Comparison of path-dependent functionals of semimartingales
aa r X i v : . [ m a t h . P R ] A ug Comparison of path-dependent functionals ofsemimartingales
Benedikt K¨opfer ∗ , Ludger R¨uschendorf Based on an extension of the martingale comparison method some compar-ison results for path-dependent functions of semimartingales are established.The proof makes essential use of the functional Itˆo calculus. A main tool is anextension of the Kolmogorov backwards equation to path-dependent functions.The paper also derives criteria for the regularity conditions of the comparisontheorems and discusses applications as to the comparison of Asian options forsemimartingale models.
The main subject of this paper is to give an extension of ordering results for path-independent functions of semimartingales based on the martingale method to path-dependentfunctions. The martingale comparison method was introduced for the comparison of path-independent functions of semimartingales in El Karoui et al. (1998) and Bellamy and Jeanblanc(2000). It was then systematized and extended in Gushchin and Mordecki (2002), Bergenthum and R¨uschendorf(2006, 2007a,b, 2008) and in K¨opfer and R¨uschendorf (2019). Essentially a comparison oflocal (differential) characteristics and the ’propagation of order’ property yield, under thecondition that the propagation operator (the value process) satisfies a Kolmogorov back-wards equation, a comparison of terminal values.In particular in Bergenthum and R¨uschendorf (2006, 2007a) and K¨opfer and R¨uschendorf(2019) general versions of the Kolmogorov backwards equation for path-independent func-tionals have been established and applied to ordering results for semimartingales w.r.t.various kinds of orderings as motivated by the problem to establish price bounds resp. ∗ A LGFG grant of the state Baden-W¨urttemberg is gratefully acknowledgedAMS 2010 subject classification: Primary: 60E15; secondary: 60E44, 60G51Key words and phrases: Path-dependent ordering, ordering of semimartingales, Kolmogorov backwardsequation, functional Itˆo calculus. isk bounds in some general class of insurance resp. financial models. Some alternative ap-proaches to related comparison results are given in Geib and Manthey (1994), El Karoui et al.(1997), Hobson (1998), Zhou (2004), Shi et al. (2005), Peng and Zhou (2006), Klein et al.(2006), Arnaudon et al. (2008), Wua and Xu (2009), Ma et al. (2010) and Criens (2019).For the extension to the ordering of path-dependent functions we make essential use ofthe functional Itˆo calculus and in particular of the functional Itˆo formula, see Bally et al.(2016). In Section 2 some necessary notions and results of this theory are collected. Thefunctional Itˆo formula allows us to extend the basic Kolmogorov backward equation to thepath-dependent framework. As a consequence we are able to derive comparison resultsfor path-dependent functions under equivalent martingale measures as well as w.r.t. semi-martingale measures. We also discuss the regularity conditions of the comparison theoremsand discuss applications as to the the comparison of Asian options for semimartingales.For further details and extensions of the comparison method we refer to the dissertationK¨opfer (2019) on which this paper is based.
In this section we recall some of the basic notions and results of the functional Itˆo calculus.This is the main tool for the extension of the martingale comparison method, to theframe of path-dependent functionals. The functional Itˆo calculus was introduced by Dupire(2009) and developed since then; see Cont and Fourni´e (2010a,b), Leventhal et al. (2013),Bally et al. (2016) and Ananova and Cont (2017). A comprehensive presentation on whichthis section is based is given in Bally et al. (2016).For the functional calculus a set of suitable functions and an appropriate notion ofderivative is needed. Let X be the canonical process on the space of c`adl`ag functionsΩ = D ([0 , T ] , R d ) and ( F t ) t ∈ [0 ,T ] be the filtration generated by it. Then any adapted real-valued process Y = ( Y t ) t ∈ [0 ,T ] may be represented as family of functionals Y ( t, · ) : Ω → R ,such that Y ( t, · ) only depends on the path stopped at t , i.e. Y ( t, ω ) = Y ( t, ω ·∧ t ). Thereforewe can view an adapted process as functional on the space of “stopped paths”. In the sequelwe use the notation ω t · := ω ·∧ t for the path stopped at t . More formally a stopped path isan equivalence class in ([0 , T ] × D ([0 , T ] , R d ) for the following equivalence relation( t, ω ) ∼ ( s, ˜ ω ) ⇔ t = s and ω t = ˜ ω s . The space of stopped paths is defined as the quotient of [0 , T ] × D ([0 , T ] , R d ) by the aboveequivalence relation:Λ dT := { ( t, ω t ); ( t, ω ) ∈ [0 , T ] × D ([0 , T ] , R d ) } = [0 , T ] × D ([0 , T ] , R d ) / ∼ . d ∞ (( t, ω t ) , ( s, ˜ ω s )) := sup u ∈ [0 ,T ] | ω u ∧ t − ˜ ω u ∧ s | + | t − s | = k ω t − ˜ ω s k ∞ + | t − s | . In the sequel we write ( t, ω ) since it is clear from the first variable at which point in timethe path is stopped. If the path is stopped at a certain point prior to t or if we want toemphasize that the path runs until t , we use the notation ( t, ω t ).The class of non-anticipative functionals is defined as follows: A non-anticipative func-tional on D ([0 , T ] , R d ) is a measurable map F : (Λ dT , d ∞ ) → ( R , B ( R )). The notion “non-anticipative” describes a functional on the path space which only depends of past values.As mentioned in Bally et al. (2016), every progressively measurable process can be repre-sented as a non-anticipative functional and conversely.To define a suitable class of non-anticipative functionals for a path dependent Itˆo formula,some regularity properties are needed, in particular the notion of continuity. Continuityof a non-anticipative functional F : Λ dT → R is defined as continuity as function betweenthe metric spaces (Λ dT , d ∞ ) and ( R , | · | ). Let C , (Λ dT ) denote the set of all continuousnon-anticipative functionals. A weaker concept is continuity at fixed times , i. e. for all t ∈ [0 , T ) the map F ( t, . ) : ( D ([0 , T ] , R d ) , k · k ∞ ) → ( RR, | · | ) is continuous. F is called left-continuous if F is continuous at fixed times and the following holds ∀ ( t, ω ) ∈ Λ dT , ∀ ε > , ∃ δ > , ∀ ( s, ˜ ω ) ∈ Λ dT , (cid:2) s < t and d ∞ (( t, ω ) , ( s, ˜ ω )) < δ (cid:3) ⇒ | F ( t, ω ) − F ( s, ˜ ω ) | < δ. The set of all left-continuous non-anticipative functionals is denoted by C , l (Λ dT ).The property of being boundedness preserving is crucial for various results in Bally et al.(2016) and a precondition for the functional Itˆo’s formula. A non-anticipative functional F : Λ dT → R is called boundedness preserving if for any compact K ⊂ R d and t < T holds ∃ C K,t > , ∀ t ≤ t , ∀ ω ∈ D ([0 , T ] , R d ) , ω ([0 , t ]) ⊂ K ⇒ | F ( t, ω ) | ≤ C K,t . Denote by B (Λ dT ) the set of boundedness preserving functionals and by C , b the set ofcontinuous boundedness preserving functionals.The derivatives which are used for the functional Itˆo calculus are the horizontal and thevertical derivative. For the horizontal derivative, a stopped path ( t, ω ) ∈ Λ dT is extendedto the interval [0 , t + h ] by its value at time t , i.e. to ( t + h, ω t ). Definition 2.1.
A non-anticipative functional F : Λ dT → R is said to be horizontallydifferentiable at ( t, ω ) ∈ Λ dT if the following limit exists D F ( t, ω ) = lim h ↓ F ( t + h, ω t ) − F ( t, ω t ) h . f F is horizontally differentiable at all ( t, ω ) ∈ Λ dT , then D F is a non-anticipative func-tional, called the horizontal derivative of F . For the vertical derivative, the stopped path at the stopping point is disturbed by aconstant x ∈ R d . For a path ω ∈ D ([0 , T ] , R d ) we denote the disturbed path by ω x,t := ω t + x [ t,T ] . Definition 2.2.
A non-anticipative functional F : Λ dT → R is said to be vertically differ-entiable at ( t, ω ) ∈ Λ dT if the map R d → R x F ( t, ω x,t ) is differentiable in . Its gradient at is called the vertical derivative of F at ( t, ω ) : ∇ ω F ( t, ω ) = ( ∇ ωi F ( t, ω ) , i = 1 , . . . , d ) , where for the standard base ( e i ) ≤ i ≤ d of R d the derivatives are defined by ∇ ωi F ( t, ω ) = lim h → F ( t, ω t + he i [ t,T ] ) − F ( t, ω t ) h . If F is vertically differentiable at all ( t, ω ) ∈ Λ dT , then ∇ ω F is a non-anticipative functionalcalled the vertical derivative of F . For each x ∈ R d , ∇ ω F ( t, ω ) .x is the directional derivative of F ( t, . ) in direction [ t,T ] x .As usual one may differentiate multiple times, if possible; we denote this by a superscript, ∇ ω , . . . , ∇ kω . Note that even if considering only continuous paths, one still has to use Λ dT for the definition of vertical differentiability to make sense.For example the non-anticipative functional F ( t, ω ) = f ( t, ω t ) with f ∈ C , ([0 , T ] × R d )has horizontal and vertical derivatives which are simply the partial (right-) derivatives of f . Thus, Definitions 2.1 and 2.2 are an extension of the notion of partial derivatives.The next definition introduces a class of regular non-anticipative functionals which issuitable for a path-wise Itˆo formula. Definition 2.3.
