Stochastic Processes and their Applications 115 (2005) 299–327
Minimal entropy preserves the Le´vy property:
how and why
Felix Esche
a, Martin Schweizer
b,aTechnische Universita¨t Berlin, Institut fu¨r Mathematik, MA 7-4, StraXe des 17. Juni 136,
D-10623 Berlin, Germany
bETH Zu¨rich, Mathematik, ETH-Zentrum, CH-8092 Zu¨rich, Switzerland
Received 26 March 2003; received in revised form 17 February 2004; accepted 24 May 2004 Available online 8 October 2004
Abstract
Let L be a multidimensional Le´vy process underP in its own filtration and consider all probability measures Qturning Linto a local martingale. The minimal entropy martingale measureQE is the uniqueQwhich minimizes the relative entropy with respect toP. We prove
thatLis still a Le´vy process underQEand explain precisely how and why this preservation of the Le´vy property occurs.
r2004 Elsevier B.V. All rights reserved. MSC:primary 60G51; 91B28; 60G48
Keywords:Le´vy processes; Martingale measures; Relative entropy; Minimal entropy martingale measure; Mathematical finance; Incomplete markets
0. Introduction
In the last years, Le´vy processes have become very popular for modelling in finance. They provide a lot of flexibility when fitting a model to observed asset prices and yet are very tractable if one needs expressions for derivative prices. One drawback is that the resulting model of a financial market is usually incomplete and
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thus has multiple martingale measures (and hence non-unique option prices). A popular approach is then to fix one particular martingale measure Q for the underlying assetsSand to price derivatives by theQ-expectation of their discounted payoff. But how should one chooseQ? Very often, this is done via the minimization of a functional over martingale measures, and the functional is in turn motivated by a dual formulation corresponding to a primal utility maximization problem; see Kallsen [15] for a list of references. Alternatively, Q might be the natural pricing measure arising from a criterion which emphasizes hedging rather than pricing aspects; this produces for instance the minimal or the variance-optimal martingale measures.
In this paper, we consider the pricing-oriented approach and we take the relative entropy of Q with respect to the original measure P as the functional to be minimized. Not only does this allow us to do many computations explicitly; one general argument for that choice is also that the resulting minimal entropy martingale measure is automatically equivalent toP. This is not so for the variance-optimal or more generally theq-optimal martingale measures.
We show that ifLis anRd-valued Le´vy process underPand ifQE minimizes the relative entropy over allQunder whichLis a local martingale, thenLis again a Le´vy process underQE:This extends a result by Fujiwara and Miyahara[8]who simply write downQE ford ¼1 and directly prove its optimality. But more important than the generalization to d41 is that we also explain precisely how this preservation happens and why QE has the structure obtained. Similarly to earlier papers by Foldes[5,6]on a different topic, the reasons are very intuitive. But the actual proofs turn out to require quite a lot of work.
The paper is structured as follows. Section 1 formulates the basic problem more precisely, states the two main results and presents the intuitive explanation mentioned above. Section 2 prepares the ground by providing a number of results from general semimartingale theory. Section 3 contains the crucial idea. It shows how one can always reduce relative entropy while preserving the martingale property by a suitable averaging procedure over certain parametersb;Y that characterizeQ. This reduces the problem from a minimization over probability measures to a minimization over non-random functions. Section 4 produces a candidate for the optimal function from the first order condition for optimality and proves that the corresponding candidate measure has indeed minimal entropy. The main result from Section 3 is then proved in Section 5 which substantiates a merely plausible reasoning with a rigorous argument. Finally, a number of proofs from Section 2 are collected in Appendix A.
1. Setup and main results
In this section, we introduce some notation, formulate the basic problem and state the two main results. Unexplained terminology used here is standard or explained in the next section.
LetðO;F;F;PÞbe a filtered probability space withF¼ ðFtÞt2T satisfying the usual
conditions and eitherT ¼ ½0;T0for some fixedT02 ð0;1Þ(finite horizon) orT ¼
½0;1Þ(infinite horizon). For a probability measureQ5locP;we denote by
ItðQjPÞ:¼EQ log dQ dP F t " # 2 ½0;þ1
the relative entropy ofQwith respect toPon Ft and callðItðQjPÞÞt2T theentropy
process of Q. For an Rd-valued F-adapted process X¼ ðXtÞt2T and a fixed dd
-matrixU, we introduce the following sets of probability measures onðO;FÞ:
QU aðXÞ:¼ Q5 loc PjUX is a localQ-martingale ;
QUeðXÞ:¼ QlocPjUX is a localQ-martingale
QUaðXÞ; QUf ðXÞ:¼ Q2QUaðXÞ jItðQjPÞo1 for allt2T ; QU
‘ðXÞ:¼ Q2QaUðXÞ jX is a Levy process underQ
:
QU
‘ ðXÞis mainly used ifXis already a Le´vy process underP. Note thatQ2QU‘ ðXÞ means thatUXis a localQ-martingale, butXitself is aQ-Le´vy process. IfUis the identity matrix, we omit the superscriptU; henceQU
s ðXÞ ¼QsðUXÞfors2 fa;e;fg;
but not fors¼‘: Elements ofQU
eðXÞare calledequivalent local martingale measures(ELMMs) for
UX. The minimal entropy martingale measure (MEMM)QEðUXÞ is defined by the
property that it minimizes the entropy process pointwise intover allQ2QUaðXÞ;i.e., QEðUXÞ is in QUaðXÞ and ItðQEðUXÞ jPÞpItðQjPÞ for all Q2QUaðXÞ and t2T:
Theminimal entropy Le´vy martingale measure QE‘ðUXÞ 2QU‘ ðXÞis similarly defined
by the property that it minimizes the entropy process pointwise in t over all Q2
QU‘ ðXÞ: We want to find QEðULÞ when L is a Le´vy process under P in its own filtrationF¼FL:
Remark. In mathematical finance, the above problem naturally arises in the
following way. Suppose we have a financial market withdrisky assets (‘‘stocks’’) and one riskless asset (‘‘bank account’’, B). We express all prices in units ofB; this is called discounting with respect to B, and the resulting discounted asset prices are denoted byS. A frequently made modelling assumption is then thatSi¼Si0EðLiÞfor someRd-valued Le´vy process L, and thenSandLhave the same ELMMs.
In economic terms, an ELMM can be interpreted as a pricing operator for financial products which is consistent with the a priori given asset prices S; see Harrison and Kreps[10]. It is also well known that the existence of some ELMM is essentially equivalent to the economically plausible and desirable property that the financial market described by S does not admit arbitrage opportunities (‘‘money pumps’’); see Delbaen and Schachermayer[3]for a precise formulation. Finally, as mentioned in the introduction, minimizing relative entropy is one possible criterion for choosing an ELMM. This explains why we are interested inQEðLÞ;the extraU will give some room for more generality.
A result called numeraire invariance provides the (economically intuitive) statement that discounting does not change anything; this is usually taken as justification for choosingB1 and directly modelling discounted prices. However, this result assumes that the filtrationFis given a priori. If we wanted to take asFthe filtration generated by asset prices, it may well make a difference if these are discounted or not as soon as the bank accountBis stochastic. Although the use of the filtration generated by prices would be desirable and is for instance advocated in Section 9.6 of Kallianpur and Karandikar [14], we follow here the standard approach in the literature to work with the filtration generated by the underlying sources of randomness; see for instance the very first pages of Karatzas and Shreve
[16]. This explains our choice F¼FL:
As already stated, our goal in this paper is now to identifyQEðULÞifLis a Le´vy process underPfor its own filtrationF¼FL;and moreover to explain exactly why QEðULÞhas the particular structure we obtain. The two main results are
Theorem A. Let L be anRd-valued P-Le´vy process forF¼FL;and U a fixed dd
-matrix. Suppose thatQUeðLÞ \QUf ðLÞ \QU‘ ðLÞa;:If QEðULÞexists,then L is a Le´vy
process under QEðULÞ:
This result explains the first part of the paper’s title since it says that the Le´vy property of L is preserved by passing from P to the minimal entropy martingale measure forUL.
Theorem B. Let L be an Rd-valued P-Le´vy process for F¼FL with Le´vy
characteristics ðb;c;KÞ; and U a fixed dd-matrix. Suppose that there exists u 2
rangeðU>Þsuch that Z Rd jxeu>xhðxÞ jKðdxÞo1; U bþcuþ Z Rd ðxeu>xhðxÞÞKðdxÞ ¼0:
Then QEðULÞexists and coincides with the Esscher martingale measure Qudefined by
dQu
dP ¼const:expðu
>
LtÞonFt for all t2T:
This showshowthe Le´vy property ofLis preserved, namely by using an Esscher transform ofPto construct a martingale measure forUL. The final third of the title, why this happens, will become clear from the proofs and constitutes a key insight contributed here.
