Comparison of path-independent functions of semimartingales
aa r X i v : . [ m a t h . P R ] A ug Comparison of path-independent functions ofsemimartingales
Benedikt K¨opfer ∗ , Ludger R¨uschendorf The martingale comparison method is extended to derive comparison resultsfor path-independent functions for general semimartingales. Our approach al-lows to dismiss with the Markovian assumption on one of the processes made inprevious literature. Main ingredients of the comparison method are extensionsof the Kolmogorov backwards equation to the non-Markovian case. Puttingthe comparison processes on the same stochastic basis allows by means of Itˆo’sformula applied to the propagation operator to conclude the comparison of theprocesses from the comparison of the semimartingale characteristics.
Mainly motivated by the problem of deriving ordering results for option prices, comparisonresults have been derived in El Karoui et al. (1998), Hobson (1998), Bellamy and Jeanblanc(2000) and Henderson (2005). Gushchin and Mordecki (2002) developed a general ap-proach to comparison results w.r.t. to convex ordering of terminal values between one-dimensional semimartingales and Markovian semimartingales based on the supermartin-gale property of a linking process - the martingale comparison method. Essentially thecomparison of local (differential) semimartigale characteristics and the ’propagation ofconvexity’ property of the Markov process imply convex ordering under the assumptionthat the propagation operator (the value process) of the Markov process satisfies a Kol-mogorov backwards equation. Some extensions of this martingale comparison method aregiven in Bergenthum and R¨uschendorf (2006, 2007a,b, 2008). In particular in these papers ∗ A LGFG grant of the state Baden-W¨urttemberg is gratefully acknowledgedAMS 2010 subject classification: Primary: 60E15; secondary: 60G44, 60G51.Key words and phrases: Path-independent ordering; ordering of semimartingales; Kolmogorov back-wards equation general version of the Kolmogorov backwards equation for Markov processes is establi-hed and extensions to multivariate processes, to further orderings and to some classes ofpath-dependent options are given.In the present paper this approach is generalized allowing to state comparison resultsbetween two general semimartingales. In comparison to the papers mentioned above, we usethe same stochastic basis (cid:0) Ω , F , ( F t ) [0 ,T ] , P (cid:1) for both semimartingales under consideration.This has the advantage that the semimartingale characteristics can be chosen more freely.In the papers above a standing assumption is that one of the processes is a Markov processsuch that the differential characteristics are functions of the space-time process. This isnot necessary if the semimartingales are on the same stochastic basis and we are able tocompare two semimartingales directly. Additionally we do not restrict the characteristicsto be absolutely continuous. So the results can be applied for example to semimartingaleswith fixed jump times.In Section 2 we specify the setting and notation. The basic tool in our paper for theproof of the comparison theorems is an extension of the Kolmogorov backwards equationfor Markov processes in Bergenthum and R¨uschendorf (2006, 2007a) to the case of specialsemimartingales. We give a formulation of these extensions as “functional equations”allowing in principle also applications different from the case of backward equations for thepricing functional.In Section 3 we derive comparison results under equivalent martingale measures (e.m.m.).For two semimartingales X and Y with corresponding e.m.m. Q and Q , we state for anintegrable function f conditions such that E Q [ f ( Y T )] ≤ E Q [ f ( X T )] . The main tool therein is the factorized conditional expectation (propagation operator) of
X G f ( t, x ) := E Q [ f ( X T ) | X t = x ] . For G f in C , we consider the basic linking process G f ( t, Y t ) which allows by Itˆo’s formulato link the semimartingale characteristics of X and Y . In the subsequent Section 4 wederive similar comparison results for two special semimartingales under P .In Section 5, we discuss the assumptions of the main comparison theorems and giveexamples of classes of semimartingales which possess the required regularity properties. Inparticular, we discuss the assumption that G f is of class C , and that G f is convex ordirectionally convex in the second variable. We conclude this paper with some examples. We consider a finite time horizon since we are interested in the comparison of the processesat fixed time points; so we can take this point as final time point. Let ( X t ) t ∈ [0 ,T ] be an2 d valued special semimartingale on a stochastic basis (Ω , F , ( F t ) [0 ,T ] , P ). Further, let X = M + B be the canonical decomposition of X into a local martingale ( M t ) t ∈ [0 ,T ] anda process of finite variation ( B t ) t ∈ [0 ,T ] . The d + 1-dimensional space-time Process ˆ X :=(( t, X t )) t ∈ [0 ,T ] then also is a special semimimartingale; thus we can choose the truncationfunction for the semimartingale characteristics to be the identity, even though it is not atruncation function in the sense of Jacod and Shiryaev (2003), see Rheinl¨ander and Sexton(2011). The local martingale part of the canonical decomposition is (0 , M ) and the finitevariation part is (id , B ). By Jacod and Shiryaev (2003, Proposition II.2.9) there exists apredictable process ˆ A := ( ˆ A t ) t ∈ [0 ,T ] ∈ A + loc such that the semimartingale characteristics ofˆ X are given as Lebesgue–Stieltjes integrals with respect to ˆ A . The characteristics ( ˆ B, ˆ C, ˆ ν )have the “good” form: ˆ B i = ˆ b i · ˆ A, ˆ C ij = ˆ c ij · ˆ A, ˆ ν ( ω, dt, dx ) = d ˆ A t ( ω ) ˆ K ω,t ( dx ) , When there is no danger of confusion, we only write differential characteristics withoutspecifying the integrator process. One candidate process ˆ A is explicitly specified, namelyˆ A = X i ≤ d Var( ˆ B i ) + X i,j ≤ d Var( ˆ C ij ) + ( | x | ∧ ∗ ˆ ν. (2.1)Altogether we obtain the following version of the canonical decomposition of the space-timeprocess ˆ X : ˆ X t = (0 , X ) + (0 , M t ) + ( t, B t ) = ˆ X + (0 , M t ) + ((ˆ b · ˆ A ) t ) . The integral in the last term is understood componentwise. Note that the change from X to ˆ X does not change the semimartingale characteristics of X , they are still containedin the last d dimensions. Only the differential characteristics change because we look fora common integrator. In the theory of Markov processes it is a common procedure toconsider the space-time process. Here the space-time process helps to connect the timederivative and the space derivatives.We start with equations which characterise C , functions of ˆ X which are local mar-tingales. Since we use Itˆo’s formula, we introduce for a function f ∈ C , ( R + × R d ) thefollowing function: H f : R + × R d × R d → R , ( t, x, y ) f ( t, x + y ) − f ( t, x ) − X i ≤ d ∂∂x i f ( t, x ) y i . (2.2)3e write dA for the measure associated to a process of finite variation A . In the sequel weuse the following notation for function classes: F i := { f : R d → R ; f is increasing } , F dcx := { f : R d → R ; f is directionally convex } , F cx := { f : R d → R ; f is convex } , F icx := { f : R d → R ; f is increasing and convex } , F idcx := { f : R d → R ; f is increasing and directionally convex } . In this section we assume first that the semimartingale X under consideration is a localmartingale. Then the canonical decomposition of ˆ X reduces toˆ X = (0 , X ) + (0 , X ) + (ˆ b · ˆ A ) = (0 , X ) + (0 , X ) + (id , . Note that here particularly (ˆ b · ˆ A ) = (id , Proposition 2.1.
Let f ∈ C , ([0 , T ] × R d ) and let X be a local martingale. Assume that:(i) ( f ( t, X t )) t ≥ is a local martingale;(ii) (cid:12)(cid:12) H f (cid:12)(cid:12) ∗ µ X ∈ A + loc .Then the following process is d ˆ A × P almost surely identical zero U t f ( t, X t − ) := ˆ b t ∂∂t f ( t, X t − ) + 12 X i,j ≤ d ˆ c ijt ∂ ∂x i x j f ( t, X t − )+ Z R d H f ( t, X t − , x ) ˆ K t ( dx ) = 0 . (2.3) Proof.