Define C , b (Λ dT ) as the set of left-continuous non-anticipative functionals F ∈ C , l (Λ dT ) such that– F is horizontally differentiable at all points ( t, ω ) ∈ Λ dT and D F is continuous at fixedtimes;– F is twice vertically differentiable and ∇ ω F, ∇ ω F ∈ C , l ;– D F, ∇ ω F, ∇ ω F ∈ B (Λ dT ) .
4n Bally et al. (2016) it is pointed out that one might use as well right continuity. Toapply the pathwise calculus to semimartingales, we use the left-continuity such that theintegrands in the pathwise Itˆo formula are predictable. For the following examples ofhorizontally and vertically differentaible functionals, see Bally et al. (2016).
Example 2.4.
1. Let g ∈ C ( R d ) and ρ : R + → R be bounded and measurable. Then anon-anticipative functional in C , ∞ b (Λ dT ) is given by F ( t, ω ) := Z t g ( ω s ) ρ ( s ) ds. The horizontal derivative is given by D F ( t, ω ) = g ( ω t ) ρ ( t ) and the vertical derivativeis ∇ ωi F ( t, ω ) = 0 .2. Let < t < · · · < t n be some points in [0 , T ] , g ∈ C ( R n × d ) and h ∈ C k ( R k ) with h (0) = 0 . Then F ( t, ω ) = h ( ω t − ω t − n ) t ≥ t n g ( ω t − , ω t − , . . . , ω t − n ) is of class C ,kb (Λ dT ) . The horizontal derivative is D F ( t, ω ) = 0 and the verticalderivative is ∇ ωi F ( t, ω ) = ∂ i h ( ω t − ω t − n ) t ≥ t n g ( ω t − , ω t − , . . . , ω t − n ) . Definition 2.3 can be extended by localization.
Definition 2.5.
A non-anticipative functional F ∈ C , b (Λ dT ) is called locally regular ifthere exists an increasing sequence ( τ k ) k ∈ N of stopping times with τ = 0 , τ k ↑ ∞ and F k ∈ C , b (Λ dT ) such that F ( t, ω ) = X k ∈ N F k ( t, ω ) [ τ k ,τ k +1 ) ( t ) . The set of all locally regular functionals is denoted by C , loc (Λ dT ) . By definition C , b (Λ dT ) ⊂ C , loc (Λ dT ); a difference is that there may be discontinuities orexplosions at the stopping times of the locally regular non-anticipative functionals.A main result in Bally et al. (2016) is a path-dependent Itˆo formula for paths of semi-martingales. Theorem 2.6 (Functional Itˆo formula) . Let X be an R d -valued semimartingale. Then for ll F ∈ C , loc (Λ dT ) and all t ∈ [0 , T ] we have almost surely F ( t, X t ) − F (0 , X ) = Z t D F ( s, X s − ) ds + 12 X ≤ i,j ≤ d Z t ∇ ωi,j F ( s, X s − ) d [ X ] cijs + X ≤ i ≤ d Z t ∇ ωi F ( s, X s − ) dX is (2.1)+ X s ∈ (0 ,t ] F ( s, X s ) − F ( s, X s − ) − X ≤ i ≤ d ∇ ωi F ( s, X s − )∆ X is . Remark 2.7.
In Bally et al. (2016) a more general version of the functional Itˆo formula isderived. Therefore the quadratic variation along a sequence of partitions and the F¨ollmerintegral is used. This is established by a non probabilistic pathwise approach, based onideas from F¨ollmer (1981). In the case of semimartingales this reduces to the quadraticvariation and the F¨ollmer integral coincides with the stochastic integral. This implies thatthe comparison results in our paper can be stated for more general processes, e.g. forfractional processes. However our approach relies on (local) martingale properties and canhence not be transferred directly.
Based on the functional Itˆo formula in this section ordering results are derived for path-dependent functions of semimartingales by an extension of the martingale comparisonmethod for the path-independent case. The first main step is to develop a version of theKolmogorov backwards equation for path-dependent functions. This equation then allowsto derive comparison results under equivalent martingale measures and w.r.t. semimartin-gale measures using the path-dependent Itˆo formula in an essential way.
In this subsection we establish a path-dependent version of the Kolmogorov backwardsequation. Let X be a (special) semimartingale on a filtered probability space (Ω , F , ( F t ) t ∈ [0 ,T ] , P ).We denote by ˆ X = (id , X ) the corresponding space-time process. Let ( B, C, ν ) be thesemimartingale characteristics of ˆ X under P and denote by ( b, c, K ) the differential char-acteristics under P with respect to an increasing process A , see Jacod and Shiryaev (2003).We denote by dA the measure associated to A and by a superscript the dimension of thesemimartingale. In the sequel we write X T for the whole path of X . For a non-anticipative6unctional F ∈ C , loc (Λ dT ), we define the increment functional H F :Λ dT × R d → R , ( t, ω, x ) F ( t, ω t − + x [ t,T ] ) − F ( t, ω t − ) − X ≤ i ≤ d ∇ ωi F ( t, ω t − ) x i . The following is a path-dependent version of the
Kolmogorov backwards equation for thecase that the underlying semimartingale is a local martingale.
Proposition 3.1.
Let F ∈ C , loc (Λ dT ) and let X be a local martingale. Assume that:(i) ( F ( t, X t )) t ≥ is a local martingale,(ii) | H F | ∗ µ X ∈ A + loc ;Then the following holds dA × P almost surely U t F ( t, X t − ) := D F ( t, X t − ) b t + 12 X i,j ≤ d ∇ ωij F ( t, X t − ) c ijt + Z R d H F ( t, X t − , x ) K t ( dx ) = 0 . (3.1) Proof.
By Itˆo’s formula for non-anticipative functionals, F has the following representation F ( t, X t ) = F (0 , X ) + Z t D F ( s, X s − ) b s dA s + X i ≤ d Z t ∇ ωi F ( s, X s − ) dX is + 12 X i,j ≤ d Z t ∇ ωij F ( s, X s − ) c ijs dA s + Z [0 ,t ] × R d F ( s, X s − + x [ s,T ] ) − F ( s, X s − ) − X i ≤ d ∇ ωi F ( s, X s − ) x i µ X ( ds, dx ) . We compensate the jump integral and combine the local martingales from the dX integralsand the compensated jump integral to a local martingale ( M t ) t ∈ [0 ,T ] . Then we have F ( t, X t ) = F (0 , X ) + M t + Z t D F ( s, X s − ) b S dA s + 12 X i,j ≤ d Z t ∇ ωij F ( s, X s − ) c ijs dA s + Z [0 ,t ] × R d H F ( s, X s − , x ) K s ( dx ) dA s . It follows that the process Z t D F ( s, X s − ) b s + 12 X i,j ≤ d ∇ ωij F ( s, X s − ) c ijs + Z R d H F ( s, X s − , x ) K s ( dx ) dA s = Z t U s F ( s, X s − ) dA s .
7s a predictable local martingale of finite variation starting in zero. As consequence thisprocess is almost surely zero. Thus, the integrand has to be dA × P almost surely zero aswell.We proceed with the case when X is a special semimartingale, which implies that theprocess ˆ X is a special semimartingale as well. Recall that we can use the identity astruncation function and hence 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 following result then states a path-dependent version of the Kolmogorov backwardsequation for special semimartingales.
Proposition 3.2.
Let F ∈ C , loc (Λ dT ) and let X be a special semimartingale. Assume that:(i) ( F ( t, X t )) t ≥ is a local martingale;(ii) | H F | ∗ µ X ∈ A + loc ;Then the following holds dA × P almost surely ¯ U t F ( t, X t − ) := D F ( t, X t − ) b t + X i ≤ d ∇ ωi F ( t, X t − ) b it + 12 X i,j ≤ d ∇ ωij F ( t, X t − ) c ijt + Z R d H F ( t, X t − , x ) K t ( dx ) = 0 . (3.2) Proof.
