LetX = (Xi)i≤dbe ad-dimensional semimartingale with characteristics (B, C, ν)relative to a given truncation function h, with continuous martingale part Xc relative to P, and with Cij = cij ãA, A increasing and predictable. Then we have the following Girsanov theorem for semimartingales (Jacod and Shiryaev (1987), III.3.24):
Theorem 6.2. Assume thatQlocP, and letXbe as above. There exist aP ⊗ B+1-measurable nonneg- ative functionY and a predictable processβ = (βi)i≤dsatisfying
|h(x)(Y −1)| ∗νt<∞Q-a.s. fort∈IR+
3The canonical decomposition of a special semimartingaleXis the unique decompositionX =X0+M +A, with the initial valueX0, a local martingaleMand a predictable process of bounded variationA, where bothM andAstart in0, that is,M0= 0andA0= 0. By Jacod and Shiryaev (1987), Corollary II.2.38, we haveM =Xc+x∗(àX−νX), whereXcis the continuous local martingale part ofX,àXis the random measure of jumps ofXandνXis the compensator ofàX. The stochastic integral with respect to the compensated random measure exists becauseX is a special semimartingale and so by Jacod and Shiryaev (1987), Proposition 2.29a,(|x|2∧ |x|)∗νXhas locally integrable variation.
X
j≤d
cijβjãAt<∞and X
j,k≤d
βjcjkβk
ãAt <∞Q-a.s. fort∈IR+,
and such that a version of the characteristics ofXrelative toQare
B0i =Bi+ P
j≤dcijβj
ãA+hi(x)(Y −1)∗ν C0=C
ν0=Y ãν
By Theorem 6.2, Y and β tell us how the characteristics of the semimartingale transform under the change of measure P ; Q. Since we assumed that the filtration is generated by the Lévy process L, Y and β tell us even more—they completely determine the change of the underlying probability measure. The reason for this is displayed below. By Jacod and Shiryaev (1987), Theorem III.4.34, the independence of increments ofLimplies that every local martingaleM has the representation property relative toL, in the sense of Jacod and Shiryaev (1987), Definition III.4.22. This means that every local martingaleM can be represented in the form
M =M0+HãLc+W∗(àL−νL) (6.8)
withH ∈L2loc(Lc)andW ∈Gloc(àL).4
If Q loc P, there is a unique (càdlàg) P-martingale Z such that Zt = dQt/dPt for all t (cf. Jacod and Shiryaev (1987), Theorem III.3.4.) It is called the density process ofQ, relative toP. SinceZ is a martingale, it can be represented in the form (6.8). Jacod and Shiryaev (1987), Theorem III.5.19, provide explicit expressions for the integrandsHandW in this case.
Z =Z0+ (Z−β)ãLc+Z−
Y −1 + Yb −a 1−a1l{a<1}
∗(àL−νL).
(6.9)
Here,β (resp.,Y) are the predictable process (resp., function) whose existence follows by the Girsanov theorem for semimartingales. The predictable processesaandYb are defined by
at(ω) :=νL(ω;{t} ×IRd), (6.10)
Ybt(ω) :=
R
IRY(ω, t, x)νL(ω;{t} ×dx) (if the integral is well-defined) +∞ otherwise.
(6.11)
In our case,Lis a Lévy process. This implies a considerable simplification of equation (6.9).
Lemma 6.3. LetLbe a Lévy process under the measureP. Consider another measureQlocP. Then the density processZt=dQ/dP has the following representation as a stochastic integral.
Z =Z0+ (Z−β)ãLc+ Z−(Y −1)
∗(àL−νL).
(6.12)
4These two set of integrands are defined in Jacod and Shiryaev (1987), III.4.3 resp. Definition II.1.27 a.
Proof. Lhas stationary and independent increments, so
ν({t} ×IR) = 0 for allt∈IR+. (6.13)
This follows from Jacod and Shiryaev (1987), II.4.3, and the fact that for processes with independent increments, quasi-left-continuity is equivalent to condition (6.13). Thusa≡0andYb ≡0, and equation (6.9) simplifies to (6.12).
Ifβ ≡0andY ≡1, then obviously the density process (6.12) vanishes up to indistinguishability. The conditions onβandY can be relaxed somewhat, admitting them to be different from zero on some null sets. We will make this precise in Lemma 6.7 below. In order to prove this, we need the following two propositions. [H2loc denotes the class of locally square integrable local martingales; see Jacod and Shiryaev (1987), Definitions I.1.33 and I.1.41.]