By Itˆo’s formula the local martingale ( f ( t, X t )) t ≥ has the following representation f ( t, X t ) = f (0 , X ) + Z t ∂∂s f ( s, X s − )ˆ b s d ˆ A s + X i ≤ d Z t ∂∂x i f ( s, X s − ) dX is + 12 X i,j ≤ d Z t ∂ ∂x i x j f ( s, X s − )ˆ c ijs d ˆ A s + Z [0 ,t ] × R d f ( s, X s − + x ) − f ( s, X s − ) − X i ≤ d ∂∂x i f ( s, X s − ) x i µ X ( ds, dx ) .
4e compensate the jump integral, which is possible by Assumption ( ii ) and Jacod and Shiryaev(2003, Proposition II.1.28). Denoting M t := X i ≤ d Z t ∂∂x i f ( s, X s − ) dX is ,N t := Z [0 ,t ] × R d f ( s, X s − + x ) − f ( s, X s − ) − X i ≤ d ∂∂x i f ( s, X s − ) x i [ µ X ( ds, dx ) − ˆ K s ( dx ) d ˆ A s ] , the processes ( M t ) t ∈ [0 ,T ] and ( N t ) t ∈ [0 ,t ] are local martingales. As consequence we obtain f ( t, X t ) = f (0 , X ) + Z t ˆ b s ∂∂s f ( s, X s − ) d ˆ A s + M t + N t + 12 X i,j ≤ d Z t ˆ c ijs ∂ ∂x i x j f ( s, X s − ) d ˆ A s + Z [0 ,t ] × R d H f ( s, X t − , x ) ˆ K s ( dx ) d ˆ A s . It follows that the process Z t ˆ b s ∂∂s f ( s, X s − ) + 12 X i,j ≤ d ˆ c ijs ∂ ∂x i x j f ( s, X s − ) + Z R d H f ( s, X t − , x ) ˆ K s ( dx ) d ˆ A s is a predictable local martingale of finite variation starting in zero and is, therefore, almostsurely zero by Jacod and Shiryaev (2003, Corollary I.3.16). Thus, the integrand has to be d ˆ A × P almost surely zero as well.We remark that f ( t, · ) ∈ F cx implies condition (ii) (see Bergenthum and R¨uschendorf(2006)).Next we obtain a similar equation in the case that X is a special semimimartingale.Note that the process ˆ X then is a special semimartingale as well and we have as truncationfunction the identity. The canonical decomposition of ˆ X has the form:ˆ X t = ˆ X + (cid:0) , X ct + x ∗ ( µ X − ν ) t (cid:1) + (ˆ b X · ˆ A ) t . The reason why we demand X to be special is that we then are able to compensate all ofthe jumps appearing in Itˆo’s formula directly. For a general semimartingale the canonicaldecomposition with a truncation function h isˆ X t = ˆ X + (cid:0) , X ct + h ∗ ( µ X − ν ) t (cid:1) + (cid:0) , ( x − h ( x )) ∗ µ X (cid:1) t + (ˆ b X · ˆ A ) t . Hence, an integral with respect to µ X is added in Itˆo’s formula. This makes an additionalassumption necessary. However, this turns the H f term into a term with a truncationfunction which leads to analog proofs; we omit details here.The following proposition is a version of the Kolmogorov backward equation for specialsemimartingales. 5 roposition 2.2. Let f ∈ C , ([0 , T ] × R d ) and let X be a special semimartingale. Assumethat:(i) ( f ( t, X t )) t ≥ is a local martingale;(ii) (cid:12)(cid:12) H f (cid:12)(cid:12) ∗ µ X ∈ A + loc .Then the following process is d ˆ A × P almost surely zero ¯ U t f ( t, X t − ) := ˆ b t ∂∂t f ( t, X t − ) + X i ≤ d ˆ b it ∂∂x i f ( t, X t − ) + 12 X i,j ≤ d ˆ c ijt ∂ ∂x i x j f ( t, X t − )+ Z R d H f ( t, X t − , x ) ˆ K t ( dx ) = 0 . (2.4) Proof.
The proof is similar to the proof of Lemma 2.1. Itˆo’s formula yields f ( t, X t ) = f (0 , X ) + Z t ∂∂s f ( s, X s − )ˆ b s d ˆ A s + X i ≤ d Z t ∂∂x i f ( s, X s − ) dX is + 12 X i,j ≤ d Z t ∂ ∂x i x j f ( s, X s − )ˆ c ijs d ˆ A s + Z [0 ,t ] × R d f ( s, X s − + x ) − f ( s, X s − ) − X i ≤ d ∂∂x i f ( s, X s − ) x i µ X ( ds, dx ) . We compensate the jumps (by Assumption ( ii )) and use the canonical decomposition of X to split the dX term into the local martingale part and the part of finite variation. Weobtain with M t = X ct + x ∗ ( µ X − ˆ ν ) t f ( t, X t ) = f (0 , X ) + Z t ∂∂s f ( s, X s − )ˆ b s d ˆ A s + X i ≤ d Z t ∂∂x i f ( s, X s − )ˆ b is d ˆ A s + X i ≤ d Z t ∂∂x i f ( s, X s − ) dM is + 12 X i,j ≤ d Z t ∂ ∂x i x j f ( s, X s − )ˆ c ijs d ˆ A s + Z [0 ,t ] × R d H f ( s, X s − , x ) h µ X ( ds, dx ) − ˆ K s ( dx ) d ˆ A s i + Z [0 ,t ] × R d H f ( s, X s − , x ) ˆ K s ( dx ) d ˆ A s
6e conclude that Z t ∂∂s f ( s, X s − )ˆ b s + X i ≤ d ∂∂x i f ( s, X s − )ˆ b is + 12 X i,j ≤ d ∂ ∂x i x j f ( s, X s − )ˆ c ijs + Z R d H f ( s, X s − , x ) ˆ K s ( dx ) d ˆ A s is a predictable local martingale of finite variation starting in zero and is therefore almostsurely zero. Hence, the integrand has to be d ˆ A × P almost surely zero as well. In this section we establish comparison results for semimartingales by the martingale com-parison method. Let X = ( X t ) t ∈ [0 ,T ] and Y = ( Y t ) t ∈ [0 ,T ] be semimartingales. Assume thatthere exist equivalent martingale measures Q and Q on (Ω , F , ( F t ) t ∈ [0 ,T ] ) for X and Y each, i.e. X is local martingale under Q and Y is a local martingale under Q . In thesequel we denote semimartingale characteristics with a superscript such that it is clear towhich process they belong. Further, denote by ˆ X := (id , X ) and ˆ Y := (id , Y ) the corre-sponding space-time processes. For an equivalent local martingale measure Q for X and ameasurable function f : ( R d , B ( R d )) → ( R , B ( R )) such that f ( X T ) ∈ L ( Q ) we introducethe pricing functional G (propagation operator) G f ( t, x ) := E Q [ f ( X T ) | X t = x ] . (3.1)Note that in the sequel the semimartingale characteristics of ˆ X are w.r.t. Q , whereas thecharacteristics of ˆ Y are w.r.t. Q . The semimartingale characteristics under the particulare.m.m. can be obtained by the Girsanov theorem, see Jacod and Shiryaev (2003, TheoremIII.3.24).The following directionally convex comparison theorem is an extension of Bergenthum and R¨uschendorf(2006, Theorem 2.3) to non-Markovian semimartingales. Theorem 3.1 (Directionally convex comparison under e.m.m.) . Let
X, Y be semimartin-gales and let X = Y = x ∈ R d almost surely. We consider a function f such that f ( X T ) ∈ L ( Q ) and f ( Y T ) ∈ L ( Q ) . Assume that(i) G f ∈ C , ([0 , T ] × R d ) and G f ( t, · ) ∈ F dcx for all t ∈ [0 , T ] ;(ii) U Xt G f ( t, Y t − ) = 0 holds dA ˆ Y × Q almost surely for all t ∈ [0 , T ] , where U Xt is theoperator defined in (2.3) with the differential semimartingale characteristics of ˆ X under Q in it; iii) (cid:12)(cid:12) H G f (cid:12)(cid:12) ∗ µ Y ∈ A + loc , where H G f is defined in (2.2) ;(iv) ( G f ( t, Y t ) − ) t ∈ [0 ,T ] is of class (DL);(v) A ˆ Y = A ˆ X ;(vi) The differential characteristics are dA ˆ Y × Q almost surely ordered, for all i, j ≤ d : c ˆ Y ijt ≤ c ˆ Xijt , Z R d g ( t, Y t − , x ) K ˆ Yt ( dx ) ≤ Z R d g ( t, Y t − , x ) K ˆ Xt ( dx ) , where the second inequality holds for all g ( t, y, · ) ∈ F dcx such that the integrals exist.Then it holds that E Q [ f ( Y T )] ≤ E Q [ f ( X T )] . If in ( vi ) the inequalities are reversed and ( G f ( t, Y t ) + ) t ∈ [0 ,T ] is of class (DL), then we havethat E Q [ f ( Y T )] ≥ E Q [ f ( X T )] . Proof.