Itˆo’s formula for non-anticipative functionals yields F ( t, X t ) = F (0 , X ) + Z t D F ( s, X s − ) b s dA s + X i ≤ d Z t ∇ ωi F ( s, X s − ) dX is + 12 X i,j ≤ d Z t ∇ ωij F ( s, X s − ) c ijs dA s + Z [0 ,t ] × R d F ( s, X s − + x [ s,T ] ) − F ( s, X s − ) − X i ≤ d ∇ ωi F ( s, X s − ) x i µ X ( ds, dx ) . We unite the local martingales into one local martingale M as in the proof of Proposition3.1. Here these are, by the canonical decomposition, the integrals with respect to X c andwith respect to the compensated jump integrals. As a result we obtain F ( t, X t ) = F (0 , X ) + M t + Z t D F ( s, X s − ) b s dA s + X i ≤ d Z t ∇ ωi F ( s, X s − ) b is dA s + 12 X i,j ≤ d Z t ∇ ωij F ( s, X s − ) c ijs dA s + Z [0 ,T ] × R d H F ( s, X s − , x ) K s ( dx ) dA s .
8o the process Z t D F ( s, X s − ) b s + X i ≤ d ∇ ωi F ( s, X s − ) b is + 12 X i,j ≤ d ∇ ωij F ( s, X s − ) c ijs + Z R d H F ( s, X s − , x ) K s ( dx ) dA s = Z t ¯ U s F ( s, X s − ) dA s . is a predictable local martingale of finite variation starting in zero implying that it is almostsurely zero. Thus, the integrand has to be dA × P almost surely zero as well. Based on the Kolmogorov backwards equations in Section 3.1 we derive path-dependentcomparison results under e.m.m.. Therefore, let X and Y be semimartingales which possessan e.m.m. each. We denote the e.m.m. and semimartingale characteristics which occur bysuperscript to make clear to which semimartingale they correspond.We introduce the path-dependent propagation operator (valuation functional) . There-fore, let f : D ([0 , T ] , R d ) → R be a measurable function then we define the valuationfunctional G f by G f ( t, ω ) := E Q X (cid:2) f ( X T ) (cid:12)(cid:12) X t = ω t (cid:3) . (3.3)This is a non-anticipative functional. Considering G f ( t, X t ) = E Q X [ f ( X T ) | σ ( X s ; s ≤ t )] , we see that this functional takes into account the complete past of the semimartingale X and that it is by construction a martingale with respect to the natural filtration generatedby X . In that case G f ( t, X t ) is a martingale and fulfills equation (3.1).Since we need to control the second vertical derivatives, we need the following path-dependent notion of convexity from Riga (2015). Definition 3.3.
A non-anticipative functional F : Λ dT → R is called vertically convex on U ⊂ Λ dT if for all ( t, ω ) ∈ U there exists a neighbourhood V ⊂ R d of such that the map V → R e → F (cid:0) t, ω t + e [ t,T ] (cid:1) (3.4) is convex. F ∈ C , (Λ T ) which is vertically convex it holds thatthe matrix of the second vertical derivative is positive semidefinite. This follows directlyfrom the definition of the vertical directional derivative in Definition 2.2, and the convexityof the function in (3.4).In the sequel also vertical directional convexity is a relevant property for the comparisonresults. We define it analogously to vertical convexity. Definition 3.4.
A non-anticipative functional F : Λ dT → R is called vertically directionalconvex on U ⊂ Λ dT if for all ( t, ω ) ∈ U there exists a neighbourhood V ⊂ R d of such thatthe map V → R e → F (cid:0) t, ω t + e [ t,T ] (cid:1) is directionally convex. For the notion of vertical directional convexity it holds that: F ∈ C , (Λ T ) is vertically directional convex on U if and only if ∇ ωij F ( t, ω ) ≥ i, j ≤ d and all ( t, ω ) ∈ U . Theorem 3.5 (Vertical directional convex comparison under e.m.m.) . Let
X, Y be semi-martingales such that X = Y = x almost surely and let f ( X T ) ∈ L ( Q X ) , f ( Y T ) ∈ L ( Q Y ) . Assume that(i) G f ∈ C , loc (Λ dT ) and G f is vertically directional convex on Λ dT ;(ii) U t G f ( t, Y t − ) = 0 holds dA × Q Y almost surely for all t ∈ [0 , T ] where the operator U is defined in (3.1) with the differential semimartingale characteristics of ˆ X under Q X ;(iii) | H G f | ∗ µ Y ∈ A + loc ;(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 Y almost surely ordered for all i, j ≤ d ;i.e. c ˆ Y ijt ≤ c ˆ Xijt , Z R d H G f ( t, Y t − , x ) K ˆ Yt ( dx ) ≤ Z R d H G f ( t, Y t − , x ) K ˆ Xt ( dx ) . hen it holds that E Q Y (cid:2) f ( Y T ) (cid:3) ≤ E Q X (cid:2) f ( X T ) (cid:3) . If the inequalities in ( vi ) are reversed and ( G f ( t, Y t ) + ) t ∈ [0 ,T ] is of class (DL), we have E Q Y (cid:2) f ( Y T ) (cid:3) ≥ E Q X (cid:2) f ( X T ) (cid:3) . Proof.
For the proof we establish that the process ( G f ( t, Y t )) t ∈ [0 ,T ] is a Q Y -supermartingale.Then it follows that E Q Y (cid:2) f ( Y T ) (cid:3) = E Q Y (cid:2) G f ( T, Y T ) (cid:3) ≤ G f (0 , x ) = E Q X (cid:2) f ( X T ) (cid:3) . Since G f ∈ C , loc (Λ dT ), we can apply Itˆo’s formula for non-anticipative functionals and obtainthat ( G f ( t, Y t )) t ∈ [0 ,T ] is a semimartingale with decomposition G f ( t, Y t )= G f (0 , x ) + Z t D G f ( s, Y s − ) b ˆ Ys dA ˆ Ys + X i ≤ d Z t ∇ ωi G f ( s, Y s − ) dY is + 12 X i,j ≤ d Z t ∇ ωij G f ( s, Y s − ) c ˆ Y ijs dA ˆ Ys + Z [0 ,t ] × R d G f ( s, Y s − + x [ t,T ] ) − G f ( s, Y s − ) − X i ≤ d ∇ ωi G f ( s, Y s − ) x i µ Y ( ds, dx ) . We compensate the jump integral and combine the local martingales into M . Keeping inmind that Y is a Q Y local martingele, this leads to G f ( t, Y t ) = G f (0 , x ) + M t + Z t D G f ( s, Y s − ) b ˆ Ys dA ˆ Ys + 12 X i,j ≤ d Z t ∇ ωij G f ( s, Y s − ) c ˆ Y ijs dA ˆ Ys + Z [0 ,t ] × R d H G f ( s, Y u − , x ) K ˆ Ys ( dx ) dA ˆ Yu . To gain the local supermartingale property we show that the following process ( Z t ) isdecreasing: Z t := Z t D G f ( s, Y s − ) b ˆ Ys + 12 X i,j ≤ d ∇ ωij G f ( s, Y s − ) c ˆ Y ijs + Z R d H G f ( s, Y s − , x ) K ˆ Ys ( dx ) dA ˆ Ys . By Assumption ( v ) we have that b ˆ Yt dA ˆ Yt = b ˆ Xt dA ˆ Yt = dt . With Assumption ( ii ) we obtain Z t = Z t X i,j ≤ d ∇ ωij 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 . vi ) the first integrand is non-positive. Thatthe second integrand is non-positive follows by Assumption ( vi ).Therefore, − Z ∈ A + loc and ( G f ( t, Y t )) t ∈ [0 ,T ] is a local Q Y -supermartingale.Finally, by Assumption ( iv ) follows that ( G f ( t, Y t )) t ∈ [0 ,T ] is a proper Q Y supermartingale.With reversed inequalities and assuming that ( G f ( t, Y t ) + ) t ∈ [0 ,T ] is of class (DL), we getthe submartingale property for ( G f ( t, Y t )) t ∈ [0 ,T ] . Remark 3.6.