In comparison to existing literature, perhaps the most characteristic feature of this paper is its combination of intuitive insight with rigorous proofs. This is best understood if we briefly explain how we obtain our results. By using Girsanov’s theorem, anyQ5locPcan be described by two parametersb;Y which are in general stochastic processes. The relative entropyItðQjPÞis then a convex functional of b
time-independent parameters obtained by averaging over o and t. Moreover, the local martingale property of UL under Q is characterized by a linear constraint betweenbandYand so is preserved by this averaging. Hence the MEMM forUL must have deterministic time-independent parameters, which means thatLis a Le´vy process under it. This explains the intuition behind Theorem A; the rigorous proof, however, must still circumvent integrability problems. We use the assumption thatF
is generated by a Le´vy process to identify a measure via its density process by its parameters.
Due to Theorem A, finding QEðULÞreduces to a classical optimization problem over deterministic time-independent quantitiesb;Y:The linear constraint from the local martingale property even eliminatesbso that only the non-random functionY needs to be varied. Formally writing down the first order conditions for optimality then produces a candidate Y;and Theorem B accomplishes the fairly straightfor-ward task of proving that the corresponding measure Qu has indeed minimal
entropy. This entire line of reasoning also makes it very transparent whyminimal entropy preserves the Le´vy property.
Remark. Conceptually, our results are similar to Foldes [5,6] who considered an investment problem with market returns given by a process R with independent increments. He proved that an optimal portfolio plan can be found in the class of deterministic strategies (and is even time-independent if R has independent and stationary increments). Like here, the main techniques used were computations based on semimartingale characteristics.
From a formal point of view, Theorem B is slightly more general than Theorem 3.1 of Fujiwara and Miyahara [8] who proved essentially this result for a finite horizon and whenLis one-dimensional (Uis then the identity matrix). Earlier work on the same problem under additional assumptions is also reviewed in Fujiwara and Miyahara [8]. It seems not quite straightforward to generalize their proof to the multidimensional case, and a number of integrability issues is also not entirely clear from their presentation. We briefly indicate below why including the matrix U is useful for applications. But the main difference to our work is that Fujiwara and Miyahara[8]simply defineQu as in Theorem B and prove directly that this is the
MEMM; there is no hint to the reader where this measure comes from.
On the other end of the scale, the paper by Chan[2]already contains the idea of computing relative entropy as a functional of the parametersb;Y;even if his setting is less general due to exponential moment conditions on L. The crucial difference here is that Chan [2] argues only heuristically (‘‘it is a little less clear’’) that a minimization over deterministic parameters is already enough. Making this intuitive idea rigorous in full generality is achieved by our Theorem A and turns out to be more involved; see Section 5.
Two immediate applications that come to mind are the following.
Example 1. Exponential Le´vy processes. Consider a model where discounted asset prices are strictly positive and given by Si¼Si0EðLiÞ; i¼1;. . .;d; for some Rd -valued Le´vy processLunderP. Since dSi¼SidLi;Sis a localQ-martingale if and
only ifLis, and soQEðSÞ ¼QEðLÞ:Hence the MEMM forSinF¼FL is given by
the Esscher measure Qu from Theorem B, provided u there exists (with
U¼identity). This generalizes Theorem 3.1 of Fujiwara and Miyahara [8] to the cased41:(Actually, Fujiwara and Miyahara[8]work withS¼S0expðLeÞfor some
Le´vy processL;e but this can be rewritten with Las above.)
Example 2. Stochastic volatility models driven by Le´vy processes. LetL be a two-dimensional Le´vy process under P for F¼FL and model the one-dimensional discounted asset price processSby
dSt¼sðt;St;L2tÞdL
1
t; (1.1)
where s:½0;T0 ð0;1Þ R!R is a measurable function such that (1.1) has a
strictly positive solution S with the property of being a local Q-martingale if and only if L1 is. This is a Le´vy version of the usual stochastic volatility models where
ðL1;L2Þis a diffusion with possibly correlated coordinates; note thatL1andL2 may well be dependent. The MEMM QEðSÞ is given byQEðL1Þ ¼QEðULÞ; whereU¼
1 0
0 0
gives the projection on the first coordinate, and QEðL1Þ can be explicitly constructed from Theorem B. See Section 4.4 of Esche [4] for a more detailed account.
2. Auxiliary results
This section presents some auxiliary results from general semimartingale theory. To facilitate reading, most proofs are relegated to the Appendix. Our basic reference is Jacod and Shiryaev [12], abbreviated JS. Without special mention, all processes take values inRd:
2.1. Semimartingales, characteristics and Girsanov’s theorem
We first fix some notation. For a semimartingaleX, we denote bymX the random measure associated with the jumps ofXand bynPthe predictableP-compensator of mX:IfWis a real-valued optional function andm a random measure, Wmis the
integral process ofWwith respect tom:Throughout the entire paper,h is a fixed but
arbitrary truncation function. Our results do not depend on the choice of h; more
precisely, we could take a differenth0and rewrite everything withh0simply replacing h throughout.
IfXis a semimartingale, we denote by ðB;C;nÞthe triplet of itsP-characteristics (relative to the truncation functionh). As in Proposition II.2.9 of JS[12], we can and always do choose a version of the form
B¼
Z
bdA; C¼
Z
cdA; nðo;dt;dxÞ ¼dAtðoÞKo;tðdxÞ; (2.1)
whereAis a real-valued predictable increasing locally integrable process,b¼ ðbitÞan
symmetric nonnegative definitedd-matrices, andKo;tðdxÞa transition kernel from ðORþ;PÞ intoðRd;BdÞwhich satisfies Ko;tðf0gÞ ¼0 and RRdð1^ jxj2ÞKo;tðdxÞp1
for allt2T:We shall also need the characteristics of a linear transformation of a semimartingale. For a Le´vy processX, this is given by Proposition 11.10 of Sato[18]. The argument for the general semimartingale case is routine and therefore omitted.
Proposition 1. Let X be a semimartingale with characteristicsðB;C;nÞand U a dd
-matrix. Then the semimartingale X~ ¼UX has the following characteristicsðB;~ C;~ ~nÞ:
~ Bt ¼UBt ðUhðxÞ hðUxÞÞ nt; ~ Ct¼UCtU>; ~ nðA1A2Þ ¼nðA1U1ðA2nf0gÞÞ for A12BðTÞ; A22Bd:
We recall Girsanov’s theorem from JS[12], Theorem III.3.24 to introduce some terminology.
Proposition 2. Let X be a semimartingale with P-characteristics ðBP;CP;nPÞ and
denote by c;A the processes from(2.1).For any probability measure Q5locP;there exist
a predictable function YX0and a predictableRd-valued processbsatisfying
jðY1Þhj nPt þ Z t 0 jcsbsjdAsþ Z t 0
b>scsbsdAso1 Q-a:s: for all t2T
and such that the Q-characteristicsðBQ;CQ;nQÞof X are given by
BQt ¼BPt þ Z t 0 csbsdAsþ ððY1ÞhÞ nPt; C Q t ¼CPt; nQðdt;dxÞ ¼Yðt;xÞnPðdt;dxÞ:
We call band Y the Girsanov parameters of Q(with respect to P relative to X).
Remark NV. Intuitively,Ydescribes how the jump distributions ofXchange when we pass from P to Q, and b together with Y determines the change in drift. CP describes the P-quadratic variation of the continuous part of X and is therefore invariant under an absolutely continuous change of measure. Note that the Girsanov parameters are not unique: From the uniqueness ofnPandnQwe only get uniqueness
ofYðo;;Þon the support ofnPðoÞ;and with this and the uniqueness ofBPandBQ
we only get uniqueness of cb for fixed c and A. However, we can choose the followingnice versionsofYandb:
First we takeYsuch thatYðo;s;xÞ ¼1 identically forðs;xÞesuppnPðoÞ:SincenP
does not charge f0g Rd;this implies Yðo;s;0Þ ¼1 identically. Next,bs is unique only ifcs is regular. Ifcs is possibly degenerate, we choosebs in the following way
(and for simplicity, we only consider the case where c is deterministic and time-independent).
Let rankðcÞ ¼rpd and let lj be the eigenvalues ofc, numbered such thatlj¼0 exactly forj4r:Sincecis nonnegative definite, there exist a diagonal matrixc~with
~
cjj¼lj and an orthogonal matrix S such that c¼ScS~ >: If b is any Girsanov parameter, thencb¼ScS~ >band sincec~is diagonal withc~jj¼0 forj4r;we can set
ðS>
bÞj¼0 forj4rwithout changingcb:So if we setgj ¼ ðS>
bÞj forjprandgj¼0 forj4rand then defineb~¼Sg;we get a new predictable processb~ withcb~¼cband
ðS>bÞ~ j¼0 for j4r: Moreover, b~ with these properties is unique. In fact, cb~ ¼cb~0
implies ScS~ >b~ ¼ScS~ >b~0 and thus cS~ >b~ ¼cS~ >b~0 which yieldsðS>bÞ~ j¼ ðS>b~0Þj for jpr by the properties of c:~ Finally, since ðS>bÞ~ j¼0¼ ðS>b~0Þj for j4r by assumption, we getS>b~ ¼S>b~0 and thusb~¼b~0:
To simplify arguments, we assume throughout that Y andb are chosen as above. Our main results do not depend on this choice.