We establish that the linking process ( G f ( t, Y t )) t ∈ [0 ,T ] is a Q -supermartingale. Thisis the key idea of the martingale comparison method. Then the assertion follows from theinequality E Q [ f ( Y T )] = E Q [ G f ( T, Y T )] ≤ G f (0 , x ) = E Q [ f ( X T )] . Since G f ∈ C , ([0 , T ] × R d ), Itˆo’s formula yields that ( G f ( t, Y t )) t ∈ [0 ,T ] is a semimartingalestarting in G f (0 , x ) with decomposition G f ( t, Y t ) = G f (0 , x ) + Z t ∂∂s G f ( s, Y s − ) b ˆ Ys dA ˆ Ys + X i ≤ d Z t ∂∂x i G f ( s, Y s − ) dY is + 12 X i,j ≤ d Z t ∂ ∂x i x j G f ( s, Y s − ) c ˆ Y ijs dA ˆ Ys + Z [0 ,t ] × R d G f ( s, Y s − + x ) − G f ( s, Y s − ) − X i ≤ d ∂∂x i G f ( s, Y s − ) x i µ Y ( ds, dx ) . We compensate the jumps (which is possible because of Assumption ( iii )) and define M t := X i ≤ d Z t ∂∂x i G f ( s, Y s − ) dY is + Z [0 ,t ] × R d H G f ( s, X s − , x ) h µ Y ( ds, dx ) − K ˆ Ys ( dx ) dA ˆ Ys i . G f ( t, Y t ) = G f (0 , x ) + Z t ∂∂s G f ( s, Y s − ) b ˆ Ys dA ˆ Ys + M t + 12 X i,j ≤ d Z t ∂ ∂x i x j G f ( s, Y s − ) c ˆ Y ijs dA ˆ Ys + Z [0 ,t ] × R d H G f ( s, Y s − , x ) K ˆ Ys ( dx ) dA ˆ Ys . To gain the local supermartingale property we show that the process Z = ( Z t ) t ∈ [0 ,T ] definedby Z t := Z t ∂∂s G f ( s, Y s − ) b ˆ Ys + 12 X i,j ≤ d ∂ ∂x i x j G f ( s, Y s − ) c ˆ Y ijs + Z R d H G f ( s, Y s − , x ) K ˆ Yu ( dx ) dA ˆ Yu (3.2)is Q almost surely non-increasing. Therefore, we use Assumption ( ii ) to replace the termwith the time derivative ∂∂s G f ( s, Y s − ) b ˆ Ys . Note that since we consider the semimartingalecharacteristics under the particular e.m.m., we have ( b ˆ Y · A ˆ Y ) t = t = ( b ˆ X · A ˆ X ) t for all t ∈ [0 , T ]. Consequently, we have by Assumption ( v ) that b ˆ Yt dA ˆ Yt = dt = b ˆ Xt dA ˆ Yt . Asconsequence we obtain for Z t Z t X i,j ≤ d ∂ ∂x i x j G f ( s, Y s − ) (cid:16) c ˆ Y ijs − c ˆ Xijs (cid:17) + Z R d H G f ( s, Y s − , x ) (cid:16) K ˆ Ys ( dx ) − K ˆ Xs ( dx ) (cid:17) dA ˆ Ys . By Assumption ( i ), G f is directionally convex in the second entry, which is equivalent tothe second partial derivatives to be nonnegative for all i, j , see M¨uller and Stoyan (2002,Theorem 3.12.2). It follows with Assumption ( vi ) that the first integrand is non-positive dA ˆ Y × Q almost surely.To see that the second integrand is non-positive dA ˆ Y × Q almost surely as well, we findthat H G f ( s, Y s − , x ) is directionally convex in x . This follows from the directional convexityof G f : ∂ ∂x i x j H G f ( s, Y s − , x ) = ∂ ∂x i x j G f ( s, Y s − + x ) − G f ( s, Y s − ) − X k ≤ d ∂∂x k G f ( s, Y s − ) x k = ∂ ij G f ( s, Y u − + x ) ≥ . With Assumption ( vi ) it follows that the second integrand is non-positive dA ˆ Y × Q almostsurely. Therefore, − Z ∈ A + loc and ( G f ( t, Y t )) t ∈ [0 ,T ] is a local Q -supermartingale.9rom the fact that G f (0 , Y ) = E Q [ f ( X T )] is integrable and from Assumption ( iv ) itfollows that ( G f ( t, Y t )) t ∈ [0 ,T ] is a proper supermartingale.If the inequalities in ( v ) are reversed and the positive part of ( G f ( t, Y t )) t ∈ [0 ,T ] is of class(DL), the process Z is in A + loc and hence G f is a Q -submartingale. Remark 3.2 (Comments on the assumptions in Theorem 3.1) . Conditions ( iii ) and ( iv ) of Theorem 3.1 are clearly unavoidable since we need the proper supermartingale propertyand in order to compensate the jumps. We comment on the other assumptions, while inSection 5 we give a more detailed discussion of the regularity conditions in this paper.1. The regularity assumption ( i ) is crucial for the applicability of Itˆo’s formula. It is acommon assumption in financial mathematics for the computation of option prices ina Markovian model by the PIDE method. For example in Cont and Tankov (2004)there are conditions given for the regularity of the functional G f in exponential L´evymodels.2. The assumption of directional convexity of G f in ( ii ) provides the positivity of thesecond derivative of G f . This is necessary for the tractability of the relevant termsin Itˆo’s formula. It is in particular fulfilled if the propagation of directional convexityproperty holds, i.e. for all directional convex functions f it holds that G f ∈ F dcx .This assumption is made e.g. in Bergenthum and R¨uschendorf (2006, 2008). It holdsin particular for processes with independent increments and for diffusion processes.3. Assumption ( ii ) allows us to obtain a connection between the differential character-istics. If X is a Markovian semimartingale and f ( X T ) ∈ L ( Q ) , then the process ( G f ( t, X t )) t ∈ [0 ,T ] is a Q -martingale if the measure Q preserves the Markov property.This is a consequence of the Markov property, G f ( t, X t ) = E Q [ f ( X T ) | X t ] = E Q [ f ( X T ) |F t ] , which is a martingale by construction. Hence, we can apply Lemma 2.1 to G f andobtain that U Xt G f ( t, X t − ) = 0 dA ˆ X × Q almost surely for all t ∈ [0 , T ] . Typicallythe differential characteristics of Markovian semimartingales are of the form a ( t, X t ) ,see C¸ inlar et al. (1980). Hence, we get that for all x ∈ supp (cid:0) ( Q ) X t − (cid:1) it holds that U Xt G f ( t, x ) = 0 dA ˆ X × Q almost surely for all t ∈ [0 , T ] as well. If a semimartingale Y fulfills supp (cid:0) ( Q ) Y t − (cid:1) ⊂ supp (cid:0) ( Q ) X t − (cid:1) , we deduce that U Xt G f ( t, Y t − ) = 0 dA ˆ X × Q almost surely for all t ∈ [0 , T ] .Note that we can replace Q by Q since we assumed that Q ∼ Q . So Markovprocesses with a “big” support are candidate processes for the semimartingale X inTheorem 3.1. n a Markovian framework, G is the transition operator of X applied to f , a wellunderstood object. This suggests that Theorem 3.1 is particularly suitable if the semi-martingale X is a Markov process w.r.t Q . In particular, previous results in literatureare special cases of Theorem 3.1.4. Assumption ( v ) seems at first glance to be a severe restriction. This condition isfulfilled for example when we compare Itˆo processes or when Y is a Girsanov trans-form of X . In the setting of this section we have Q ∼ Q since we assumed bothmeasures to be equivalent to P . So this theorem is in particular useful if we compareone semimartingale under different e.m.m. In that case also the comparison of thecharacteristic simplifies since the quadratic variation of the continuous martingalepart is unchanged.Further, the integrator in a good version of the semimartingale characteristics can bechosen more or less freely. Only existence, not uniqueness is stated in Jacod and Shiryaev(2003, Proposition II.2.9). So if we consider two semimartingales X and Y underthe same measure we can, analogously to equation (2.1) , find a joint integrator for agood version, for example the process A = X i ≤ d Var( B ˆ Xi ) + X i,j ≤ d Var( C ˆ Xij ) + ( | x | ∧ ∗ ν ˆ X + X i ≤ d Var( B ˆ Y i ) + X i,j ≤ d Var( C ˆ Y ij ) + ( | x | ∧ ∗ ν ˆ Y .