1. Instead of demanding that the kernels are ordered for H G f , we couldalso have demanded that they are ordered for a bigger function class, for examplefor all functions which are directionally convex. Note that by vertical directionalconvexity of G f , H G f is directionally convex in x .2. Compared to previous papers on this topic we do not need the propagation of orderproperty. The propagation of order means that the propagation operator maps partic-ular function classes, like (directional) convex functions or increasing functions, intothemselves. Since we consider a single function we only assume that the propagtionoperator maps this function into the class of vertically directional convex functions. Next we consider the case that G f is a vertically convex function. Theorem 3.7 (Vertical convex comparison under e.m.m.) . Let
X, Y be semimartingaleswith X = Y = x almost surely and let f ( X T ) ∈ L ( Q X ) , f ( Y T ) ∈ L ( Q Y ) . Assume that(i) G f ∈ C , loc (Λ dT ) and G f is vertically convex;(ii) – ( v ) of Theorem 3.5 hold;(vi) The differential characteristics are dA ˆ Y × Q Y almost surely ordered: c ˆ Yt ≤ psd c ˆ Xt , Z R d H G f ( t, Y t − , x ) K ˆ Yt ( dx ) ≤ Z R d H G f ( t, Y t − , x ) K ˆ Xt ( dx ) . Then it holds that E Q Y (cid:2) f ( Y T ) (cid:3) ≤ E Q X (cid:2) f ( X T ) (cid:3) . If the inequalities in ( vi ) are reversed and ( G f ( t, Y t ) + ) t ∈ [0 ,T ] is of class (DL), we have E Q Y (cid:2) f ( Y T ) (cid:3) ≥ E Q X (cid:2) f ( X T ) (cid:3) . roof. We show that ( G f ( t, Y t )) t ∈ [0 ,T ] is a Q Y -supermartingale. Analogously to the proofof Theorem 3.5 we need to show, that dA ˆ Y × Q Y a.s.12 X i,j ≤ d ∇ ωij 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) ≤ . (3.5)Then the assertion follows since the other terms in the functional Itˆo formula are localmartingales. As in the proof of Theorem 3.4 in K¨opfer and R¨uschendorf (2019) we get bypositive definiteness, that the eigendecomposition of the matrix − ( c ˆ Ys − c ˆ Xs ) = c ˆ Xs − c ˆ Ys has the form ( P k ≤ d λ k e ik e jk ) i,j ≤ d with eigenvalues λ k ≥ e k . We obtainequality of the first process above with − X k ≤ d λ k X i,j ≤ d ∇ ωij G f ( s, Y s − ) e ik e jk = − X k ≤ d λ k e ′ k ∇ ω G f ( s, Y s − ) e k , which is non-positive dA ˆ Y × Q Y almost surely due to the positive semidefiniteness of thematrix ∇ ω G f .The second integrand is non-positive dA ˆ Y × Q Y almost surely by Assumption ( vi ). WithAssumption ( iv ) it follows that ( G f ( t, Y t )) t ∈ [0 ,T ] is a proper supermartingale.If the inequalities in ( vi ) are reversed and ( G f ( t, Y t ) + ) t ∈ [0 ,T ] is of class (DL), we have that( G f ( t, Y t )) t ∈ [0 ,T ] is a submartingale.With the help of the key inequality of the proofs above, we can state a corollary whichdoes not need the assumption of vertical convexity or vertical directional convexity butonly uses the inequality in (3.5) for a comparison result. Corollary 3.8 (General comparison condition under e.m.m.) . Let
X, Y be semimartingalesand let X = Y = x almost surely. Further let f be such that f ( X T ) ∈ L ( Q X ) and f ( Y T ) ∈ L ( Q Y ) . Assume that G f ∈ C , loc (Λ dT ) and that Assumptions ( ii ) – ( v ) of Theorem3.5 hold. Further, let dA ˆ Y × Q Y almost surely inequality (3.5) hold. Then we obtain E Q Y (cid:2) f ( Y T ) (cid:3) ≤ E Q X (cid:2) f ( X T ) (cid:3) If the inequality is reversed and ( G f ( t, Y t ) + ) t ∈ [0 ,T ] is of class (DL), then we obtain E Q Y (cid:2) f ( Y T ) (cid:3) ≥ E Q X (cid:2) f ( X T ) (cid:3) . Proof.
The process Z from the proof of Theorem 3.5 is by inequality (3.5) decreasing andhence G f is a supermartingale. The inverse inequality follows since Z then is increasingand hence G f is a submartingale. 13he Girsanov transform can be used to compare the expectation under different e.m.m.This leads to the path-dependent version of Corollary 3.8 in K¨opfer and R¨uschendorf(2019). By Girsanov’s theorem only the compensator of the jump measure changes,the predictable quadratic variation of the continuous martingale part and the increas-ing process of a good version of the semimartingale characteristics remain the same, cf.Jacod and Shiryaev (2003, Theorem III.3.24). Corollary 3.9 (Comparison of e.m.m.) . Let X be a semimartingale. Let Q and Q be equivalent local martingale measures for X . We denote the particular semimartingalecharacteristics of X by superscript. Assume that f ( X T ) ∈ L ( Q ) ∩ L ( Q ) and that(i) G f ∈ C , loc (Λ dT ) ,(ii) U Xt G f ( t, X t − ) = 0 , dA ˆ X × Q almost surely where U Xt is defined as in (3.1) 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 H G f ( t, X t − , x ) K t ( dx ) ≤ Z R d H G f ( t, X t − , x ) K t ( dx ) . Then it holds E Q (cid:2) f ( X T ) (cid:3) ≤ E Q (cid:2) f ( X T ) (cid:3) . If the inequality in ( v ) is reversed and ( G f ( t, X t ) + ) t ∈ [0 ,T ] is of class (DL), then: E Q (cid:2) f ( X T ) (cid:3) ≥ E Q (cid:2) f ( X T ) (cid:3) . Proof.
This follows with help of the functional Itˆo formula in a similar way as in the pathindependent case replacing the horizontal derivative of G f by the vertical derivatives. Thisreplacement is possible by Assumption ( ii ). The following results are versions of Theorems 3.5 and 3.7 under the semimartingale mea-sure P . Let X and Y be special semimartingales. Then the space-time processes ˆ X andˆ Y are special semimartingales and we can choose for both semimartingales the same in-tegrator process A for a good version of the semimartingale characteristics, for details seeK¨opfer (2019, Section 4.4.). 14e adapt the non-anticipative value functional G f from equation (3.3) to P : G f ( t, ω ) := E [ f ( X T ) | X t = ω t ] . In the path-independent comparison under P in K¨opfer and R¨uschendorf (2019) it is as-sumed that G f ( t, · ) is an increasing function for all t ∈ [0 , T ] in order to control the firstpartial derivative. To control the first vertical derivative of non-anticipative functionals weintroduce vertical monotonicity. Definition 3.10.
A non-anticipative functional F : Λ dT → R is called vertically monotone on U ⊂ Λ dT if for all ( t, ω ) ∈ U there exists a neighbourhood V ⊂ R d of such that the map V → R e → F ( t, ω t + e [ t,T ] ) is monotone in e . This definition guarantees that the first vertical derivative is non-negative or non-positiveif it exists.
Theorem 3.11 (Vertically increasing and vertically directional convex comparison underP) . Let
X, Y be special semimartingales and let X = Y = x almost surely. Consider afunction f ∈ L ( P X T ) ∩ L ( P Y T ) and assume that(i) G f ∈ C , loc (Λ dT ) and G f is vertically directionally convex and vertically increasing on Λ dT ;(ii) ¯ U t G f ( t, Y t − ) = 0 holds dA × P almost surely for all t ∈ [0 , T ] , where ¯ U is defined asin (3.2) with the characteristics of ˆ X ;(iii) | H G f | ∗ 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: b ˆ Y it ≤ b ˆ Xit ,c ˆ Y ijt ≤ c ˆ Xijt , Z R d H G f ( t, Y t − , x ) K ˆ Yt ( dx ) ≤ Z R d H G f ( t, Y t − , x ) K ˆ Xt ( dx ) . 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 get E [ f ( Y T )] ≥ E [ f ( X T )] . roof. Analogously to the comparison under equivalent martingale measures we establishthat ( G f ( t, Y t )) t ∈ [0 ,T ] is a supermartingale. Therefore, using the functional Itˆo formula wehave to verify that dA × P almost surely it holds X i ≤ d ∇ ωi G f ( s, Y s − ) (cid:16) b ˆ Y is − b ˆ Xis (cid:17) + 12 X i,j ≤ d ∇ ωij 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) ≤ . This process however is non-positive dA × P almost surely by Assumption ( v ) and usingthat G f is vertically increasing and vertically directional convex. Assumption ( iv ) thenyields the proper supermartingale property.If the inequalities in ( v ) are reversed and ( G f ( t, Y t ) + ) t ∈ [0 ,T ] is of class (DL), G f is asubmartingale.Next we transfer the comparison result to the case when G f is vertically convex andvertically increasing. Theorem 3.12 (Vertically increasing and vertically convex comparison under P) . Let
X, Y be special semimartingales and let X = x = Y almost surely. Let f ∈ L ( P X T ) ∩ L ( P Y T ) . Assume that(i) G f ∈ C , (Λ dT ) and G f is vertically convex and vertically increasing on Λ T ;(ii) – ( iv ) of Theorem 3.11 hold;(v) The differential characteristics are dA × P almost surely ordered for all i ≤ d : b ˆ Y it ≤ b ˆ Xit ,c ˆ Yt ≤ psd c ˆ Xt , Z R d H G f ( t, Y t − , x ) K ˆ Yt ( dx ) ≤ Z R d H G f ( t, Y t − , x ) K ˆ Xt ( dx ) . Then it holds that E [ f ( Y T )] ≤ E [ f ( X T )] . If in ( v ) the inequalities are reversed and ( G f ( t, Y t ) + ) t ∈ [0 ,T ] is of class (DL), we get E [ f ( Y T )] ≥ E [ f ( X T )] . roof. Again using the functional Itˆo formula we have to verify, that dA × P a.s. X i ≤ d ∇ ωi G f ( s, Y s − ) (cid:16) b ˆ Y is − b ˆ Xis (cid:17) + 12 X i,j ≤ d ∇ ωij 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) ≤ . The first term is non positive due to Assumption ( v ) and the fact that G f is verticallyincreasing in the second variable. The remaining part is non-positive as in the proof ofTheorem 3.7. By Assumption ( iv ) it follows that ( G f ( t, Y t )) t ∈ [0 ,T ] is a proper supermartin-gale.If the inequalities in ( v ) are reversed and ( G f ( t, Y t ) + ) t ∈ [0 ,T ] is of class (DL), then G f is asubmartingale.As before the key inequality of the proof can be used to formulate a comparison resultwithout the assumption of vertical convexity and vertical monotonicity on the functional G f . Corollary 3.13 (General comparison condition under P) . Let
X, Y be special semimartin-gales and let X = x = Y almost surely. Let f ∈ L ( P X T ) ∩ L ( P Y T ) . Assume that G f ∈ C , loc (Λ dT ) and that ( ii ) – ( iv ) of Theorem 3.11 hold. Further assume that dA × P almost surely X i ≤ d ∇ ωi G f ( s, Y s − ) (cid:16) b ˆ Y is − b ˆ Xis (cid:17) + 12 X i,j ≤ d ∇ ωij 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) ≤ . (3.6) Then it holds that E [ f ( X T )] ≤ E [ f ( Y T )] . If inequality (3.6) is reversed and ( G f ( t, Y t ) + ) t ∈ [0 ,T ] is of class (DL), then E (cid:2) f ( X T ) (cid:3) ≥ E (cid:2) f ( Y T ) (cid:3) . Proof.