2.2. Le´vy processes
LetRbe a probability measure onðO;FÞandXa stochastic process null at 0 with RCLL paths and adapted to a filtration Fsatisfying the usual conditions underR. ThenXis called anðR;FÞ-Le´vy processif for allspt;the random variableXtXsis
independent ofFs underRand has a distribution under Rwhich depends only on
ts:(This is called a PIIS by JS in Section II.4.) If there is only a processXwith independent and stationary increments underR, we callXanR-Le´vy process, take as
F theR-augmentation of the filtration generated byX and denote this by FX; this
satisfies the usual conditions since a Le´vy process is a Feller process. ForR¼P;we even sometimes drop the mention ofP.
EveryF-Le´vy process is anF-semimartingale (JS[12, Corollary II.4.19]), and anF -martingale if and only if it is a local F-martingale [11, Theorem 11.46]. X is an
ðR;FXÞ-Le´vy process if and only if ER½expðiu>ðXtXsÞÞ jFXs ¼ER½expðiu>XtsÞ
for allu2Rd andspt:According to JS[12, Theorem II.4.15 and Corollary II.4.19], a semimartingale Xnull at 0 is a Le´vy process if and only if its characteristics are deterministic and linear in time, i.e.,Bt¼bt;Ct ¼ctandnðdt;dxÞ ¼KðdxÞdt;where
b2Rd; c is a symmetric nonnegative definite dd-matrix and K is a s-finite measure onðRd;BdÞwithKðf0gÞ ¼0 andRRdð1^ jxj2ÞKðdxÞo1:(Note thatKiss -finite becausejðxÞ ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1^ jxj2 is40K-a.e. and inL2ðKÞ:We needs-finiteness later to use Fubini’s theorem.) The constant triplet ðb;c;KÞ coincides with the Le´vy
characteristics from the Le´vy–Khinchine representation of the infinitely divisible
distribution ofX1;and we see that for a Le´vy processX, the compensatornPof the
jump measuremX satisfiesnPðftg RdÞ ¼0 P-a.s. for allt2T:
2.3. A converse of Girsanov’s theorem for Le´vy processes
Proposition 2 generally describes a measure Q5locP via parameters b;Y;and we want to express the density processZQexplicitly in terms ofb;Y:This works ifXhas the weak property of predictable representation (as in [11], Definition 13.13; in Section III.4 of JS [12], this is called ‘‘all local martingales have the representation property relative toX’’). As usual, we denote byEðYÞthe stochastic exponential of a semimartingaleY. Putting together Section II.6, Theorem III.4.34, Theorem III.5.19, Corollary III.5.22 and Proposition III.5.10 from JS[12]and usingnPðo;ftg RÞ ¼0 for allt2T leads to:
Proposition 3. Let L be a P-Le´vy process forF¼FL: If QlocP with Girsanov
para-metersb;Y;the density process of Q with respect to P is given by ZQ¼EðNQÞwith
NQt ¼
Z t
0
bsdLcsþ ðY1Þ ðmLnPÞt for t2T: (2.2)
While Proposition 2 gives a description of measuresQ5locPin terms of Girsanov parameters b;Y; we also need to go the other way round. We want to start with given quantities b;Y and find a measure Q5locP which has b;Y as Girsanov parameters. In the setting of Le´vy processes, Proposition 3 makes this look almost straightforward because if we defineNQfromb;Y as in (2.2), the natural candidate forQshould haveZ:¼EðNQÞas density process. However, two problems remain: we must verify that the localP-martingaleZis a trueP-martingale, and then we need to prove the existence of a probability measureQwith the given martingaleZas density process. The first point needs conditions onb;Y:The second is easily solved for a finite time horizon T02 ð0;1Þ by setting dQ¼ZT0dP; no matter what the
underlying spaceOis. For an infinite time horizon, existence ofQstill follows if we work on the canonical path space O:¼Dð½0;1Þ;RdÞ ¼:Dd with F:¼BðDdÞ and appeal to Lemma 18.18 of Kallenberg[13].
We start this program with a technical result. Its proof is purely analytic and therefore omitted; see Section 2.3 of Esche[4].
Lemma 4. The functions f;g:½0;1Þ !R defined by fðyÞ:¼ylogy ðy1Þ (with 0 log 0:¼0)and gðyÞ:¼ ð1pffiffiffiyÞ2 are convex and satisfy0pgðyÞpfðyÞfor all yX0:
Proposition 5. Let L be a P-Le´vy process for F¼FL with Le´vy characteristics
ðb;c;KÞ:If ¯bis a predictable process and ¯Y40 a predictable function such that
EP exp Z t 0 1 2b¯ > sc ¯bsþ Z Rd fðY¯ðs;xÞÞKðdxÞ ds o1 for all t2T; (2.3)
then ¯Y1 is integrable with respect tomLnP;and ¯Z:¼EðN¯Þwith
¯ Nt:¼ Z t 0 ¯ bsdLcsþ ðY¯ 1Þ ðmLnPÞt; t2T (2.4)
is a strictly positive P-martingale onT:
Proof. See Appendix A. &
Ifb¯andY¯ are deterministic and time-independent, we obtain from Proposition 5:
Corollary 6. Let L be a P-Le´vy process forF¼FL with Le´vy characteristicsðb;c;KÞ:
If ¯b2Rd and ¯Y:Rd ! ð0;1Þ is a measurable function with
Z
Rdf
ðY¯ðxÞÞKðdxÞo1; (2.5)
The next result now starts with given quantities b¯;Y¯ and identifies these as Girsanov parameters of a measure Q:¯ As pointed out above, there is only one candidate forQ;¯ whose existence is ensured as soon asEðN¯Þis a true P-martingale and we either have a finite time horizon or work on the path spaceDd:
Proposition 7. Let L be a P-Le´vy process for F¼FL with P-Le´vy characteristics
ðb;c;KÞ: Let ¯b be a predictable process and ¯Y40 a predictable function such that
¯
Y1 is integrable with respect to mLnP; and define ¯N¼Rb¯
sdLcsþ ðY¯ 1Þ
ðmLnPÞ: If there is a probability measure ¯QlocP with density process ZQ¯ ¼
¯
Z:¼EðN¯Þ;then ¯band ¯Y are the Girsanov parameters of ¯Q:
Proof. See Appendix A. &
For future reference, we explicitly state the following result. (If we are only interested in constructing the Le´vy measure Q;¯ an alternative proof could use a combination of Sato [18], Theorem 8.1 and Corollary 11.6, with JS [12, Theorem
IV.4.39], but would not be much shorter.)
Corollary 8. Let P be a probability measure onO¼Dd withF ¼BðDdÞ;coordinate
process L andF¼FL:Suppose that L is a P-Le´vy process with P-Le´vy characteristics
ðb;c;KÞ: For any ¯b2Rd and any measurable function Y¯ :Rd ! ð0;1Þ with
R
RdfðY¯ðxÞÞKðdxÞo1; there exists a probability measure ¯Q
loc
P on ðO;FÞ with
Girsanov parameters ¯b;Y and such that L is a ¯¯ Q-Le´vy process with ¯Q-Le´vy characteristics
bQ¯ ¼bþc ¯bþ
Z
Rd
hðxÞðY¯ðxÞ 1ÞKðdxÞ; cQ¯ ¼c; KQ¯ðdxÞ ¼Y¯ðxÞKðdxÞ:
ForT ¼ ½0;T0with T02 ð0;1Þ;this holds for any probability spaceðO;F;PÞand any
P-Le´vy process L ifF¼FLandF ¼FT0:
Proof. Combining Corollary 6 and Lemma 18.18 of Kallenberg[13]gives a measure ¯
QlocPwith Girsanov parametersb¯;Y¯ by Proposition 7. The Q-characteristics of¯ L are then given by Proposition 2, and since they are deterministic and linear in time,L
is aQ-Le´vy process.¯ &
2.4. Martingale measures for Le´vy processes
As seen above, a measure Q5locP can be described via two quantities b;Y that determine the characteristics of X under Q from those under P. By JS [12, Proposition II.2.29],Xis a localQ-martingale if and only ifBQþ ðxhðxÞÞ mX is.
Since this gives a relation betweenbandY, a martingale measureQforXshould be determined by a single quantity Y, and for aQ-Le´vy process, this should further reduce to a deterministic time-independent function.
To make these ideas more precise, letLbe aP-Le´vy process andUa fixeddd -matrix. For a given measureQ5locPwith Girsanov parametersb;Y;we want to give conditions on b;Y for UL to be a local Q-martingale. We denote by nPðdt;dxÞ ¼
KðdxÞdttheP-compensator of the jump measuremLofL. For technical reasons, we need to characterize the Q-integrability of large jumps of UL to in a different manner, and this is achieved by the following result.