Proposition 6.4. Consider a local martingale X ∈ H2loc. Let H be a predictable process such that H2ã hX, Xi= 0up to indistinguishability. Then the stochastic integralHãXexists and
HãX= 0 up to indistinguishability.
Proof. According to Jacod and Shiryaev (1987), Theorem 4.40 d, hHãX, HãXi=H2ã hX, Xi,
and this process vanishes up to indistinguishability. This means that the local martingaleM := HãX is orthogonal to itself, and we haveM = M0 = 0 up to indistinguishability. See Jacod and Shiryaev (1987), Lemma 4.13 a.
Remark: Since continuous local martingales are locally bounded, the class of continuous local martin- gales is contained inH2loc. Thus Proposition 6.4 applies to all continuous local martingales.
Proposition 6.5. If V ∈ Gloc(à)with ν({V 6= 0}) = 0P-a.s., then for the stochastic integrals with respect to the compensated random measureà−νwe have
V ∗(à−ν) = 0 up to indistinguishability.
Proof. According to Jacod and Shiryaev (1987), Theorem II.1.33 a, we haveV ∗(à−ν)∈ Hloc2 , with hV ∗(à−ν), V ∗(à−ν)i= (V −Vb)2∗νt+X
s≤t
(1−as)(Vbs)2, (6.14)
where a and Vb are defined analogously to (6.10) and (6.11). For P-almost all ω, aã(ω) ≡ 0 and Vb(ω,ã,ã) ≡ 0. This implies that the predictable quadratic variation (6.14) of M := V ∗(à−ν) is equal to0, that is, the local martingaleM is orthogonal to itself. By Jacod and Shiryaev (1987), Lemma I.4.13 a,M =M0 = 0up to an evanescent set.
Corollary 6.6. IfV, W ∈Gloc(à)withν({V 6=W}) = 0P-a.s., then V ∗(à−ν) =W∗(à−ν) up to indistinguishability.
Proof. Gloc(à)is a linear space and the mappingV 7→V ∗(à−ν)is linear onGloc(à)up to indistin- guishability (cf. Jacod and Shiryaev (1987), remark below II.1.27.) HenceV −W ∈Gloc(à), and
V ∗(à−ν)−W∗(à−ν) = (V −W)∗(à−ν).
But by Proposition 6.5, this vanishes up to indistinguishability.
The following lemma uses Propositions 6.4 and 6.5 to examine a change-of-measure problem.
Lemma 6.7. LetLbe a Lévy process with respect to the probability measureP. Assume that the stochas- tic basis is generated byLand the null sets. LetQbe a probability measure which is locally absolutely continuous w.r.t.P, and letβ andY be the predictable process (resp. function) associated according to Theorem 6.2 with the change of probabilityP ;Q. IfR∞
0 β(ω, s)dhLc, Lci = 0P-almost-surely and ν(ω;{Y(ω,ã,ã)6= 1}) = 0P-almost-surely, thenQ=P.
Proof. According to equation (6.12), the conditions imply that Z ≡ Z0 up to indistinguishability. But Z0 = 1 P-a.s., since under the assumptions above,F0consists only of null-sets and their complements.
ThusZ ≡Z0 = 1P-a.s.
In addition to this result from the theory of stochastic processes, we need the following lemma. It states that a measure on IR is uniquely characterized by the values of its bilateral Laplace transform on an interval of the real line.
Lemma 6.8. LetG(dx)andH(dx)be measures on(IR,B1). If Z
IR
exp(ux)G(dx) = Z
IR
exp(ux)H(dx)<∞ for allufrom a non-empty finite interval(a, b)⊂IR, thenH=G.
Proof. Setc:= (a+b)/2,d:= (b−a)/2, and define measuresG0andH0by G0(dx) := exp(cx)G(dx), H0(dx) := exp(cx)H(dx).
Then Z
IR
exp(vx)G0(dx) = Z
IR
exp(vx)H0(dx)<∞ (6.15)
for all v ∈ (−d, d). In particular, takingv = 0 shows that G0 and H0 are finite (positive) measures with equal mass, and without loss of generality we can assume that this mass is1. Thus we can apply the theory of probability distributions. Equation (6.15) says that the moment generating functions ofG0 andH0coincide on the interval(−d, d). By well-known results about moment generating functions (cf.
Billingsley (1979), p. 345), this implies thatG0 =H0and henceG=H.