As in the proof of Theorem 3.12 inequality (3.6) implies that ( G f ( t, Y t )) t ∈ [0 ,T ] is asupermartingale or submartingale respectively. The comparison results in Section 3 need various properties of the valuation functional G f ,like continuity, vertical/horizontal differentiability and convexity. In this section we give17ome results establishing these regularity properties and some applications to comparisonresults.We first discuss the regularity of G f . For notational simplicity we consider the processesunder the semimartingale measure P .An example for a vertically differentiable conditional expectation is given in Riga (2015,Proposition 4.4) who states conditions such that the conditional expectation of a path-dependent function of a semimartingale can be represented as horizontally differentiablenon-anticipative functional. The underlying process is a stochastic exponential defined bythe SDE dS t = S t σ t dB t , where B is a standard Brownian motion and ( σ t ) t ∈ [0 ,T ] is a non-negative adapted processsuch that S is a L -martingale.We modify this approach to transfer it to non-continuous processes. Therefore, weconsider the probability space (Ω , ( F t ) t ∈ [0 ,T ] , F , P ), where Ω = D ([0 , T ] , R d ), F is the Borelsigma-field and ( F t ) t ∈ [0 ,T ] is the filtration generated by the canonical process, X t ( ω ) = ω ( t ).We assume that the canonical process is a semimartingale.In the center of our considerations in the previous section is the valuation functional G f : Λ dT → R , G f ( t, ω ) := E (cid:2) f ( X T ) (cid:12)(cid:12) X t = ω t (cid:3) . In the setting of this section this is the same as the expectation w.r.t. factorized conditionalprobability of X T given F t due to the fact that ( F t ) t ∈ [0 ,T ] is the natural filtration. If thespace of c`adl`ag functions is equipped with the Skorokhod topology there exists a regularversion of the conditional probability of X T given X t since D ([0 , T ] , R d ) then is a Polishspace. However, for the sup norm this is not valid anymore, see Billingsley (1968). Weassume in the sequel that a regular version of the conditional probability exists as in thecase of processes with continuous paths. Then G f takes the form G f ( t, ω ) = Z D ([0 ,T ] , R d ) f (˜ ω ) P X T | X t = ω t ( d ˜ ω ) . and, therefore, the horizontal and vertical differentiability is mainly a question of corre-spondent differentiability of the kernel P X T | X t .Since the metric in the space of stopped paths uses in the path component the sup norm,we need a tool to handle the sup norm of a semimartingale. This motivates the use of theclass of H semimartingales (for details see Protter (2005)). Without loss of generality weassume that all semimartingales in this section start in zero. For simplicity we considerone-dimensional semimartingales. Let X be a semimartingale; then there exists at least18ne decomposition X = M + B , where M is a local martingale and B is of finite variation.Denoting for 1 ≤ p ≤ ∞ j p ( M, B ) := (cid:13)(cid:13)(cid:13)(cid:13) [ M ] T + Z T | dB s | (cid:13)(cid:13)(cid:13)(cid:13) L p , then the H p norm of X is defined as k X k H p = inf X = M + B j p ( M, B ) , where the infimum is taken over all possible semimartingale decompositions of X .By Protter (2005, Chapter V, Theorem 2) the H p -norm allows to dominate the sup normof X . This is a consequence of Burkholder’s inequalities and is an important tool in thesequel. For 1 ≤ p ≤ ∞ there exists a constant c p such that for any semimartingale X with X = 0 we have for X ∗ := sup t ∈ [0 ,T ] | X t | the inequality k X ∗ k L p ≤ c p k X k H p . (4.1)The following definition reminds the concatenation operators as introduced in Riga (2015).In comparison we use a slightly different definition since we want the c`adl`ag functions tomeet in t . Definition 4.1.
The family of concatenation operators ( ⊕ t ) t ∈ [0 ,T ] is defined by ⊕ t : D ([0 , T ] , R ) × D ([0 , T ] , R ) → D ([0 , T ] , R ) , ( ω, ω ′ ) ω ⊕ t ω ′ := ω [0 ,t ) + ( ω t + ω ′ − ω ′ t ) [ t,T ] . The idea of the following theorem is to use Lipschitz continuity and independent incre-ments to dominate the increments of the function under consideration. Then we are ableto show the continuity and vertical and horizontal differentiability of G f . Theorem 4.2.
Let X be a semimartingale with finite H norm and independent incrementswithout fixed times of discontinuity. Further, let f : ( D ([0 , T ] , R ) , k·k ∞ ) → R be a Lipschitzcontinuous functional such that E [ | f ( X T ) | ] < ∞ . Assume that for any ω ∈ D ([0 , T ] , R ) and any t ∈ [0 , T ] the function g ( · , t, ω ) : R → R e f ( ω + [ t,T ] e ) (4.2) is twice continuously differentiable in zero such that the derivatives are Lipschitz continuousin ω . Further, assume that for every ω, ω ′ ∈ D ([0 , T ] , R ) the function l ( · , ω, ω ′ ) : [0 , T ] → R t f ( ω ⊕ t ω ′ ) (4.3)19 s continuously right differentiable with derivative which is Lipschitz continuous in ω . Thenit follows that G f ∈ C , b (Λ T ) . Proof.