5. The inequalities between the differential characteristics are the key for the comparisonresult. In the proof we can see that it suffices to check the inequality between thekernels for the function H G f only. Instead of directional convexity we can use anyfunction class F such that H G f ∈ F for an ordering of the kernels.We emphasize that in general K ˆ X is not the kernel of the semimartingale character-istics of ˆ X under Q . However, c ˆ X is the process from the differential characteristicsof ˆ X under Q since we use equivalent measures. This follows from the Girsanovtheorem. Effectively we do not compare the semimartingale characteristics of ˆ X and ˆ Y under Q . We compare under Q the semimartingale characteristics we get underthe particular e.m.m. If X and Y are already local martingales we compare the differ-ential characteristics under the same measure P . This is a special case of the theoremabove and we will discuss the comparison under P in Section 4 in more detail.6. We could also demand that the inequalities in ( vi ) hold Q almost surely for all t ∈ [0 , T ] . However, the choice of the product measure is more general, even if itseems more complicated at first glance. . The assumption that X and Y start in the same point can be easily achieved by shiftingone of the semimartingales. Depending on the aim of the comparison of X and Y this might not be reasonable. Then we can replace this assumption by demanding G f (0 , y ) ≤ G f (0 , x ) . Next we derive an ordering result when the functional G f is a convex function in x .Therefore, we use the positive semidefinite order for matrices, also called Loewner order.Remind that for A, B ∈ R d × d A is said to be smaller than B in the positive semidefiniteorder, if the matrix B − A is positive semidefinite, i.e. for all x ∈ R it holds x ′ ( B − A ) x ≥ A ≤ psd B if A is smaller than B in this order.The following convex comparison theorem extends Theorem 2.6 in Bergenthum and R¨uschendorf(2006). Theorem 3.3 (Convex comparison under e.m.m.) . Let
X, Y be semimartingales and let X = Y = x ∈ R d almost surely. Let f ∈ L (cid:0) ( Q ) X T (cid:1) ∩ L (cid:0) ( Q ) Y T (cid:1) and assume that(i) G f ∈ C , ([0 , T ] × R d ) and G f ( t, · ) ∈ F cx for all t ∈ [0 , T ] ;(ii) - ( v ) of Theorem 3.1 hold;(vi) The differential characteristics are dA ˆ Y × Q almost surely ordered: c ˆ Yt ≤ psd c ˆ Xt , Z R d g ( t, Y t − , x ) K ˆ Yt ( dx ) ≤ Z R d g ( t, Y t − , x ) K ˆ Xt ( dx ) , where the second inequality holds for all g ( t, y, · ) ∈ F cx such that the integrals exist.Then it holds E Q [ f ( Y T )] ≤ E Q [ f ( X T )] . If the inequalities in ( vi ) are reversed and ( G f ( t, Y t ) + ) t ∈ [0 ,T ] is of class (DL), we get E Q [ f ( Y T )] ≥ E Q [ f ( X T )] . Proof.
We show that ( G f ( t, Y t )) t ∈ [0 ,T ] is a Q -supermartingale. Similarly as in the proofof Theorem 3.1 we need to show, that the process Z t X i,j ≤ d ∂ ∂x i x j G f ( s, Y s − ) (cid:16) c ˆ Y ijs − c ˆ Xijs (cid:17) + Z R d H G f ( s, Y s − , x ) (cid:16) K ˆ Ys ( dx ) − K ˆ Xs ( dx ) (cid:17) dA ˆ Ys
12s non-increasing dA ˆ Y × Q almost surely. By Assumption ( vi ) the matrix − ( c ˆ Yt − c ˆ Xt ) = c ˆ Xt − c ˆ Yt is positive semidefinite for fixed ( ω, t ). Thus, the eigendecomposition has the form( P k ≤ d λ k e ik e jk ) i,j ≤ d with eigenvalues λ k ≥ e k . We get that the firstintegrand has the form − X k ≤ d λ k X i,j ≤ d ∂ ∂x i x j G f ( s, Y s − ) e ik e jk = − X k ≤ d λ k e ′ k ∂ ∂x i x j G f ( s, Y s − ) e k which is non-positive dA ˆ Y × Q almost surely due to the positive semidefiniteness of theHessian matrix of G f .Analogously to the proof of Theorem 3.1 we have that H G f ( s, Y s − , x ) is convex in x sincethe second derivative in direction of x is ∂ ∂x i x j H G f ( s, Y s − , x ) = ∂ ∂x i x j G f ( s, Y s − + x ) − G f ( s, Y s − ) − X k ≤ d ∂∂x k G f ( s, Y s − ) x k = ∂ ∂x i x j G f ( s, Y s − + x ) . Therefore, the Hessian matrix of H G f is positive semidefinite and it follows that H G f isconvex in x . Consequently, the second integrand is non-positive dA ˆ Y × Q almost surelyby Assumption ( vi ). By Assumption ( iv ) it follows that ( G f ( t, Y t )) t ∈ [0 ,T ] is a proper su-permartingale.If the inequalities in ( vi ) are reversed and ( G f ( t, Y t ) + ) t ∈ [0 ,T ] is of class (DL), then( G f ( t, Y t )) t ∈ [0 ,T ] is a submartingale. Remark 3.4.
As seen in the proofs, the key inequality is X i,j ≤ d ∂ ∂x i x j G f ( s, Y s − ) (cid:16) c ˆ Y ijs − c ˆ Xijs (cid:17) + Z R d H G f ( s, Y s − , x ) (cid:16) K ˆ Yu ( dx ) − K ˆ Xu ( dx ) (cid:17) ≤ dA ˆ Y × Q almost surely. Thus, we can replace the ordering assumption on the semimartin-gale characteristics by this inequality. Then also the (directional) convexity of G f is notnecessary anymore.This is a starting point for ordering results of other function classes (Bergenthum and R¨uschendorf(2007a)). Based on this inequality and under the assumption of propagation of order, thereis given a table with conditions on the semimartingale characteristics for the comparisonof further function classes. The classes investigated therein are increasing, supermodular,convex and directionally convex as well as increasing supermodular, increasing convex andincreasing directionally convex functions. The considerations in Remark 3.4 lead to the following corollary giving a comparisonresult under more general conditions on G f .13 orollary 3.5 (general comaprison under e.m.m.) . Let
X, Y be semimartingales and let X = Y = x ∈ R d almost surely. Let f ∈ L (cid:0) ( Q ) X T (cid:1) ∩ L (cid:0) ( Q ) Y T (cid:1) and assume that G f ∈ C , ([0 , T ] × R d ) and that ( ii ) – ( v ) of Theorem 3.1 hold. Further, let dA ˆ Y × Q almostsurely X i,j ≤ d ∂ ij G f ( s, Y s − ) (cid:16) c ˆ Y ijs − c ˆ Xijs (cid:17) + Z R d H G f ( s, Y s − , x ) (cid:16) K ˆ Ys ( dx ) − K ˆ Xs ( dx ) (cid:17) ≤ . (3.3) Then it holds that E Q [ f ( X T )] ≤ E Q [ f ( Y T )] . (3.4) If the term in (3.3) is non-negative and ( G f ( t, Y t ) + ) t ∈ [0 ,T ] is of class (DL), we get thereverse inequality in (3.4) .Proof. Equation (3.3) is chosen in such a way that the process Z defined in (3.2) is non-increasing or non-decreasing and the assertion follows as in Theorem 3.1. Remark 3.6.
