [ eXc, Y ]Q= [ eXc, Y ]P= [Xc, Y ]P. Likewise we have hH ·PXc − Z · 0 Hs>csHsds, Y iQ= [H ·PX c − Z · 0 Hs>csHsds, Y ]Q = [H ·PXc− Z · 0 Hs>csHsds, Y ]P = [H ·PXc, Y ] P = H · [Xc, Y ]P.
Theorem 1.7 and Lemma 1.8 yield that H ·PXc t− Z t 0 Hs>csHsds