We obtain by the independence of increments that G f ( t, ω ) = E [ f ( X T ) | X t = ω t ]= E (cid:2) f ( ω ⊕ t X T ) | X t = ω t (cid:3) = E (cid:2) f (cid:0) ω [0 ,t ) + ( ω t + X T − X t ) [ t,T ] (cid:1) |F t (cid:3) ( ω )= E (cid:2) f ( ω ⊕ t X T ) (cid:3) . Since f is Lipschitz continuous, it follows that for all ω, ω ′ ∈ D ([0 , T ] , R ) there exists a c > | f ( ω ) − f ( ω ′ ) | ≤ c k ω − ω ′ k ∞ . We show the continuity of G f by sequentialcontinuity. Let (( t n , ω n )) n ∈ N ⊂ Λ T converge to ( t, ω ) , ∈ Λ T . Then we have | G f ( t, ω ) − G f ( t n , ω n ) | = (cid:12)(cid:12) E (cid:2) f ( ω ⊕ t X T ) (cid:3) − E (cid:2) f ( ω n ⊕ t n X T ) (cid:3) (cid:12)(cid:12) ≤ E (cid:2)(cid:12)(cid:12) f ( ω ⊕ t X T ) − f ( ω n ⊕ t n X T ) (cid:12)(cid:12)(cid:3) ≤ cE (cid:2) k ( ω ⊕ t X T ) − ( ω n ⊕ t n X T ) k ∞ (cid:3) (4.4) ≤ cE (cid:2) k ( ω − ω n ) [0 ,t ∧ t n ) k ∞ + k (cid:0) ω t + X T − X t − ω n (cid:1) [ t,t n ) k ∞ + k (cid:0) ω nt n + X T − X t n − ω (cid:1) [ t n ,t ) k ∞ + k (cid:0) ω t + X T − X t − ω nt n − X T + X t n (cid:1) [ t ∨ t n ,T ] k ∞ (cid:3) . This can be further dominated by cE [ k ( ω − ω n ) [0 ,t ∧ t n ) k ∞ + k ( ω t − ω n ) [ t,t n ) k ∞ + k ( X T − X t ) [ t,t n ) k ∞ + k ( ω nt n − ω ) [ t n ,t ) k ∞ + k ( X T − X t n ) [ t n ,t ) k ∞ + k ( ω t − ω nt n ) [ t ∨ t n ,T ] k ∞ + k ( X t n − X t ) [ t ∨ t n ,T ] k ∞ (cid:3) . Note that for fix n only one of the indicator functions [ t n ,t ) and [ t,t n ) differs from zero.We consider the first term on the right-hand side. It is clearly bounded from above by k ω t − ( ω n ) t n k ∞ which tends to zero by d ∞ convergence. As consequence we obtain E (cid:2) k ( ω − ω n ) [0 ,t ∧ t n ) k ∞ (cid:3) ≤ E (cid:2) k ω t − ( ω n ) t n k ∞ (cid:3) = k ω t − ( ω n ) t n k ∞ → . The same argument yields convergence to zero for the other terms containing ω and ω n .Next we consider the expectation E (cid:2) k ( X T − X t ) [ t,t n ) k ∞ (cid:3) . The process therein X n :=(( X T − X t ) [ t,t n ) ) t ∈ [0 ,T ] is a semimartingale starting in zero, hence we can apply (4.1) with p = 1 to obtain E (cid:2) k ( X T − X t ) [ t,t n ) k ∞ (cid:3) ≤ c k X n k H . X = M + A be a semimartingale decomposition of X . Then after a restriction to M n :=(( M T − M t ) [ t,t n ) ) t ∈ [0 ,T ] and A n := (( A T − A t ) [ t,t n ) ) t ∈ [0 ,T ] we get that X n = M n + A n is a semimartingale decomposition of X n . Since for each n and ω ∈ Ω the path X n ( ω ) isjust a shifted piece of the path of X ( ω ), we have that [ M n ] t ≤ [ M ] t and | A nt | ≤ | A t | for all t ∈ [0 , T ]. This means that we can dominate the H norm of all X n by the H norm of X which is finite by assumption. Dominated convergence and right continuity then leads tolim n →∞ E (cid:2) k ( X T − X t ) [ t,t n ) k ∞ (cid:3) = E " lim n →∞ sup s ∈ [ t,t n ) | X s − X t | = 0 . Analogously we get thatlim n →∞ E (cid:2) k ( X T − X t n ) [ t n ,t ) k ∞ (cid:3) = E " lim n →∞ sup s ∈ [ t n ,t ) | X s − X t n | ≤ E " lim n →∞ sup s ∈ [ t n ,t ] | X s − X t n | = E [ | ∆ X t | ]= 0 . The last equality follows from the assumption that there are no fixed times of discontinuity.It remains to show that E [ k ( X t n − X t ) [ t ∨ t n ,T ] k ∞ ] also tends to zero. Therefore, note that E (cid:2) k ( X t n − X t ) [ t ∨ t n ,T ] k ∞ (cid:3) = E [ | ( X t n − X t ) | ] ≤ E " sup s ∈ [ t n ,t ] | X s − X t n | + sup s ∈ [ t,t n ] | X s − X t | . The terms on the right-hand side are both bounded by the H norm of X . It follows bydominated convergence that this tends to zero. Thus, G f is continuous.Next we show that G f is vertically differentiable. We consider the vertical differencequotient of G f G f ( t, ω h,t ) − G f ( t, ω ) h = 1 h (cid:16) E h f ( ω h,t ⊕ t X T ) i − E (cid:2) f ( ω ⊕ t X T ) (cid:3)(cid:17) = 1 h E (cid:2) f ( ω ⊕ t X T + h [ t,T ] ) − f ( ω ⊕ t X T ) (cid:3) . Since f is Lipschitz continuous, dominated convergence yields ∇ ω G f ( t, ω ) = E (cid:20) ∂∂e g ( e, t, ω ⊕ t X T )(0) (cid:21) . g is assumed to be Lipschitzcontinuous in ω and get by dominated convergence ∇ ω G f ( t, ω ) = lim h → ∇ ω G f ( t, ω h,t ) − ∇ ω G f ( t, ω ) h = E " lim h → ∂∂e g ( e, t, ω h,t ⊕ t X T )(0) − ∂∂e g ( e, t, ω ⊕ t X T )(0) h = E (cid:20) ∂ ∂e g ( e, t, ω ⊕ t X T )(0) (cid:21) . We are left to show that ∇ ω G f and ∇ ω G f are (left-)continuous. In fact we have continuitywhich follows as the continuity of G f from Lipschitz continuity.We now turn to the horizontal differentiability. Therefore, we consider the horizontaldifference quotient. G f ( t + h, ω t ) − G f ( t, ω t ) h = E (cid:2) f ( ω t ⊕ t + h X T ) (cid:3) − E (cid:2) f ( ω t ⊕ t X T ) (cid:3) h . From the Lipschitz continuity of f it follows as in (4.4) that the difference is bounded bythe H norm of X . With dominated convergence it follows for h ↓ D G f ( t, ω ) = E (cid:20) ∂ + ∂t l ( t, ω t , X T ) (cid:21) . The continuity of the derivative now follows from the Lipschitz continuity of the derivativeof l .It remains to show that G f is boundedness preserving. Therefore, let be K ⊂ R becompact and t fixed. We need to show the existence of a constant C K,t > t ≤ t and all ω ∈ D ([0 , T ] , R ) we have ω ([0 , t ]) ⊂ K ⇒ | G f ( t, ω ) | ≤ C K,t . Since K is compact it is bounded; let k be this bound. We obtain from (4.4) and theconsiderations thereafter that | G f ( t, ω ) − G f (0 , | ≤ cE [2 k ω t k ∞ + 2 k X k H ] ≤ c (2 k + 2 k X k H ) =: ˜ C. The term G f (0 ,
0) is just E [ f ( X T )] which is finite. So we get by the choice C K,t =˜ C + | E [ f ( X T )] | that G f is boundedness preserving. Remark 4.3.
1. The Lipschitz continuity helps to show continuity and to apply dom-inated convergence. H¨older continuity as condition on the functions above works aswell. . The property to be boundedness preserving is a local property; it depends on t . In theproof we have seen that under the conditions of Theorem 4.2 G f is even “globally”boundedness preserving.3. By Jacod and Shiryaev (2003, Corollary II.4.18) the property “without fixed timesof continuity” is for processes with independent increments equivalent to quasi-left-continuity of X .4. The functions g and l from equations (4.2) and (4.3) provide the vertical and hori-zontal differentiability. If only one of the functions has the demanded properties, westill get G f ∈ C , b (Λ T ) or G f ∈ C , b (Λ T ) . We give an example for a semimartingale and the integral functional from Example 2.4which fulfill the conditions of Theorem 4.2.
Example 4.4.