Equation (3.3) arises in a similar form in the field of model uncertaintyin financial mathematics under the notion volatility misspecification, see El Karoui et al.(1998). There it is assumed that a market participant uses a model to price and hedge aEuropean option which does not coincide with the real evolution of the underlying. Thenthe left side of inequality (3.3) indicates the so-called tracking error which is the differenceof the real price of the option and the price derived by the model of the market participant.
As stated in Remark 3.2 these kind of theorems are especially useful if we compare asingle semimartingale under different equivalent martingale measures We now turn to thisspecial case.By Girsanov’s theorem only the compensator of the jump measure changes while thepredictable quadratic variation of the continuous martingale part and the increasing processof a good version of the semimartingale characteristics remain the same.
Corollary 3.7 (Comparison of e.m.m.) . Let X be a semimartingale, let Q and Q beequivalent martingale measures for X and denote the particular semimartingale character-istics of X by superscript. Assume that f ∈ L (cid:0) ( Q ) X T (cid:1) ∩ L (cid:0) ( Q ) X T (cid:1) and that(i) G f ∈ C , ([0 , T ] × R d ) and G f ( t, · ) ∈ F dcx (or G f ( t, · ) ∈ F cx ) for all t ∈ [0 , T ] ;(ii) U Xt G f ( t, X t − ) = 0 dA ˆ X × Q almost surely where U Xt is defined in (2.3) , here withsemimartingale characteristics of X under Q ;(iii) (cid:12)(cid:12) H G f (cid:12)(cid:12) ∗ µ X ∈ A + loc ;(iv) ( G f ( t, X t ) − ) t ∈ [0 ,T ] is of class (DL); v) The kernels K and K are dA ˆ X × Q almost surely ordered for all t ∈ [0 , T ] : Z R d g ( t, X t − , x ) K t ( dx ) ≤ Z R d g ( t, X t − , x ) K t ( dx ) , where the inequality holds for all g ( t, y, · ) ∈ F dcx (or g ( t, y, · ) ∈ F cx ) such that theintegrals exist.Then we obtain E Q [ f ( X T )] ≤ E Q [ f ( X T )] . If the inequalities in ( vi ) are reversed and ( G f ( t, X t ) + ) t ∈ [0 ,T ] is of class (DL), we get E Q [ f ( X T )] ≥ E Q [ f ( X T )] . Proof.
This follows directly from Theorem 3.1 and Theorem 3.3.
Remark 3.8.
As remarked in Corollary 3.5 we do not need the assumption of convexity anddirectional convexity of G f since the terms with second partial derivatives vanish in Itˆo’sformula. This is due to the fact that the semimartingale characteristic from the continuousmartingale part c remains the same when we change the measure. Assumption ( v ) thenneeds to be adapted to H G f as in Corollary 3.5. Having in mind that it suffices to haveinequality of the kernels for H G f , the result also follows by means of Itˆo’s formula. Afterthe partial derivative in time is replaced, the only remaining term of finite variation thenis non-increasing by assumption ( v ) . We turn in this section to the case, when we regard both semimartingales under P . As inSection 2 we restrict ourselves to special semimartingales.We begin with a version of the comparison of processes X and Y in Theorem 3.1 underthe same semimartingale measure P . Let X and Y be special semimartingales, then theprocesses ˆ X and ˆ Y are special semimartingales and we can choose for both semimartingalesthe same process A for a good version of the semimartingale characteristics under P ,see Remark 3.2. Further, we choose the identity as truncation function. The canonicaldecomposition of ˆ X has the form:ˆ X t = ˆ X + (cid:0) , X ct + x ∗ (cid:0) µ X − ν (cid:1) t (cid:1) + ( b ˆ X · A ) t , Analogously we have such a decomposition for ˆ Y and we use superscripts to point out towhich process the characteristics belong. The functional G f is defined in equation (3.1)15s conditional expectation under the equivalent martingale measure Q and needs to beadapted. We define the valuation operator G f ( t, x ) := E [ f ( X T ) | X t = x ] , where the conditional expectation is now with respect to P . Since in this setting the driftpart is not only the identity we need to control additionally the first derivatives in Itˆo’sformula. We accomplish this by assuming that G f ( t, · ) is an increasing function for all t ∈ [0 , T ] in the poitwise ordering on R d . Theorem 4.1 (Increasing directionally convex comparison under P ) . Let
X, Y be specialsemimartingales and let X = Y = x almost surely and let f ∈ L ( P X T ) ∩ L ( P Y T ) .Assume that(i) G f ∈ C , ([0 , T ] × R d ) and G f ( t, · ) ∈ F idcx for all t ∈ [0 , T ] ;(ii) ¯ U Xt G f ( t, Y t − ) = 0 holds dA × P almost surely for all t ∈ [0 , T ] where ¯ U Xt is definedin (2.4) with the characteristic of ˆ X in it;(iii) (cid:12)(cid:12) H G f (cid:12)(cid:12) ∗ µ Y ∈ A + loc ;(iv) ( G f ( t, Y t ) − ) t ∈ [0 ,T ] is of class (DL);(v) The differential characteristics are dA × P almost surely ordered for all i, j ≤ d : b ˆ Y it ≤ b ˆ Xit ,c ˆ Y ijt ≤ c ˆ Xijt , Z R d g ( t, Y t − , x ) K ˆ Yt ( dx ) ≤ Z R d g ( t, Y t − , x ) K ˆ Xt ( dx ) , where the last inequality holds for all g ( t, y, · ) ∈ F idcx such that the integrals exist.Then it holds E [ f ( Y T )] ≤ E [ f ( X T )] . If the inequalities in ( v ) are reversed and ( G f ( t, Y t ) + ) t ∈ [0 ,T ] is of class (DL), we obtain that E [ f ( Y T )] ≥ E [ f ( X T )] . roof. Analogously to the comparison under equivalent martingale measures we show that( G f ( t, Y t )) t ∈ [0 ,T ] is a P -supermartingale. Itˆo’s formula yields G f ( t, Y t ) = G f (0 , x ) + Z t ∂∂s G f ( s, Y s − ) b ˆ Ys dA s + X i ≤ d Z t ∂∂x i G f ( s, Y s − ) dY is + 12 X i,j ≤ d Z t ∂ ∂x i x j G f ( s, Y s − ) c ˆ Y ijs dA s + Z [0 ,t ] × R d G f ( s, Y s − + x ) − G f ( s, Y s − ) − X i ≤ d ∂∂x i G f ( s, Y s − ) x i µ Y ( ds, dx ) . We compensate the jumps and use the canonical decomposition of ˆ Y . This leads to G f ( t, Y t ) = G f (0 , x ) + Z t ∂∂s G f ( s, Y s − ) b ˆ Ys dA s + X i ≤ d Z t ∂∂x i G f ( s, Y s − ) b ˆ Y is dA s + 12 X i,j ≤ d Z t ∂ ∂x i x j G f ( s, Y s − ) c ˆ Y ijs dA s + M t + Z [0 ,t ] × R d H G f ( s, Y s − , x ) K ˆ Ys ( dx ) dA s , where M is the local martingale from the integrals with respect to the continuous mar-tingale part of Y and the compensated jumps. With similar arguments as in the proof ofTheorem 3.1 it suffices to show, that the following process Z t is non-increasing dA × P almost surely: Z t := Z t X i ≤ d ∂∂x i G f ( s, Y s − ) (cid:16) b ˆ Y is − b ˆ Xis (cid:17) + 12 X i,j ≤ d ∂ ∂x i x j G f ( s, Y s − ) (cid:16) c ˆ Y ijs − c ˆ Xijs (cid:17) + Z R d H G f ( s, Y s − , x ) (cid:16) K ˆ Ys ( dx ) − K ˆ Xs ( dx ) (cid:17) dA s . The first term in the integral is non-positive dA × P almost surely because of Assumption ( v )and the fact that G f is increasing in x . The remaining terms are non-positive dA × P almostsurely which can be seen as in the proof of Theorem 3.1. Assumption ( iv ) yields the propersupermartingale property.If the inequalities in ( v ) are reversed and ( G f ( t, Y t ) + ) t ∈ [0 ,T ] is of class (DL), the process( G f ( t, Y t )) t ∈ [0 ,T ] is a submartingale. Remark 4.2.