Lemma 9. Let L be a P-Le´vy process, U a fixed dd-matrix and Q5locP with
Girsanov parametersb;Y:If EQ½fðYÞ nPto1 for all t2T;we have for all t2T
jUxhðUxÞj nQt o1 Q-a:s: if and only if
jUðxYhÞj nPto1 Q-a:s:; ð2:6Þ
jUxhðUxÞj nQt 2L1ðQÞ if and only if
jUðxYhÞj nPt 2L1ðQÞ: ð2:7Þ Proof. See Appendix A. &
Remarks. (1) We shall see in Lemma 12 thatEQ½fðYÞ nPto1for allt2T holds in
particular ifQlocPhas a finite entropy process.
(2) By JS[12, Proposition II.1.28],jUxhðUxÞj nQisQ-integrable if and only if jUxhðUxÞj mL¼ jxhðxÞj mULis, and the latter means that the large jumps of
UL are Q-integrable. For QlocP with finite entropy process, this is by Lemma 9
equivalent toQ-integrability ofjUðxYhÞj nPwhich turns out to be a technically
more convenient condition.
Proposition 10. Let L be a P-Le´vy process for F¼FL with P-Le´vy characteristics
ðb;c;KÞ;and U a fixed dd-matrix. Let Q5locP with Girsanov parametersb;Y and
such that EQ½fðYÞ ntPo1for all t2T:Then UL is a local Q-martingale if and only
if we have Q-a.s. for all t2T bothjUðxY hÞj nP
to1and U bþcbtþ Z RdðxYðt;xÞ hðxÞÞKðdxÞ ¼0: (2.8)
Condition(2.8)is called the martingale condition for UL.
Proof. See Appendix A. &
Remarks. (1) The martingale condition is independent of the choice of the
truncation function. In fact, if we replace hby some h0;then b is replaced byb0¼
bRRdðhðxÞ h0ðxÞÞKðdxÞ (see JS [12, Proposition II.2.24]) and (2.8) holds with
ðb0;c;KÞrelative toh0:
(2) IfUis regular, (2.8) is equivalent to
bþcbtþ
Z
Rd
ðxYðt;xÞ hðxÞÞKðdxÞ ¼0 Q-a.s. for allt2T: (2.9)
This is the martingale condition as it appears in Bu¨hlmann et al. [1], Chan [2], Fujiwara and Miyahara [8]or Section VII.3 of Shiryaev[19], among others. Note that (2.9) requires that the appearing integral is well-defined; this needsZ
Rd
which is equivalent to jUðxYhÞj nPto1 Q-a.s. for t2T: Actually, not all authors are equally careful or explicit about verifying this condition. However, this does matter; see the comment following ‘‘Pseudo-Proposition 14’’ below.
3. Reducing relative entropy
In this section, we show how the entropy process of anyQlocPin a Le´vy filtration can be reduced by averaging Girsanov parameters. Since this preserves the linear constraint imposed by the local martingale property, the MEMM, if it exists, must preserve the Le´vy property. For reasons of integrability, this is not exactly true, but it does give the correct intuition.
To minimize repetitions, we assume throughout this section that L is a P-Le´vy
process forF¼FLwith P-Le´vy characteristicsðb;c;KÞ;and U is a fixed dd-matrix.
We start by computing the entropy process of a given Qin terms of its Girsanov parameters.
Lemma 11. Fix a probability measure QlocP with Girsanov parametersb;Y and finite
entropy processðItðQjPÞÞt2T;and denote by Z¼ZQ¼EðNÞits density process with
respect to P. The canonical decomposition of the P-submartingale ZlogZ is ZlogZ¼
MþA with M¼ Z Zð1þlogZÞdNþ ðZfðYÞÞ ðmLnPÞ; A¼1 2 Z ZdhNci þ ðZfðYÞÞ nP¼:A0þA00: Moreover,A0 t and A 00
t are P-integrable for all t2T:
Proof. It is straightforward to check thatZlogZ is aP-submartingale because the entropy process ofQis finite-valued. By the product rule, we have
dðZlogZÞ ¼ZdðlogZÞ þ ðlogZÞdZþd½Z;logZ (3.1) and the explicit expression forZ¼EðNÞyields
logZt ¼Nt 1 2hN ci tþ X spt ðlogð1þDNsÞ DNsÞ ¼:Nt 1 2hN ci tþDt; (3.2)
where the sum is absolutely convergent for all t2T: In fact, jDNsj412 only for
finitely manyspt;and forjxj2p12we havejlogð1þxÞ xjpconst:jxj2 so that X spt jlogð1þDNsÞ DNsjIfjDNsjp12gpconst: X spt jDNsj2pconst:½Nto1:
To compute the d½Z;logZ-term in (3.1), we use dZ¼ZdNand (3.2) to get d½Z;logZ ¼Zd½N;logZ ¼Zðd½N 1
2d½N;hN
Since hNci is continuous, ½N;hNci vanishes, and since D is of finite variation, we have ½N;Dt ¼X spt DNsDDs¼ X spt DNsðlogð1þDNsÞ DNsÞ: (3.4)
This sum is absolutely convergent sincePsptjDNsDDsj ¼R0tjd½N;Dsjpð½NtÞ 1 2ð½D
tÞ 1 2
by the Kunita–Watanabe inequality, and since PsptðDNsÞ2p½Nt converges as well,
we may decompose the sum in (3.4) and get
½N;D ¼X s DNslogð1þDNsÞ X s ðDNsÞ2: This yields ½N;logZ ¼ ½N X s ðDNsÞ2þ X s DNslogð1þDNsÞ ¼ hNci þX s DNslogð1þDNsÞ; or in terms of (3.3) ½Z;logZ ¼ Z ZdhNci þX s ZsDNslogð1þDNsÞ:
Putting all this together and using dZ¼ZdN;we finally get a decomposition
ZlogZ¼ Z Zð1þlogZÞdNþ1 2 Z ZdhNci þX s ZsðDDsþDNslogð1þDNsÞÞ ¼ Z Zð1þlogZÞdNþ1 2 Z ZdhNci þX s Zsfð1þDNsÞ ¼:M0þA0þV; ð3:5Þ
whereM0is a localP-martingale,A0is continuous and increasing, andVis increasing sinceDNs41 and fX0:However, Vis not predictable so that (3.5) is not yet the
canonical decomposition. ButV¼ZlogZM0A0 is locallyP-integrable since all terms on the right-hand side are. Moreover, DNs¼ ðYðs;DLsÞ 1ÞIfDLsa0g and Yðs;0Þ ¼1 yieldsfð1þDNsÞ ¼fðYðs;DLsÞÞIfDLsa0gand therefore
V¼ ðZfðYÞÞ mL¼ jZfðYÞj mL;
sinceZfðYÞX0:BecauseVis locallyP-integrable, we obtain from JS[12, Proposition II.1.28]that
ðZfðYÞÞ mL¼ ðZfðYÞÞ ðmLnPÞ þ ðZfðYÞÞ nP; soðZfðYÞÞ nPis theP-compensator ofVand we end up with
This is now in fact the canonical decomposition since the first term is a local P-martingale and the second is predictable and of finite variation.
AsA0andA00are both nonnegative, the final assertion follows if we prove thatA
t
is P-integrable for each t2T: But ZlogZ is a P-submartingale with ZtlogZt 2
L1ðPÞ since ItðQjPÞo1; and so the family fZtlogZtjtptis a stopping timeg is uniformly integrable becausee1pZ
tlogZtpEP½ZtlogZtjFt:ThusðZlogZÞtis
a P-submartingale of class ðDÞ and so the increasing process in its unique
Doob–Meyer decomposition is P-integrable. By uniqueness, ðZlogZÞt ¼MtþAt and thereforeEP½At ¼EP½At1o1: &
The next result provides us with a number of important integrability properties.
Lemma 12. For QlocP with finite entropy process and Girsanov parametersb;Y;the
following random variables are Q-integrable for all t2T:
ðaÞ Z t 0 b> scbsds; ðbÞ Z t 0 jbsjds; ðcÞfðYÞ nPt; ðdÞ Z t 0 Yðs;xÞds for x2suppK:
Moreover,the entropy process of Q with respect to P is explicitly given by
ItðQjPÞ ¼ 1 2EQ Z t 0 ðbsÞ>cbsds þEQ½fðYÞ nPt: (3.6)
Proof. (a) The quadratic variationhNcit¼R0tb>scbsdsis the same underPandQ.