Let X be a compound Poisson process with finite H norm. Then it has nofixed times of discontinuity, see Jacod and Shiryaev (2003, II.4.3). Further, let ˜ f : R → R be Lipschitz continuous and twice continuously differentiable with Lipschitz continuousderivatives and let f be the integral f ( ω ) := R T ˜ f ( ω t ) dt . Assume that E [ | f ( X T ) | ] < ∞ .Then f is Lipschitz continuous in ω ∈ D ([0 , T ] , R ) . This is consequence of the Lipschitzcontinuity of ˜ f : | f ( ω ) − f ( ω ′ ) | ≤ Z T | ˜ f ( ω t ) − ˜ f ( ω ′ t ) | dt ≤ c Z T | ω t − ω ′ t | dt ≤ cT k ω − ω ′ k ∞ . Further, the function g from equation (4.2) is twice continuously differentiable in zero.To see this fix s ∈ [0 , T ] and ω ∈ D ([0 , T ] , R ) ; then we have by Lipschitz continuity anddominated convergence ∂∂e g ( e, s, ω )(0) = lim h → f ( ω + [ s,T ] h ) − f ( ω ) h = lim h → R T ˜ f ( ω t + [ s,T ] h ) − ˜ f ( ω t ) dth = lim h → R Ts ˜ f ( ω t + h ) − ˜ f ( ω t ) dth = Z Ts ˜ f ′ ( ω t ) dt. his expression is Lipschitz continuous in ω since we assumed ˜ f ′ to be Lipschitz continuous.Analoguously we get ∂ ∂e g ( e, s, ω )(0) = Z Ts ˜ f ′′ ( ω t ) dt, which is Lipschitz continuous in ω as well. Thus, g fulfills the conditions of Theorem 4.2.For the function l from (4.3) we show now the right differentiability. Therefore, fix ω, ω ′ ∈ D ([0 , T ] , R ) , then ∂ + ∂t l ( t, ω, ω ′ ) = lim h ↓ f ( ω ⊕ t + h ω ′ ) − f ( ω ⊕ t ω ′ ) h = lim h ↓ h (cid:18)Z t + h ˜ f ( ω s ) ds + Z Tt + h ˜ f ( ω ′ s − ω ′ t + h + ω t ) ds − Z t ˜ f ( ω s ) ds − Z Tt ˜ f ( ω ′ s − ω ′ t + ω t ) ds (cid:19) = lim h ↓ h (cid:18)Z t + ht ˜ f ( ω s ) − ˜ f ( ω ′ s − ω ′ t + ω t ) ds + Z Tt + h ˜ f ( ω ′ s − ω ′ t + h + ω t ) − ˜ f ( ω ′ s − ω ′ t + ω t ) ds (cid:19) = lim h ↓ h (cid:18)Z T ˜ f ( ω ′ s − ω ′ t + h + ω t ) − ˜ f ( ω ′ s − ω ′ t + ω t ) ds − Z t + h ˜ f ( ω ′ s − ω ′ t + h + ω t ) ds + Z t + ht ˜ f ( ω s ) − f ( ω ′ s − ω ′ t + ω t ) ds + Z t ˜ f ( ω ′ s − ω ′ t + ω t ) ds + Z t + ht ˜ f ( ω ′ s − ω ′ t + ω t ) ds (cid:19) = Z T ˜ f ′ ( ω ′ s − ω ′ t + ω t ) ∂ + ∂t ω ′ t ds − ˜ f ( ω t ) ˜ f ′ ( ω t ) ∂ + ∂t ω ′ t + ˜ f ( ω t ) . The first term results from dominated convergence, the second term is the right derivativeof the integral R u ˜ f ( ω ′ s − ω ′ u + ω t ) ds . For a compound Poisson process, the path ω ′ = X isright differentiable and it follows that on such paths D G f = ˜ f ( ω t ) .That G f is boundedness preserving follows as in the proof of Theorem 4.2. Altogether wehave that G f ∈ C , b (Λ T ) . From this example one can see that in this setting the function l can cause problems formore general semimartingales since in the derivation a right derivative of the future pathoccurred. In fact this proceeding works fine for semimartingales of finite variation sincethey are differentiable almost everywhere. But since integrals over path independent func-tions of semimartingales are not of finite variation, we need other conditions for horizontaldifferentiability. 24 xample 4.5. Let B be a Brownian motion. We consider the function f ( ω ) := R T ˜ f ( ω t ) dt from Example 4.4. In contrast to the previous example we only assume that ˜ f is bounded.Then we have by the Markov property and the strong continuity of the corresponding transi-tion semigroup ( T t ) ≤ t ≤ T that the transition semigroup is differentiable in time. It followsthat G f is horizontally differentiable: D G f ( t, ω ) = lim h ↓ E [ f ( ω t ⊕ t + h B T ) − f ( ω t ⊕ t B T )] h = lim h ↓ E [ R T ˜ f (( ω t ⊕ t + h B T ) s ) − ˜ f (( ω t ⊕ t B T ) s ) ds ] h = lim h ↓ R Tt E [ ˜ f (( ω t ⊕ t + h B T ) s ) − ˜ f (( ω t ⊕ t B T ) s )] dsh = lim h ↓ R Tt T ( s − h ) ∧ ˜ f ( ω t ) − T s ˜ f ( ω t ) dsh = Z Tt ∂∂s T s ˜ f ( ω t ) ds. The most important part in the proof of Theorem 4.2 is that we are able to reducethe conditional expectation to a normal expectation. This is a consequence of the inde-pendent increments. Since Markov processes have conditionally independent increments,we can obtain a similar result. In fact in the following proposition we derive for a Fellersemimartingale X under the assumption that f is an integral function that G f ∈ C , b (Λ dT ).We recall the notion of Feller processes, cf. Ethier and Kurtz (2005). A semigroup( T t ) t ≥ on C ( R d ) is called a Feller semigroup if it is strongly continuous. In particular,Feller semigroups map C ( R d ) to C ( R d ). Sometimes also the bigger class C b ( R d ) is usedin the definition of Feller semigroups, we denote this by C b -Feller semigroup. A Markovprocess is called a Feller process if the corresponding transition semigroup is a Feller semi-group. If X is in addition a semimartingale we call it a Feller semimartingale. Note thatby this definition Feller processes are always time-homogeneous.Examples of C b -Feller processes are L´evy processes. For L´evy processes it holds that thetransition semigroup is of the form T t f ( x ) = Z R f ( y + x ) p t ( dy ) . Here p t is the distribution of X t . From this equation we see that T t f inherits the bound-edness and continuity of f . Proposition 4.6.
Let X be a C b -Feller semimartingale with strongly continuous transitionsemigroup ( T t ) ≤ t ≤ T . Further, let ˜ f : R → R be bounded and continuous such that T t ˜ f ∈ C , . We consider the function f : ( D ([0 , T ] , R ) , k · k sup ) → R to be the integral functional f ( ω ) = R T ˜ f ( ω t ) dt . Then it holds that G f ∈ C , b (Λ T ) . roof. By the time-homogeneity and the Markov property we obtain G f ( t, ω ) = E [ f ( X T ) | X t = ω t ]= E (cid:20) Z Tt ˜ f ( X s ) ds (cid:12)(cid:12)(cid:12)(cid:12) X t = ω t (cid:21) + Z t ˜ f ( ω s ) ds = E (cid:20) Z T − t ˜ f ( X s ) ds (cid:12)(cid:12)(cid:12)(cid:12) X = ω t (cid:21) + Z t ˜ f ( ω s ) ds = Z T − t T s ˜ f ( ω t ) ds + Z t ˜ f ( ω s ) ds. (4.5)We show the continuity of G f by sequential continuity. Let ( t n , ω n ) converge in Λ T to( t, ω ). Then we have | G f ( t, ω ) − G f ( t n , ω n ) | = (cid:12)(cid:12)(cid:12)(cid:12) E (cid:20) Z T − t ˜ f ( X s ) ds (cid:12)(cid:12)(cid:12)(cid:12) X = ω t (cid:21) − E (cid:20) Z T − t n ˜ f ( X s ) ds (cid:12)(cid:12)(cid:12)(cid:12) X = ω nt n (cid:21) + Z t ˜ f ( ω s ) ds − Z t n ˜ f ( ω ns ) ds (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T − ( t ∨ t n )0 E h ˜ f ( X s ) (cid:12)(cid:12)(cid:12) X = ω t i − E h ˜ f ( X s ) (cid:12)(cid:12)(cid:12) X = ω nt n i ds + Z T − ( t ∧ t n ) T − ( t ∨ t n ) E h ˜ f ( X s ) (cid:12)(cid:12)(cid:12) X = ω t i { t ≥ t n } − E h ˜ f ( X s ) (cid:12)(cid:12)(cid:12) X = ω nt n i { t n ≥ t } ds + Z t ∧ t n ˜ f ( ω s ) − ˜ f ( ω ns ) ds + Z t ∨ t n t ∧ t n ˜ f ( ω s ) { t ≥ t n } − ˜ f ( ω ns ) { t n ≥ t } ds (cid:12)(cid:12)(cid:12)(cid:12) We first take a closer look at the integrals not depending on X . (cid:12)(cid:12)(cid:12)(cid:12)Z t ∧ t n ˜ f ( ω s ) − ˜ f ( ω ns ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z t ∧ t n | ˜ f ( ω s ) − ˜ f ( ω ns ) | ds ≤ Z t | ˜ f ( ω s ) − ˜ f ( ω ns ) | ds. This converges to zero by dominated convergence using the continuity of ˜ f . Let c be thebound of ˜ f , then we have (cid:12)(cid:12)(cid:12)(cid:12)Z t ∨ t n t ∧ t n ˜ f ( ω s ) { t ≥ t n } − ˜ f ( ω ns ) { t n ≥ t } ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z t ∨ t n t ∧ t n (cid:12)(cid:12)(cid:12) ˜ f ( ω s ) { t ≥ t n } − ˜ f ( ω ns ) { t n ≥ t } (cid:12)(cid:12)(cid:12) ds ≤ c ( t ∨ t n − t ∧ t n ) . X . For the firstterm we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T − ( t ∨ t n )0 E h ˜ f ( X s ) (cid:12)(cid:12)(cid:12) X = ω t i − E h ˜ f ( X s ) (cid:12)(cid:12)(cid:12) X = ω nt n i ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z T − t (cid:12)(cid:12)(cid:12) E h ˜ f ( X s ) (cid:12)(cid:12)(cid:12) X = ω t i − E h ˜ f ( X s ) (cid:12)(cid:12)(cid:12) X = ω nt n i(cid:12)(cid:12)(cid:12) ds This converges to zero since E h ˜ f ( X s ) (cid:12)(cid:12)(cid:12) X = ω t i is continuous in ω and bounded by theFeller property.The last term tends to zero as follows. Let ˜ c be the bound of E h ˜ f ( X s ) (cid:12)(cid:12)(cid:12) X = ω t i whichexists by the Feller property. Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T − ( t ∧ t n ) T − ( t ∨ t n ) E h ˜ f ( X s ) (cid:12)(cid:12)(cid:12) X = ω t i { t ≥ t n } − E h ˜ f ( X s ) (cid:12)(cid:12)(cid:12) X = ω nt n i { t n ≥ t } ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ˜ c ( T − ( t ∨ t n ) − T + ( t ∧ t n )) → . We now turn to the vertical differentiability of G f . G f ( t, ω h,t ) − G f ( t, ω ) h = 1 h (cid:18) E (cid:20) Z T − t ˜ f ( X s ) ds (cid:12)(cid:12)(cid:12)(cid:12) X = ω t + h (cid:21) + Z t ˜ f ( ω s ) ds − E (cid:20) Z T − t ˜ f ( X s ) ds (cid:12)(cid:12)(cid:12)(cid:12) X = ω t (cid:21) − Z t ˜ f ( ω s ) ds (cid:19) = 1 h (cid:18)Z T − t E h ˜ f ( X s ) (cid:12)(cid:12)(cid:12) X = ω t + h i − E h ˜ f ( X s ) (cid:12)(cid:12)(cid:12) X = ω t i ds (cid:19) = 1 h (cid:18)Z T − t T s ˜ f ( ω t + h ) − T s ˜ f ( ω t ) ds (cid:19) Since T t ˜ f ∈ C , by assumption, we obtain ∇ ω G f ( t, ω ) = Z ( T − t )0 ∂∂x T s ˜ f ( ω t ) ds. Analog we receive for the second derivative ∇ ω G f ( t, ω ) = Z ( T − t )0 ∂ ∂x T s ˜ f ( ω t ) ds.