1. Since we assumed the functional G f to be increasing in the secondvariable it is not necessary anymore that Y = X . If X = x ≥ y = Y , the upermartingale property and the fact that G f is increasing in the second variablestill yield the inequality of the expectations, E [ f ( Y T )] = E [ G f ( T, Y T )] ≤ G f (0 , y ) ≤ G f (0 , x ) = E [ f ( X T )] . When G f is a submartingale, we need to impose X = x ≤ y = Y for an analogstatement.2. In contrast to the last section we compare in this framework the original semimartin-gale characteristics, cf. Remark 3.2. The following comparison result for processes X and Y in the case that G f is increas-ing and convex in x is the analogon of the comparison result in Theorem 3.3 under asemimartingale measure P . Theorem 4.3 (Increasing convex comparison under P) . Let
X, Y be special semimartin-gales and let x = X ≥ Y = y almost surely. Let f ∈ L (cid:0) P X T (cid:1) ∩ L (cid:0) P Y T (cid:1) and assumethat(i) G f ∈ C , ([0 , T ] × R d ) and G f ( t, · ) ∈ F icx for all t ∈ [0 , T ] ;(ii) - ( iv ) of Theorem 4.1 hold;(v) The differential characteristics are dA × P almost surely ordered for all t ∈ [0 , T ] andall i ≤ d : b ˆ Y it ≤ b ˆ Xit ,c ˆ Yt ≤ psd c ˆ Xt , Z R d g ( t, Y t − , x ) K ˆ Yt ( dx ) ≤ Z R d g ( t, Y t − , x ) K ˆ Xt ( dx ) , where the last inequality holds for all g ( t, y, · ) ∈ F cx such that the integrals exist.Then we have that E [ f ( Y T )] ≤ E [ f ( X T )] . If in ( v ) the inequalities are reversed, x = X ≤ Y = y and ( G f ( t, Y t ) + ) t ∈ [0 ,T ] is of class(DL), we get E [ f ( Y T )] ≥ E [ f ( X T )] . roof. We show, that the process Z t X i ≤ d ∂∂x i G f ( s, Y s − ) (cid:16) b ˆ Y is − b ˆ Xis (cid:17) + 12 X i,j ≤ d ∂ ∂x i x j G f ( s, Y s − ) (cid:16) c ˆ Y ijs − c ˆ Xijs (cid:17) + Z R d H G f ( s, Y s − , x ) (cid:16) K ˆ Ys ( dx ) − K ˆ Xs ( dx ) (cid:17) dA ˆ Ys is non-increasing P almost surely. Then the assertion follows as in Theorem 4.1. Thefirst term in the integral is non-positive due to Assumption ( v ) and the fact that G f isincreasing in the second variable. The remaining part is non-positive similarily as in theproof of Theorem 3.3. By Assumption ( iv ) it follows that ( G f ( t, Y t )) t ∈ [0 ,T ] is a propersupermartingale.If the inequalities in ( v ) are reversed, x = X ≤ Y = y and ( G f ( t, Y t ) + ) t ∈ [0 ,T ] is of class(DL), it is a submartingale.As under e.m.m., the key inequality of the proof can be used to replace the convexityassumption by a general form of conditions. Corollary 4.4 (General comparison condition) . Let
X, Y be special semimartingales, let x = X ≥ Y = y and let f ∈ L (cid:0) P X T (cid:1) ∩ L (cid:0) P Y T (cid:1) . Assume that G f ∈ C , ([0 , T ] × R d ) and that assumptions ( ii ) – ( iv ) of Theorem 4.1 hold. Further, assume that dA × P almostsurely X i ≤ d ∂∂x i G f ( s, Y s − ) (cid:16) b ˆ Y is − b ˆ Xis (cid:17) + 12 X i,j ≤ d ∂ ∂x i x j G f ( s, Y s − ) (cid:16) c ˆ Y ijs − c ˆ Xijs (cid:17) + Z R d H G f ( s, Y s − , x ) (cid:16) K ˆ Ys ( dx ) − K ˆ Xs ( dx ) (cid:17) ≤ . (4.1) Then it holds that E [ f ( X T )] ≤ E [ f ( Y T )] . If the inequality (4.1) is reversed, x = X ≤ Y = y and ( G f ( t, Y t ) + ) t ∈ [0 ,T ] is of class(DL), we get E [ f ( X T )] ≥ E [ f ( Y T )] .Proof. Similarly as in the proof of Theorem 4.3 we obtain that under these assumptions( G f ( t, Y t )) t ∈ [0 ,T ] is a supermartingale (submartingale if the inequality is inverse). In this section we describe several approaches to establish the regularity conditions of thecomparison results and give examples. The focus is on the question of differentiability and19onvexity. It is clear that the regularity and (directional) convexity condition on G f onlydepend on the semimartingale which is used in the definition of G f .For probability measures P and Q on a space (Ω , F ) and a class of integrable functions F , P is said to be smaller than Q in the integral stochastic order generated by F , P (cid:22) F Q if Z f dP ≤ Z f dQ, for all f ∈ F . So in terms of integral stochastic orders we state in Section 3 conditions for the ordering( Q ) Y T (cid:22) F ( Q ) X T and in Section 4 for the ordering P Y T (cid:22) F P X T for some function class F . The question arises what function classes are well fitting with theses conditions. Sofar we did not impose conditions on the function f under consideration but only on thefunctional G f .The section is organized as follows. We first discuss approaches from the theory ofMarkov processes to conclude differentiability of G f . In particular for smooth functions adirect argument for differentiability can be given. Therefore, we recapitulate some insightsfrom the PIDE method in option pricing. Then we proceed with another ansatz in theframework of integral stochastic orders. Afterwards we deal with the issue of convexity anddirectional convexity of G f and in particular remind some approaches from the literatureand give corresponding references. We conclude this section with some explicit examplesof processes which can be compared with the theorems of this paper. There are various approaches in the literature to establish the regularity assumptions on G f . In general, the representation G f ( t, x ) = E [ f ( X T ) | X t = x ] = Z R d f ( y ) P X T | X t = x ( dy ) . (5.1)suggests that differentiability of G f is mainly an issue of the conditional distribution. ForMarkov processes this question is a well studied object. Also for the computation of optionprices this question has been investigated in many papers in particular in connection withthe PIDE method. For L´evy processes differentaibility can be shown by a convolutionargument (see Cont and Tankov (2004), Glau (2010)). Assume that a L´evy process X t possesses a density p t with respect to the Lebesgue measure. Due to the temporal and20patial homogeneity of L´evy processes we can simplify the conditional expectation G f ( t, x ) = E [ f ( X T ) | X t = x ]= E [ f ( X T − t + x )]= Z R d f ( y + x ) p T − t ( y ) dy = Z R d ˜ f ( − x − y ) p T − t ( y ) dy = ˜ f ∗ p T − t ( − x ) , (5.2)where ˜ f ( x ) = f ( − x ). If the density p is twice continuously differentiable in x so is G f .In Cont and Tankov (2004, Proposition 3.12) conditions are given so that the density of aL´evy process is smooth. More generally we can check the number of times of continuousdifferentiability of a density with the help of its associated characteristic function, see Sato(1999, Proposition 28.1). If for the characteristic function ˆ µ of a measure µ it holds that R R d | z | n | ˆ µ ( z ) | dz < ∞ for a n ∈ N , then µ has a C n Lebesgue density.For smooth functions satisfying some Lipschitz conditions, smoothness of G f can beshown directly (without smoothness of the density). For a smooth Lipschitz continuousfunction f , we obtain by dominated convergence ∂∂x i G f ( t, x ) = lim h → E [ f ( X T − t + x + he i ) − f ( X T − t + x )] h = E (cid:20) lim h → f ( X T − t + x + he i ) − f ( X T − t + x ) h (cid:21) = E (cid:20) ∂∂x i f ( X T − t − x ) (cid:21) . Similarly we get ∂ ∂x i x j G f ( t, x ) = E [ ∂ ∂x i x j f ( X T − x )] if the derivative of f is Lipschitzcontinuous. This is continuous if ∂ ∂x i x j f is Lipschitz continuous.In this conection it is of interest