Hence Lemma I.3.12 of JS[12]and Lemma 11 give EQ½hNcit ¼EP½ZthNcit ¼EP Z t 0 ZsdhNcis ¼2EP½A0to1:
(b) Let r¼rankðcÞ and lj be the eigenvalues of c, numbered such that lj¼0 exactly forj4r:Chooseb as in Remark NV so thatbs¼Sgs withgjs¼0 forj4r:
Then b> scbs¼g>sc~gs¼ Pr j¼1l jjgj sj2 and b i s¼ Pr j¼1S ijgj s so that Rt 0jb i sjdsp Pr j¼1jS ijj Rt 0jg j
sjds: Hence it suffices to show that
Rt
0jg
j
sjds is Q-integrable, and this follows
from part (a) since
EQ 1 t Z t 0 jgjsjds 2 pEQ 1 t Z t 0 jgjsj2ds pconst:EQ Z t 0 Xr j¼1 ljjgjsj2ds " # ¼const:EQ Z t 0 b> scbsds :
(c) As in part (a), Lemma I.3.12 of JS and Lemma 11 yield EQ½fðYÞ nPt ¼EP Zt Z t 0 Z Rd fðYðs;xÞÞKðdxÞds ¼EP Z t 0 Zs Z Rdf ðYðs;xÞÞKðdxÞds ¼EP½ðZfðYÞÞ nPt ¼2EP½A00to1:
(d) Since EQ½RRd½0;tfðYÞKðdxÞds ¼EQ½fðYÞ nPto1 by part (c), we obtain
EQ½R0tfðYðs;xÞÞdso1 for x2suppK by Fubini’s theorem. Because f is convex,
Jensen’s inequality yields
f EQ 1 t Z t 0 Yðs;xÞds pEQ 1 t Z t 0 fðYðs;xÞÞds o1 forx2suppK; and asfðyÞo1impliesyo1;the assertion follows.
To obtain (3.6), note thatZlogZ¼MþAandMt is a uniformly integrable P-martingale by the last argument in the proof of Lemma 11. So parts (a) and (c) give
ItðQjPÞ ¼EP½ZtlogZt ¼EP½At ¼ 1 2EQ Z t 0 ðbsÞ>cbsds þEQ½fðYÞ nPt: &
Now we can prove that relative entropy is reduced by averaging Girsanov parameters.
Theorem 13. Suppose that QlocP with ITðQjPÞo1for some T 2 ð0;1Þ;and define
b‘¼ 1 T EQ Z T 0 bQs ds ; Y‘ðxÞ ¼ 1 T EQ Z T 0 YQðs;xÞds for x2suppK:
(a) There exists a probability measure Q‘P on FT0 with Girsanov parametersb
‘
and Y‘;which satisfies I
T0ðQ
‘jPÞpI
T0ðQjPÞ;and such that the restriction of L to
the interval½0;T0is a Q‘-Le´vy process.
(b) LetO¼Dd withF ¼BðDdÞ;coordinate process L andF¼FL:Then there exists a
probability measure Q‘locP onðO;FÞwith Girsanov parametersb‘ and Y‘;which
satisfies ITðQ‘jPÞpITðQjPÞ;and such that L is a Q‘-Le´vy process on½0;1Þ:
(c) For Q‘ constructed as above, ITðQ‘jPÞ ¼ITðQjPÞ if and only if both bQ: ¼b‘
Pl-a.e. on O ½0;T and YQð;xÞ ¼Y‘ðxÞ Pl-a.e. on O ½0;T; for all
x2suppK;i.e., if and only if L is a Q-Le´vy process on½0;T:
Proof. By Lemma 12, b‘ and Y‘ are well-defined, and Corollary 8 yields the existence of Q‘ with Girsanov parameters b‘;Y‘ and the Q‘
forLbecause Z Rd fðY‘ðxÞÞKðdxÞp Z Rd EQ 1 T Z T 0 fðYQðs;xÞÞds KðdxÞ ¼ 1 TEQ½fðY QÞ nP To1 ð3:7Þ
by the definition of Y‘; Jensen’s inequality, Fubini’s theorem for nonnegative functions and part (c) of Lemma 12. Moreover, (3.6) gives
ITðRjPÞ ¼ 1 2ER Z T 0 ðbRsÞ>cbRs ds þER½fðYRÞ nPT forR2 fQ;Q‘g;
and we claim that
EQ Z T 0 ðbQsÞ>cbQs ds XE Q‘ Z T 0 ðb‘Þ>cb‘ds ¼Tðb‘Þ>cb‘; (3.8) EQ½fðYQÞ nPTXEQ‘½fðY‘Þ nPT ¼T Z RdfðY ‘ðxÞÞKðdxÞ; (3.9)
with equality if and only ifbQ: ¼b‘ Pl-a.e. onO ½0;Tand YQð;xÞ ¼Y‘ð;xÞ
Pl-a.e. on O ½0;T; for all x2suppK: For brevity, we omit to say ‘‘on
O ½0;T’’ below.
Now (3.9) is simply (3.7); sincefis strictly convex, equality holds if and only if we haveYQð;xÞ ¼Y‘ð;xÞPl-a.e., for allx2suppK:For the proof of (3.8), we use the notation of Remark NV and define g~s¼
ffiffiffi
~
c
p
S>bQs so that ðbQsÞ>cbQs ¼ j~gsj2:
Jensen’s inequality then givesT1R0Tj~gsj2dsXjT1
RT 0 g~sdsj2 and therefore EQ 1 T Z T 0 ðbQsÞ>cbQs ds XEQ 1 T Z T 0 ~ gsds 2 " # X 1 TEQ Z T 0 ~ gsds 2 ;
equality holds if and only ifg~(or, equivalently,bQ) is constantPl-a.e. But 1 TEQ Z T 0 ~ gsds ¼pffiffiffic~S> 1 TEQ Z T 0 bQs ds ¼pffiffiffic~S>b‘ by the definitions of g~ and b‘ and therefore EQ½R0TðbQsÞ
>
cbQs dsXTðb‘Þ>cb‘; with equality if and only ifbQ: ¼b‘ Pl-a.e. This proves (b) and (c). To obtain (a), we argue with T ¼T0 if IT0ðQ
‘jPÞo1; otherwise, we use (3.6) to get IT0ðQ
‘jPÞ ¼T0 T ITðQ
‘jPÞo1: &
Using the description of the localQ-martingale property ofULin Proposition 10 yields
‘‘Pseudo-Proposition 14’’. Suppose Q‘ is constructed from Q as in Theorem13.If UL
is a local martingale under Q,it is still a local martingale under Q‘:
‘‘Pseudo-Proof’’. By construction, the Girsanov parameters of Q‘ are obtained by averaging those ofQ. But the local martingale property ofULis characterized by the
linear constraint (2.8) between Girsanov parameters, and this is preserved by averaging. &
We have put ‘‘Pseudo-Proposition 14’’ and its ‘‘Pseudo-Proof’’ in quotation marks because they are not necessarily true as they stand. More precisely, we need Fubini’s theorem to prove that (2.8) is preserved by averaging, and this requires the additional assumption (onQ) thatEQ½jUðxY hÞj nPTo1:Hence the subsequent
‘‘Pseudo-Proof’’ of the next result is also flawed. Nevertheless, Theorem A itself is true, and we shall provide a proper proof in Section 5. The current presentation has been chosen to highlight the key idea behind the argument.
Theorem A. Let L be a P-Le´vy process for F¼FL; and U a fixed dd-matrix.
Suppose thatQUeðLÞ \QUf ðLÞ \QU‘ðLÞa;:If QEðULÞexists,then L is a Le´vy process
under QEðULÞ:
‘‘Pseudo-Proof’’. For brevity, write QE forQEðULÞ: If the assertion were not true, we could use Theorem 13 to construct ðQEÞ‘ which would be a local martingale measure for UL by ‘‘Pseudo-Proposition 14’’ and satisfy ITððQEÞ‘jPÞoITðQEjPÞ
for some T 2 ð0;1Þ by part (c) of Theorem 13, in contradiction to the optimality ofQE: &
4. Identifying the minimal entropy martingale measure
In this section, we give a very explicit representation for the MEMMQEðULÞ;and one important point is to make transparent where this comes from. We have seen in Theorem A that QEðULÞ;if it exists, preserves the Le´vy property of L. Instead of minimizing relative entropy over all ELMMs forUL, it should thus be sufficient to minimize only over those which in addition preserve the Le´vy property ofL. (That this is indeed enough is proved in Corollary 20 in Section 5.) We use this intuition to derive by partly formal arguments a candidate forQEðULÞ;and then we prove that this candidate gives indeed the optimal measure. For simplicity, we give the derivation for the caseL¼UL whereUis the identity matrix, and for brevity, we often writeQE forQEðULÞandQUs forQUs ðLÞ;wheres2 fa;e;f; ‘g:
To find a candidate forQE;we start with anyQinQeU\QUf \QU‘ because this is where QE should lie. Since Q is in QU‘; it has deterministic time-independent Girsanov parametersb2Rd andY :Rd ! ð0;1Þ:AsQ2QUf ;(3.6) gives an explicit expression for ItðQjPÞ in terms ofb;Y; and asQ2QUe; the martingale condition
(2.8) or (2.9) relatesbandYby
cb¼ b
Z
Rd
ðxYðxÞ hðxÞÞKðdxÞ ¼:bkðYÞ: (4.1) If we takecregular for simplicity, we can solve (4.1) forband plug into (3.6) to get
ItðQjPÞ ¼ 1 2ðbþkðYÞÞ >c1ðbþkðYÞÞ þZ Rd fðYðxÞÞKðdxÞ t¼:¯IðYÞt: (4.2)
As explained intuitively in Section 2.4, we have now parametrizedQby a functionY and want to minimize the functional¯IðYÞ:IfYis optimal, we obtain for anyYand alle40
0p¯IðYþeðYYÞÞ ¯IðYÞ
¼ Z ðfðYþeðYYÞÞ fðYÞÞdK þe Z xðYYÞdK > c1 bþ Z ðxYhÞdK þ1 2e 2 Z xðYYÞdK>c1 Z xðYYÞdK
by using (4.2) and the expression forkðYÞ:Now divide byeand letetend to 0 to get 0p Z f0ðY ÞðYYÞdKþ Z xðYYÞdK > c1 bþ Z ðxYhÞdK for allY:
The particular choiceY:¼ ð1dÞY withd40 leads to
0¼ Z xYdK > c1 bþ Z ðxYhÞdK þ Z f0ðYÞYdK ¼ Z ðb>xþlogYÞYdK; whereb ¼bðYÞ ¼ c1ðbþ R
ðxYhÞdKÞis the optimalbfrom the martingale condition (4.1) and we have used f0ðyÞ ¼
logy: As Y40; we thus should have logYðxÞ b>x¼0 or
YðxÞ ¼eb>x ðat least on the support of KÞ:
Hence the optimal measure Q should have Girsanov parameters b
¼u and YðxÞ ¼eu>
x for some u 2Rd which must be determined from the martingale
condition
bþcuþ
Z
Rd
ðxeu>xhðxÞÞKðdxÞ ¼0:
This recipe gives our candidate forQE:To make it even more explicit, we define as in Corollary 6 Z:¼Eðu>
L
cþ ðY1Þ ðmLnPÞÞ and find by formal calculations
thatZ
t ¼expðu>L
c
t þ ðu>xÞ mLt const:tÞwhich suggests that the density process
ofQE should be of the form ZQtE ¼const:ðtÞeu >
Lt:Hence we expectQE to be a so-called Esscher measure.