that in the theory of stochastic orders it has beenestablished that several stochastic orders can be generated by classes of smooth functions,see M¨uller and Stoyan (2002). Therefore, in the framework of L´evy processes, we getdirectly from equation (5.2) that G f is smooth in x with no further assumptions on thedensity. In particular, the directionally convex order and increasing directionally convexorder are integral stochastic orders which are generated by smooth functions.For the differentiability in t we consider a time-homogeneous Markov process X . Assumethat its transition operators ( T t ) t ≥ form a strongly continuous semigroup. Then we have G f ( t, · ) = T T − t f ( · ) and by semigroup theory G f is differentiable in t . The continuityof the derivative then follows because the semigroup of a Markov process ( T t ) ≤ t ≤ T withgenerator A can be represented as solution to an evolution problem( c.f. Ethier and Kurtz212005)), i.e. for f ∈ D ( A ) such that Af ∈ D ( A ) it holds: ddt T t f = T t Af. (5.3)The term Af does not depend on time and T t is assumed to be continuous. In the time-inhomogeneous case we can achieve continuous differentiability in time analogously withthe theory of evolution systems. Then we obtain an equation similar to (5.3).In the case of L´evy processe, this holds for functions in C ( R d ), i.e. continuous functionsvanishing at infinity. For these functions the transition operators of a L´evy process form astrongly continuous semigroup, see Sato (1999, Theorem 31.5) . Concerning convexity and directional convexity the propagation of order property for a func-tion class F has turned out to be useful, i.e. f ∈ F implies G f ∈ F . Therefore properties for G f , as for example convexity, can be inferred from those of f . For papers in this field see e.g.Bergmann et al. (1996), Martini (1999), El Karoui et al. (1998), Bellamy and Jeanblanc(2000), Gushchin and Mordecki (2002) and Bergenthum and R¨uschendorf (2006, 2007a).The results in this direction use various methods as establishing a Cauchy problem for G f in the case of one-dimensional diffusions and reduction to the non-crossing property by theFeynman–Kac formula (see Bergmann et al. (1996) and El Karoui et al. (1998)) or using in-dependent increments as Gushchin and Mordecki (2002). In Bergenthum and R¨uschendorf(2006, 2007a) an approximation argument is used. A coupling aproach is used in Hobson(1998). A detailed exposition of these approaches is given in K¨opfer (2019) In the sequel we give some explicit examples for semimartingales which fulfill the conditionsof the comparison results. Throughout this section we assume that G f is continuouslydifferentiable in time and focus on the differentiability in space.We begin with the assumption that X is a one-dimensional L´evy process. Then by theconsiderations above we conclude that convexity and directional convexity is propagated.Also the differentiability in time is straightforward if we restrict ourselves to functions in C ( R d ). In addition we can use a result from Sato (1999) which characterises the supportof a one-dimensional L´evy process. Example 5.1 (Theorems 4.1 and 4.3 for R -valued L´evy processes and Itˆo semimartingales) . We consider a comparison under P . Let X be an R -valued L´evy process with differentialsemimartingale characteristic ( b, c , K ) , where b ∈ R , c ∈ R + and K is a L´evy measure.We assume that X is a type C L´evy process, i.e. either c > or R | x |≤ | x | K ( dx ) = ∞ . dditionally the L´evy measure is assumed to fulfill Z | x | > | x | K ( dx ) < ∞ (5.4) and lim inf ε ↓ ε − c Z ε − ε | x | K ( dx ) > , (5.5) where c ∈ (0 , . The first condition assures that X is a special semimartingale, seeRheinl¨ander and Sexton (2011, Lemma 4.5). The second condition yields the existenceof a smooth density, see Cont and Tankov (2004, Proposition 3.12).Further, let Y = ( Y t ) ≤ t ≤ T be a special Itˆo semimartingale in the sense of C¸ inlar et al.(1980), i.e. a special semimartingale of the form Y t = y + Z t β s ds + Z t δ s dB s + Z t Z R y ˜ µ Y ( ds, dy ) . Here y ∈ R , β and δ are adapted processes such that the integral exist, ( B t ) ≤ t ≤ T is astandard Brownian motion and ˜ µ Y is the compensated jump measure of Y . The compen-sator is of the form ν ( dt, dx ) = dt n t ( dx ) .It follows that the identity is an integrator for a good version of the semimartingale char-acteristics for X and Y . Also we can use the identity in the good version of ˆ X and ˆ Y and do not need to adapt it as in Remark 3.2. Besides the differential characteristics of X and Y are the differential characteristics which occur in the space dimensions of ˆ X and ˆ Y respectively. We consider an integrable increasing convex or increasing directionally convexfunction f .From the independence of increments of X we obtain propagation of increasing convexityand increasing directional convexity. By (5.5) X possesses a smooth Lebesgue density andit follows that G f ∈ C , . Thus, condition ( i ) of Theorems 4.1 and 4.3 is fulfilled.From the Markov property of X we get that the functional G f can be written as G f ( t, X t ) = E [ f ( X T ) | X t ] = E [ f ( X T ) |F t ] , which is a martingale by construction. Therefore, Lemma 2.2 can be applied. Further, since X is a type C L´evy process, we have by Sato (1999, Theorem 24.10) that supp( P X t ) = R for all t ≥ . Hence, we get for all x ∈ R and all t ∈ [0 , T ] that ¯ U Xt G f ( t, x ) = 0 ,dt × P almost surely. It follows that ¯ U Xt G f ( t, Y t − ) = 0 , t × P almost surely for all t ∈ [0 , T ] . Altogether condition ( ii ) is fulfilled. This holdsindependently from the choice of Y , see Remark 3.2. This consideration shows that typeC L´evy processes are candidates for the semimartingale X in the theorems stated in thischapter. Further, we assume the conditions ( iii ) and ( iv ) .If now the differential characteristics are dt × P almost surely ordered, β t ≤ bδ t ≤ c Z R g ( t, Y t − , x ) ν Yt ( dx ) ≤ Z R g ( t, Y t − , x ) K ( dx ) , for all g ( t, y, · ) ∈ F idcx or F icx such that the integrals exist, we get by Theorem 4.1 and 4.3that E [ f ( Y T )] ≤ E [ f ( X T )] . Note that since we assumed the semimartingales to be one-dimensional, the second inequal-ity yields also the inequality in the positive semidefinite order.As usual a reverse ordering in the differential characteristics provides the inverse inequality.
Remark 5.2.
1. The condition that X is of type C can be modified. It is used to obtainthat supp (cid:0) P X t (cid:1) = R for all t ∈ [0 , T ] . In Sato (1999, Theorem 24.10) there isanother condition for this support, namely ∈ supp( ν ) , supp( ν ) ∩ (0 , ∞ ) = ∅ and supp( ν ) ∩ ( −∞ , = ∅ .2. If we are interested in the integral stochastic order generated by increasing direc-tionally convex functions, we can omit the condition on the L´evy measure (5.5) .By M¨uller and Stoyan (2002, Theorem 3.12.9) this stochastic order is generated byinfinitely differentiable directionally convex functions and we hence do not need asmooth density. This is a consequence of the convolution argument in the last sec-tion, equation (5.2) . Next we give an example for a comparison of a L´evy process and an Itˆo semimartingaleby Theorems 3.1 and 3.3.
Example 5.3 (Comparison of R -valued L´evy processes and Itˆo semimartingales) . Let X be an R -valued L´evy process as in Example 5.1, without assuming inequality (5.4) ; we donot need that the semimartingales are special. Further, let Y be an Itˆo semimartingale: Y t = y + Z t β s ds + Z t δ s dB s + Z t Z | y |≤ y ˜ µ Y ( ds, dy ) + Z t Z | y |≥ yµ Y ( ds, dy ) , where y ∈ R , β and δ are adapted processes such that the integral exist, ( B t ) t ∈ [0 ,T ] is astandard Brownian motion and ˜ µ is the compensated jump measure of Y . As before the ompensator is of the form ν ( dt, dx ) = dt n t ( dx ) . We consider an integrable convex ordirectionally convex function f .We assume the existence of e.m.m. for X and Y . As we have seen in Example 5.1, it isadvantageous if X is a L´evy process under Q . Therefore, we assume that Q is structurepreserving, i.e. X remains a L´evy process under Q . By Rheinl¨ander and Sexton (2011,Theorem 4.21) we see directly that the differential semimartingale characteristics of X under Q are given by (0 , c , hK ) , where h : R → R + is a Borel measurable function suchthat Z R ( p h ( x ) − K ( dx ) < ∞ . This choice of Q leads to the validity of Assumption ( i ) . Assumption ( ii ) is achieved as inExample 5.1 since Q ∼ Q . Further conditions ( iii ) and ( iv ) are assumed to be in force.The change of measure does not affect the integrator of a good version of the semi-martingale characteristics. This follows directly form Girsanov’s theorem for semimartin-gales, cf. Jacod and Shiryaev (2003, Theorem III.3.24). Hence, the integrator remainsthe identity. Consequently, the differential characteristics of Y alter to (0 , δ , Zη ) , where Z : Ω × R + × R → R is non-negative, P ⊗ B ( R ) -measurable and fulfills the conditions ofJacod and Shiryaev (2003, Theorem III.3.24). If now dt × Q almost surely the differentialcharacteristics are ordered, δ t ≤ c , Z R g ( t, Y t − , x ) Z t ( x ) η t ( dx ) ≤ Z R g ( t, Y t − , x ) h ( x ) K ( dx ) , for all g ( t, y, · ) ∈ F idcx or F icx such that the integrals exist, we obtain from Theorem 3.1,3.3 that E Q [ f ( Y T )] ≤ E Q [ f ( X T )] . So far we only considered one-dimensional examples because we then obtain that thesupport is the whole space R . We now give an example in higher dimensions. Example 5.4 (Comparison of Markovian special Itˆo semimartingales and special Itˆo semi-martingales) . Let X be a d -dimensional Markovian special Itˆo semimartigale. Then itsdifferential charateristics with respect to the Lebesgue measure are deterministic functionsof time and state b it = b i ( t, X t − ) ,c ijt = c ij ( t, X t − ) ,K ω,t ( dx ) = K ( t, X t − ( ω ))( dx ) . e compare X to an special Itˆo semimartingale Y . Therefore, let Y be as in Example5.1. Assume that supp (cid:0) P Y t (cid:1) ⊂ supp (cid:0) P X t (cid:1) for all t ∈ [0 , T ] . This case is considered inBergenthum and R¨uschendorf (2006, 2007a). Let f be an increasing convex or increasingdirectionally convex integrable function.For the regularity of G f in time we assume that the transition operators form a stronglycontinuous evolution system, then the differentiability follows as in equation (5.3) . Forthe regularity in the space variable, we assume that the transition probabilities are regularenough, see Section 5.1. Further, we assume that X propagates increasing convexity orincreasing directional convexity.Recall that by the Markov property of X , G f ( t, X t ) is a martingale. From the conditionon the support of Y we gain condition ( ii ) . Assumptions ( iii ) and ( iv ) are imposed.Now the dt × P almost sure ordering of the differential semimartingale characteristicsfor all i, j ≤ d , β it ≤ b i ( t, X t − ) ,δ ij t ≤ c ij ( t, X t − ) , Z R d g ( t, Y t − , x ) ν Y ( dx ) ≤ Z R d g ( t, Y t − , x ) K ( t, X t − )( dx ) , (5.6) for all g ( t, y, · ) ∈ F idcx or F icx such that the integrals exist, yields by Theorem 4.1, 4.3 that E [ f ( Y T )] ≤ E [ f ( X T )] when f is integrable and increasing directionally convex. If f is integrable and increasingconvex we need to exchange the second inequality in (5.6) to δ t ≤ psd c ( t, X t − ) to gain thesame inequality. Note that this example differs a bit from Bergenthum and R¨uschendorf (2006, 2007a).There the semimartingales are defined on different probability spaces. As consequence ofthis setting on the right-hand side of the Inequalities (5.6) the differential semimartingalecharacteristics of X have to be evaluated at Y t − . This is not necessary in the frame-work here. However, we consider in the proofs f ( X T ) conditioned on X t = Y t andhence in this setting we could interchange X and Y . This then leads to inequalities asin Bergenthum and R¨uschendorf (2006, 2007a).Next we give an application of Theorems 4.1, 4.3 to the case when the integrator A inthe good version of the semimartingale characteristics is not the identity. Example 5.5 (Comparison result for extended Grigelionis Processes) . We assume thesemimartingale X to be an extended Grigelionis process. This sort of processes are usedfor example in Kallsen (1998). A special semimartingale is called an extended Grigelionisprocess if there exists a discrete set Θ ⊂ R + \ { } so that the increasing process of a good ersion of the semimartingale characteristics is given by A t = t + X s ≤ t Θ ( s ) . Intuitively this definition means that we have an Itˆo semimartingale plus jumps at fixedtimes. The semimartingale characteristics ( B, C, ν ) of X then have the form B it = Z t b is ds + X s ∈ Θ ∩ [0 ,t ] b is ,C ijt = Z t c ijs ds,ν ([0 , t ] × G ) = Z t K s ( G ) ds + X s ∈ Θ ∩ [0 ,t ] K s ( G ) for any G ∈ B ( R d ) , where ( b t ) t ∈ [0 ,T ] is a predictable R d -valued process, ( c t ) t ∈ [0 ,T ] is a predictable R d × d -valuedprocess and K is a transition kernel from (Ω × R + , P ) into ( R d , B ( R d )) . Further, weassume that X is a Markov process with transition probabilities that are regular enough asin Example 5.4.We compare X to another extended Grigelionis process Y with characteristics ( ˜ B, ˜ C, ˜ ν ) with respect to ˜ A t = t + P s ≤ t ˜Θ ( s ) . We assume that supp( P Y t ) ⊂ supp( P X t ) for all t ∈ [0 , T ] . To apply Theorems 4.1 and 4.3, we need to find a common integrator for a goodversion of the semimartingale characteristics. Choosing A ′ = A + ˜ A , cf. Remark 3.2, thedifferential characteristics of X change to (cid:16) b ( ˜Θ \ Θ) c , c, K ( ˜Θ \ Θ) c (cid:17) , the differential charac-teristics of Y change accordingly. Let f be an increasing convex or increasing directionallyconvex integrable function.The differentiability of G f follows as in Example 5.4. Further we assume the propagationof order. Hence, we have condition ( i ) . Condition ( ii ) follows from the Markov propertyand the assumptions on the supports; conditions ( iii ) and ( iv ) are imposed.Then an ordering of the semimartingale characteristics yields an ordering of expectations.For more details see K¨opfer (2019). In Corollary 3.7 we mentioned already the particular simplification if we consider asemimartingale under two different e.m.m. We apply this simplification in the case of aL´evy process.
Example 5.6 (Comparison of e.m.m. for a one-dimensional L´evy process) . Let X be a one-dimensional type C L´evy process and Q and Q e.m.m. of X . We assume that X possessesa smooth Lebesgue-density under Q , for example by imposing inequality (5.5) . Further,we assume that Q is structure preserving (see Example 5.3). No further restrictions are ut on Q . Let f be an integrable function not necessarily convex or directional convex,see Remark 3.8.As seen in Example 5.3 in this case condition ( i ) and ( ii ) of Corollary 3.7 hold. Condition ( iii ) and ( iv ) are assumed. If we assume that the L´evy measures are dt × Q almost surelyordered, Z R H G f ( t, Y t − , x ) K t ( dx ) ≤ Z R H G f ( t, Y t − , x ) K ( dx ) , then we obtain from Corollary 3.7 E Q [ f ( X T )] ≤ E Q [ f ( X T )] . References
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