To explain this more carefully, we start with a P-Le´vy process L with P-Le´vy characteristicsðb;c;KÞand fix add-matrixU. We define
A:¼ fu2RdjEP½eu >L
and recall from Theorem 25.17 of Sato[18]that CðuÞ:¼b>uþ1 2u > c uþ Z Rd ðeu>x1 ðu>xÞIfjxjp1gÞKðdxÞ is well-defined onAand thatEP½eu
>L
t ¼etCðuÞ foru2A:Due to the Le´vy structure of LunderP, it is easy to see thatZu
t:¼expðu>LttCðuÞÞis a strictly positive
P-martingale and therefore the density process of a measureQulocPonðDd;BðDdÞÞby Lemma 18.18 of Kallenberg[13]. Any suchQu is calledEsscher measure(for L with
parameter u). IfQu is in addition a martingale measure forUL, we callQu Esscher
martingale measure for UL.
The next result collects some simple properties of Esscher measures.
Lemma 15. Fix u2Aand let Qube an Esscher measure with parameter u. Then:
(a) L is a Le´vy process under Qu:
(b) The Girsanov parameters of Qu are given bybu¼u;YuðxÞ ¼eu>x
:
(c) If Quis in addition an Esscher martingale measure for UL and u2rangeðU>Þ;the
entropy process of Qu is finite-valued and given by ItðQujPÞ ¼ tCðuÞ for all
t2T:
Proof. (a) See Shiryaev[19], Theorem VII.3c.1.
(b) Ifb;Y are the Girsanov parameters ofQu;part (a) implies thatbis a constant andY ¼YðxÞis a deterministic function. Proposition 3 yields
Zut ¼EðNuÞt ¼Eðb>Lcþ ðY1Þ ðmLnPÞÞt
and the explicit formula for the stochastic exponential gives logZut ¼b>Lc t 1 2b >cbtþ ðY1Þ ðmLnPÞ tþ X spt ðlogð1þDNusÞ DNusÞ ¼u>LttCðuÞ
by the definition ofZu:Comparing the continuous local martingale parts of the two
representations yields b¼u; and since DNut ¼YðDLtÞ 1; comparing the jumps
impliesu>DL
t ¼logYðDLtÞso that we getYðxÞ ¼eu >x
on the support ofK. (c) Writeu¼U>u:~ By part (a) and Proposition 1,ULis both a Le´vy process and a local martingale underQuand hence a trueQu-martingale. Becauseu2A;this gives
ItðQujPÞ ¼EQu½logZut ¼EQu½u~>ULttCðuÞ ¼ tCðuÞo1: &
To prove that our candidate is indeed optimal, we use the following Le´vy version of Proposition 3.2 in Grandits and Rheinla¨nder [9]. It tells us that the Esscher martingale measure forULis optimal inQU‘ if it exists. Note that we do not assume thatQE exists.
Lemma 16. Let L be a P-Le´vy process. If there exists an Esscher martingale measure
Qu for UL with u2rangeðU>Þ;then ItðQujPÞpItðRjPÞfor all R2QU‘ and for all
Proof. From part (c) of Lemma 15, we know that ItðQujPÞ ¼ tCðuÞo1: Write
u¼U>u~and fixR2QU
‘ :ThenULis underRa Le´vy process and a local martingale, hence a true martingale, and because relative entropy is nonnegative, we get as in the proof of Lemma 15
ItðRjPÞ ¼ItðRjQuÞ þER½logZut ¼ItðRjQuÞ tCðuÞ
XtCðuÞ ¼ItðQujPÞ: &
Now we can prove that the heuristically derived recipe for our candidate produces indeed the minimal entropy martingale measureQEðULÞ:
Theorem B. Let L be a P-Le´vy process forF¼FLwith Le´vy characteristicsðb;c;KÞ;
and U a fixed dd-matrix. Suppose that there exists u2rangeðU>Þsuch that
Z Rd jxeu>xhðxÞ jKðdxÞo1; (4.3) U bþcuþ Z Rd ðxeu>xhðxÞÞKðdxÞ ¼0: (4.4)
Then both the Esscher measure Qu and the minimal entropy martingale measure
QEðULÞexist and coincide.
Proof. Existence of Qu follows if we show that u2A;and by Theorem 25.17 of
Sato[18], this holds if and only ifRfjxj41geu>
xKðdxÞo1:But withh
0ðxÞ:¼ jxjIfjxjp1g;
we easily getjh0ðxÞ hðxÞjpconst:ð1^ jxj2Þand therefore
Z fjxj41g eu>xKðdxÞp Z fjxj41g jxjeu>xKðdxÞ þ Z fjxjp1g jxðeu>x1Þ jKðdxÞ ¼ Z Rd jxeu>xh 0ðxÞ jKðdxÞ pZ Rd jxeu>xhðxÞ jKðdxÞ þ Z Rd jh0ðxÞ hðxÞ jKðdxÞo1
by (4.3) and the properties ofK. By part (b) of Lemma 15, the Girsanov parameters ofQu areb
¼u andYðxÞ ¼eu
>
x:Hence (4.3) and (4.4) are the conditions from
Proposition 10 for UL to be a local Qu-martingale so that Qu2QU
e \QUf \QU‘ by part (c) of Lemma 15. Lemma 16 implies that Qu has minimal entropy
among all Q2QU
e \QUf \QU‘ so that QE‘ðULÞ exists and coincides with Qu: But then Corollary 20 below implies that QEðULÞ exists as well and QEðULÞ ¼
QE‘ðULÞ ¼Qu: &
Remark. The derivation of our candidate for QEðULÞ suggests in particular that finding the MEMM for a Le´vy process can be reduced to adeterministic optimization
the functional ^ Iðb;YÞ:¼1 2b > cbþ Z Rd fðYðxÞÞKðdxÞ
which by (3.6) equals I1ðQjPÞ for the measure Q loc
P with Girsanov parameters
b;Y:Denote byHthe class of all pairs ðb;YÞsatisfying Z Rdf ðYðxÞÞKðdxÞo1; (4.5) Z Rd jxYðxÞ hðxÞjKðdxÞo1; (4.6) U bþcbþ Z Rd ðxYðxÞ hðxÞÞKðdxÞ ¼0: (4.7)
By Corollary 8, (4.5) is the condition for the existence of Q with I1ðQjPÞo1;
whereas (4.6) and (4.7) come from the martingale condition in Proposition 10. If we set YuðxÞ:¼eu>
x for u2Rd; purely analytic arguments show that if there is some
u2rangeðU>Þ withðu;YuÞ 2H;then I^ðu;YuÞpI^ðb;YÞ for all ðb;YÞ 2H:The
crucial point is to prove 0p1 2ðbuÞ >cðbuÞ þ Z Rd YuðxÞf YðxÞ YuðxÞ KðdxÞ ¼ 1 2b > cbþ Z Rd fðYðxÞÞKðdxÞ 1 2u > cuþ Z Rd fðYuðxÞÞKðdxÞ ;
where the first inequality is obvious and the second corresponds to the probabilistic argument in the proof of Lemma 16. For details, we refer to Section 4.3 of Esche[4].
5. A proper proof of Theorem A
In this section, we give a rigorous proof of Theorem A.Throughout the section,L is
a P-Le´vy process forF¼FLwith Le´vy characteristicsðb;c;KÞ;and U is a fixed dd
-matrix. The basic idea is the assertion of ‘‘Pseudo-Proposition 14’’ that the local
martingale property of UL under Qis preserved under an averaging of Girsanov parameters. However, we can rigorously prove this only if the big jumps ofULare
Q-integrable. To make this precise, we define for a semimartingale Xa new set of
martingale measures by
QUintðXÞ:¼ fQ2QUaðXÞ jEQ½jUðxYhÞj ntPo1for allt2Tg
and writeQU
s ¼QUs ðLÞfors2 fa;e;f;int; ‘g:As pointed out after Lemma 9,Qbeing
inQU
int is equivalent toQ-integrability ofjxhðxÞj mUL;the sum over large jumps
of UL, if Q has a finite entropy process. But for proof purposes, the above
Remark. AnyQ2QUe \QfU\QU‘ is also inQUint:In fact, part (c) of Lemma 12 and the proof of Proposition 10 show thatjUðxYhÞj nPis finite-valued; this uses only
Q2QUe \QUf :If alsoQ2Q‘U;thenYis deterministic, hence so isjUðxYhÞj nP;
and then finiteness is the same as Q-integrability.
The correct version of ‘‘Pseudo-Proposition 14’’ is now
Proposition 17. Let Q2QU
e \QUf \QUint with Girsanov parameters b;Y: Then Q‘
constructed from Q in Theorem13is inQUe \QUf \QU‘ so that UL is still a local Q‘
-martingale.
Proof. Letb‘;Y‘be the Girsanov parameters ofQ‘:Theorem 13 givesQ‘locP;that L is a Q‘-Le´vy process and ITðQ‘jPÞo1 for some T 2 ð0;1Þ: Since b‘;Y‘ are
deterministic and time-independent, this implies ItðQ‘jPÞ ¼tI1ðQ‘jPÞo1 for all
t2T and it only remains to show thatULis a localQ‘-martingale. By Proposition 10, we need to verify thatRRdjUðxY‘ðxÞ hðxÞÞ jKðdxÞo1and that b‘;Y‘ satisfy the martingale condition (2.8).
Using the definition ofY‘;Jensen’s inequality and Fubini’s theorem yields Z Rd jUðxY‘ðxÞ hðxÞÞ jKðdxÞ ¼ Z Rd EQ 1 T Z T 0 UðxYðs;xÞ hðxÞÞds KðdxÞ p1 T EQ½jUðxYhÞj n P To1
since Q2QUint: This allows us now to use Fubini’s theorem forUðxYðs;xÞ hðxÞÞ
and combine this with (2.8) forb;Y to conclude
U bþcb‘þ Z Rd ðxY‘ðxÞ hðxÞÞKðdxÞ ¼1 TEQ Z T 0 U bþcbsþ Z RdðxYðs;xÞ hðxÞÞKðdxÞ ds ¼0 so thatb‘;Y‘ satisfy the martingale condition forUL as well. &
If Qis not in QUint; we do not know if Q‘ from Theorem 13 preserves the local
martingale property ofUL. The key idea for using Proposition 18 in a proper proof of Theorem A is thus to argue that the martingale measures inQU
e \QUf \QUint are
dense in the set of all martingale measures QU
e \QUf in a suitable sense. This is
achieved by
Proposition 18. Let Q2QUe \QUf and supposeQeU\QUf \QU‘a;:Then there exists a sequenceðQnÞn2NinQUe \QUf \QUintwithlimn!1ItðQnjPÞ ¼ItðQjPÞfor all t2T: Proof. Choose Q¯ 2QU
e \QUf \Q‘U so that Q¯ 2QUint by the remark before
Proposition 17. Denote byb;Y the Girsanov parameters ofQand write the density process as ZQ¼EðNÞ with N¼RbdLcþ ðY1Þ ðmLnPÞ by Proposition 3. Analogous quantities with a bar-refer to Q:¯ Because UL is a local Q-martingale,
Q-integrable with localizing sequenceðtnÞn2N:To constructQn which agrees withQ
up totn and withQ¯ afterwards, we define forn2N bns ¼bsI10;tnUþb¯IUtn;11;
Ynðs;xÞ ¼Yðs;xÞI10;tnUþY¯ðxÞIUtn;11;
and setNn¼RbndLcþ ðYn1Þ ðmLnPÞ andZn¼EðNnÞ: It is straightforward to check thatNn¼NtnþN¯ N¯tn andZn¼ZI1
0;tnUþ Ztn ¯ Ztn ¯ ZIUtn;11:Hence Zn is a strictly positive martingale starting at 1 and there existsQnlocPwith density process Zn: (For T ¼ ½0;1Þ; we work on the path space Dd as usual.) It follows from Proposition 7 thatbn;Yn are the Girsanov parameters of Qn; and Qn¼Q on Ftn sinceZntn ¼Ztn:We claim thatQ
n is a local martingale measure forUL withQn2
QUint:In fact, the definition ofYn yields
jUðxYnhÞj nPtpjUðxYhÞj nPt þ jUðx ¯YhÞj nPto1
Qn-a.s. for allt2T
by Proposition 10 sinceQ;Q¯ 2QUe;andbn;Yn satisfy (2.8) by construction so that Qnis inQUe as well. Moreover, usingQn¼QonFtn;the fact thatY¯ is deterministic and time-independent, andR0tIUtn;11dsptyields
EQn½jUðxYnhÞj nP t pEQ½jUðxYhÞj nPtn þt Z Rd jUðx ¯YðxÞ hðxÞÞjKðdxÞo1
by the choice oftn and sinceQ¯ 2QUint:HenceQn is inQUe \QUint as claimed above.
It remains to show that each Qn is in QU
f and the convergence of ItðQnjPÞ to
ItðQjPÞ:From Lemma 12, we know that
ItðRjPÞ ¼ 1 2ER Z t 0 ðbRsÞ>cbRs ds þER½fðYRÞ nPt forR2 fQ;Q ng; (5.1)
and becauseQn¼Qon FtnandtnisFtn-measurable, we get from the definition of
bn that EQn Z t 0 ðbnsÞ>cbnsds ¼EQ Z t 0 b>scbsI10;tnUðsÞds þb¯>c ¯bEQ½ðttnÞþ !EQ Z t 0 b>scbsds
by monotone convergence sincetn" 1Q-a.s. In the same way, the definition ofYn
yields EQn½fðYnÞ nP t ¼EQ Z t 0 Z Rdf ðYðs;xÞÞI10;tnUðsÞKðdxÞds þ Z Rd fðY¯ðxÞÞKðdxÞEQ½ðttnÞþ
!EQ Z t 0 Z Rdf ðYðs;xÞÞKðdxÞds ¼EQ½fðYÞ nPt
by monotone convergence, and in view of (5.1), this completes the proof. &
The next result shows that for the approximating martingale measures in Proposition 18, we also get convergence of entropies for the corresponding ‘‘Le´vyfied’’ measures.
Proposition 19. In the setting of Proposition18,let Q‘and Qn;‘¼ ðQnÞ‘be constructed
as in Theorem 13 for some T 2 ð0;1Þ: Then limn!1ItðQn;‘jPÞ ¼ItðQ‘jPÞ for all
t2T:
Proof. SinceQ‘;Qn;‘have deterministic and time-independent Girsanov parameters, ItðRjPÞ ¼ 1 2ðb RÞ> cbRþ Z Rd fðYRðxÞÞKðdxÞ t forR2 fQ‘;Qn;‘g
by Lemma 12 and so it is enough to prove thatbn;‘!b‘ andRRdfðYn;‘ðxÞÞKðdxÞ ! R
RdfðY
‘ðxÞÞKðdxÞ:
Denote by b;Y and bn;Yn the Girsanov parameters of Q and Qn: By the
construction ofbn;‘ andbn;and sinceQn¼Qon Ftn andtn isFtn-measurable, we have bn;‘¼EQn 1 T Z T 0 bnsds ¼EQ 1 T Z T 0 bsI10;tnUðsÞds þ1 T b¯EQ½ðTtnÞ þ !EQ 1 T Z T 0 bsds ¼b‘
by monotone convergence for the second and dominated convergence for the first term, because jR0TbsI10;tnUðsÞdsjp
RT
0 jbsjds2L1ðQÞ by part (b) of Lemma 12. In
the same way, we obtainfðYn;‘ðxÞÞ !fðY‘ðxÞÞfor all x2suppK by using part (d) of Lemma 12 and continuity off. To find a K-integrable dominating function for fðYn;‘ðxÞÞ;we use the definition ofYn;‘;Jensen’s inequality for the convex function fX0;the definition ofYn;and again thatQn¼QonFtnandFtn-measurability oftn to obtain fðYn;‘ðxÞÞpEQn 1 T Z T 0 fðYnðs;xÞÞds ¼EQ 1 T Z T 0 fðYðs;xÞÞI10;tnUðsÞds þ1 TfðY¯ðxÞÞEQ½ðTtnÞ þ pEQ 1 T Z T 0 fðYðs;xÞÞds þfðY¯ðxÞÞ:
ButQandQ¯ both have finite relative entropy and since fX0;we can use Fubini’s
theorem and part (c) of Lemma 12 to get Z RdEQ Z T 0 fðYðs;xÞÞds KðdxÞ ¼EQ Z Rd Z T 0 fðYðs;xÞÞds KðdxÞ ¼EQ½fðYÞ nPTo1:
In the same way, we get K-integrability of fðY¯ðxÞÞ:Hence dominated convergence yieldsRRdfðYn;‘ðxÞÞKðdxÞ !
R
RdfðY‘ðxÞÞKðdxÞ;and this completes the proof. & Now we can finally prove Theorem A which we recall for the convenience of the reader.
Theorem A. Let L be a P-Le´vy process for F¼FL; and U a fixed dd-matrix.
Suppose thatQUeðLÞ \QUf ðLÞ \QU‘ðLÞa;:If QEðULÞexists,then L is a Le´vy process
under QEðULÞ:
Proof. For brevity, we write QE forQEðULÞ:IfLis not a Le´vy process underQE; there exists T2 ð0;1Þ such that L is not a QE-Le´vy process on ½0;T: For the measure QE;‘¼ ðQEÞ‘ obtained from Theorem 13, we then have ITðQE;‘jPÞoITðQEjPÞ:However, this is not yet a contradiction to the optimality
of QE; we do not know whether UL is a local martingale under QE;‘ since QE is perhaps not inQUint:But ifðQE;nÞn2Nis the sequence inQUe \QUf \QUint forQE from Proposition 19 andQE;n;‘¼ ðQE;nÞ‘are the corresponding Le´vy martingale measures forUL obtained from Theorem 13, Proposition 19 yields
lim
n!1ITðQ
E;n;‘jPÞ ¼I
TðQE;‘jPÞoITðQEjPÞ:
So fornsufficiently large we haveITðQE;n;‘jPÞoITðQEjPÞandQE;n;‘2QUe \Q U f by
Proposition 18 which is the desired contradiction. &
In view of Theorem A, it seems clear that we should be able to find QEðULÞby minimizing relative entropy only overLe´vymartingale measures. This is indeed true:
Corollary 20. Let L be a P-Le´vy process for F¼FL; and U a fixed dd-matrix.
Suppose thatQUeðLÞ \QUf ðLÞ \QU‘ ðLÞa;:If QE‘ðULÞ exists, then QEðULÞexists as
well and coincides with QE‘ðULÞ:In particular,we have QE‘ðULÞ locP:
Proof. We again omit writing ðLÞand ðULÞ for brevity. IfQE exists, it is in QUe \
QUf \QU‘ by Theorem A. Then we must have QE ¼QE‘; and it also follows from Theorem 2.2 of Frittelli[7]thatQElocP:
SupposeQE does not exist. Then there is someT 2 ð0;1Þand someQ2QU f with
ITðQjPÞoITðQE‘ jPÞ: Since Q U e \Q
U
f a; and ITð jPÞ is convex, we may assume
that Q2QUe as well (otherwise replace Q by ð1eÞQþeQ0 for some
Q02QU
e \Q U
f ). Construct Q
‘ from Q via Theorem 13, the sequence ðQnÞ n2N
Proposition 19 yields lim
n!1ITðQ
n;‘jPÞ ¼I
TðQ‘jPÞpITðQjPÞoITðQE‘ jPÞ and thus ITðQn;‘jPÞoITðQE‘ jPÞ for large n. But since Q
n;‘2QU e \Q U f \Q U ‘ ; this contradicts the optimality ofQE‘;and soQE does exist. &
Remark. Theorem A implies that in order to determineQEðULÞit suffices to find a martingale measure which is optimal inQUf \QU‘ ;and Corollary 20 shows that this measure must be locally equivalent to P. Hence we have to look for the optimal measure in QUe \QUf \QU‘ so that the assumption QUe \QUf \QU‘ a; is entirely natural.
Acknowledgements
Financial support by the Deutsche Forschungsgemeinschaft through the Graduiertenkolleg ‘‘Stochastische Prozesse und probabilistische Analysis’’ at the Technical University of Berlin is gratefully acknowledged. MS also thanks Jan Kallsen for some useful comments.
Appendix A
This section collects a number of proofs omitted from the body of the paper for better reading.
Proof of Proposition 5. By Lemma 4 and sincefðY¯Þ nPX0;we have for allt2T
gðY¯Þ nPtpfðY¯Þ nPtpexp Z t 0 1 2b¯ > sc ¯bsþ Z Rd fðY¯ðs;xÞÞKðdxÞ ds : (6.1) Soð1pffiffiffiffiY¯Þ2nPis locallyP-integrable by (2.3), and JS[12, Theorem II.1.33]gives the integrability ofY¯ 1 with respect tomLnP:By (2.3),Rb¯>sc ¯bsdsis also locally
P-integrable so thatb¯ is integrable with respect toLcandN¯ is well-defined. SinceN¯
is a local P-martingale and (2.4) is its decomposition into continuous and purely discontinuous parts, DN¯t¼ ðY¯ðt;DLtÞ 1ÞIfDLta0g41 P-a.s. since Y¯40: Hence
¯
Z¼EðN¯Þ is a strictly positive local P-martingale, and a true P-martingale if EP½EðN¯Þt ¼1 for every bounded stopping timet:But iftpt0for some deterministic
t02 ð0;1Þ;thenEðN¯Þt¼EðN¯ t0
ÞtandM:¼N¯t0
is again a localP-martingale null at 0 withDM41:So if we defineAby At:¼ 1 2hM ci tþ X spt ðð1þDMsÞlogð1þDMsÞ DMsÞ fortpt0
and show thatAadmits a predictableP-compensatorBwithEP½expðBt0Þo1;then
integrable P-martingale and therefore EP½EðN¯Þt ¼EP½EðMÞt ¼1; which will end the proof.
To find theP-compensatorBofA, note thathN¯ci ¼ hRb¯dLci ¼Rb¯>
sc ¯bsdsso that At ¼ 1 2 Z t 0 ¯ b>sc ¯bsdsþX spt ðY¯ðs;DLsÞlogY¯ðs;DLsÞ Y¯ðs;DLsÞ þ1ÞIfDLsa0g ¼ 1 2 Z t 0 ¯ b>sc ¯bsdsþfðY¯Þ mLt fortpt0: Now jfðY¯Þj nP
t ¼fðY¯Þ nPt isP-integrable for allt2T by (6.1) and (2.3), and so
we get from JS [12, Proposition II.1.28] that fðY¯Þ is integrable with respect to
mLnPand that fðY¯Þ ðmLnPÞ ¼fðY¯Þ mLfðY¯Þ nP:HenceBt¼12R0tb¯
>
sc ¯bsds þfðY¯Þ nPt is the P-compensator of A, and we have E½expðBt0Þo1 by
assump-tion (2.3). &
Proof of Proposition 7. By assumption, the density process ZQ¯ ¼EðN¯Þis a strictly
positive P-martingale. On the other hand, Proposition 2 gives us a predictable function Y^X0 and a predictable process b^ with Rt
0b^
>
scb^sdso1 and jðY^ 1Þhj nP
to1 P-a.s. for all t2T; and we know that Z
¯
Q¼EðNQ¯Þ with NQ¯ ¼Rb^ sdLcsþ ðY^ 1Þ ðmLnPÞ by Proposition 3. So since EðNQ¯Þ ¼EðN¯Þ40;we have NQ¯ ¼N¯
or, equivalently, V1t:¼
Z t
0
ðb¯sb^sÞdLcs¼ ðY^ Y¯Þ ðmLnPÞt¼:V2t; t2T:
As V1 is a continuous and V2 a purely discontinuous local P-martingale, we get V10V2; and this implies b^¼b¯ and Y^ ¼Y¯: In fact, hV1i ¼Rðb¯sb^sÞ>
cðb¯sb^sÞds0 yields ðb¯sb^sÞ>ScS~ >ðb¯
sb^sÞ ¼0 P-a.s. for alls2T;
and becauseb¯andb^are chosen as in Remark NV, this impliesðS>ðb¯
sb^sÞÞj¼0 for
jprankðcÞandðS>b^
sÞj ¼0¼ ðS>b¯sÞj forj4rankðcÞ:Hence we get S>ðb¯sb^sÞ ¼0
and thusb¯s¼b^sP-a.s. for alls2T:Moreover, applying JS[12, Theorem II.1.33]to the square-integrableP-martingale V2 yields
0¼ hV2it¼ ðY^ Y¯Þ2nPt ¼ Z t 0 Z Rd ðY^ðs;xÞ Y¯ðs;xÞÞ2KðdxÞds
P-a.s. for allt2T
so that Y^ðs;xÞ ¼Y¯ðs;xÞnP-a.e.,P-a.s. Thusb¯ andY¯ are the Girsanov parameters
ofQ:¯ &
Proof of Lemma 9. We claim that we have for every truncation function h the estimates
jUhðxÞ hðUxÞj nQt pconst:ðtþfðYÞ nPtÞ; (6.2)