27o compute the horizontal derivative we take a look at the horizontal differential quotient. G f ( t + h, ω t ) − G f ( t, ω ) h = 1 h (cid:18) E (cid:20)Z T − t − h ˜ f ( X s ) ds | X = ω t (cid:21) + Z t ˜ f ( ω s ) ds + h ˜ f ( ω t ) − E (cid:20)Z T − t ˜ f ( X s ) ds | X = ω t (cid:21) − Z t ˜ f ( ω s ) ds (cid:19) = ˜ f ( ω t ) − h Z T − tT − t − h T s ˜ f ( ω t ) ds → ˜ f ( ω t ) − T T − t ˜ f ( ω t ) . It remains to show that G f is boundedness preserving. This follows directly from therepresentation (4.5) since ˜ f and T t ˜ f are both bounded by the Feller property.Next we consider functions which depend of the average of a semimartingale X . Suchfunctions are used in financial mathematics in the framework of Asian options. Example 4.7.
Let X be a semimartingale of finite variation with independent increments,finite H norm and without fixed times of discontinuity. Further, define I t := R t X s ds .We consider a function of the form ˜ f ( T I T ) , where ˜ f : R → R is an integrable function.We assume that ˜ f is twice differentiable with Lipschitz continuous derivatives. The path-dependent function corresponding to ˜ f is f : D ([0 , T ] , R ) → R ω ˜ f (cid:18) T Z T ω t dt (cid:19) . Since the identity on R is Lipschitz continuous, we get as in Example 4.4 that T I T isLipschitz continuous in ω . It follows that f is Lipschitz continuous in ω as well. Denoteby c ˜ f the Lipschitz constant of ˜ f and by c I the Lipschitz constant of T I T . Then it follows | f ( ω ) − f ( ω ′ ) | = (cid:12)(cid:12)(cid:12)(cid:12) ˜ f (cid:18) T Z T ω t dt (cid:19) − ˜ f (cid:18) T Z T ω ′ t dt (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ˜ f (cid:12)(cid:12)(cid:12)(cid:12) T Z T ω t dt − T Z T ω ′ t dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ c ˜ f c I k ω − ω ′ k . We consider the function g from equation (4.2) . Fix s ∈ [0 , T ] and ω ∈ D ([0 , T ] , R ) , thenwe get for the derivative ∂e g ( e, s, ω )(0) = lim h → f ( ω + [ s,T ] h ) − f ( ω ) h = lim h → ˜ f (cid:16) T R T ω t dt + T − sT h (cid:17) − ˜ f (cid:16) T R T ω t dt (cid:17) h = T − sT ˜ f ′ (cid:18) T Z T ω t dt (cid:19) . This is Lipschitz continuous in ω by the same argument as above. For the second deriva-tive we get ∂ ∂e g ( e, s, ω )(0) = ( T − s ) T ˜ f ′′ (cid:18) T Z T ω t dt (cid:19) . So g meets the conditions of Theorem 4.2. For the horizontal derivative, we get D G f ( t, ω )= lim h ↓ E (cid:2) f ( ω t ⊕ t + h X T ) − f ( ω t ⊕ t X T ) (cid:3) h = lim h ↓ h E " ˜ f T Z t ω s ds + hω t + Z Tt + h X s − X t + h + ω t ds !! − ˜ f T Z t ω s ds + Z Tt X s − X t + ω t ds !! = lim h ↓ h E " ˜ f T Z t ω s ds + Z Tt X s + ω t ds − ( T − t − h ) X t + h − Z t + ht X s ds !! − ˜ f T Z t ω s ds + Z Tt X s + ω t ds − ( T − t ) X t !! . We set R t ω s ds + R Tt X s + ω t ds := x and obtain by dominated convergence D G f ( t, ω )= E (cid:20) lim h ↓ h ˜ f (cid:18) T (cid:18) x − ( T − t − h ) X t + h − Z t + ht X s ds (cid:19)(cid:19) − ˜ f (cid:18) T ( x − ( T − t ) X t ) (cid:19)(cid:21) . his can be further computed as D G f ( t, ω )= E (cid:20) lim h ↓ h ˜ f (cid:18) T (cid:18) x − ( T − t ) X t − ( T − t )( X t + h − X t ) + hX t + h − Z t + ht X s ds (cid:19)(cid:19) − ˜ f (cid:18) T ( x − ( T − t ) X t ) (cid:19)(cid:21) = E (cid:20) ˜ f ′ ( x − ( T − t ) X t ) T ∂ + ∂t X t (cid:21) . Again we see that in the horizontal derivative a right derivative of the path occurs.This is the reason to restrict to finite variation semimartingales. In this case the Markovproperty does not help since for a Markov process X the functional G f reduces to G f ( t, ω ) = E (cid:20) ˜ f (cid:18) T I T (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X t = ω t (cid:21) = E (cid:20) ˜ f (cid:18) T (cid:18)Z t ω s ds + Z Tt X s ds (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X t = ω t (cid:21) . This representation does not allow to use the transition operators since the function in theconditional expectation depends on the whole path after t , whereas the transition operatorsonly depend on the process at t .The comparison results in Section 3 also need vertical convexity, vertical directionalconvexity and vertical monotonicity. In particular independent increments are useful toestablish these properties as shown in the following example. Example 4.8.
Let X be a semimartingale with independent increments. Then G f is ofthe form G f ( t, ω ) = E [ f ( ω ⊕ t X T )] (see Theorem 4.2). To establish vertical convexity, weneed to show that G f ( t, ω + e [ t,T ] ) is convex as function in e in a neighbourhood of . Forthe functional f ( ω ) := R T ˜ f ( ω t ) dt as in Example 4.4, we obtain G f ( t, ω + e [ t,T ] ) = E (cid:2) f ( ω ⊕ t X T + e [ t,T ] ) (cid:3) = Z t ˜ f ( ω s ) ds + E (cid:20)Z Tt ˜ f ( X s + e ) ds (cid:21) . Thus, if ˜ f is convex or f is vertically convex, we obtain that G f is vertically convex. Analogstatements hold for directional convexity and monotonicity. We finally apply the regularity results in this section to obtain a comparison result fora path-dependent function between a L´evy process and an Itˆo semimartingale. Concretelythe following example is based on Theorem 3.11.30 xample 4.9.
Let X be a type C L´evy process (see Sato (1999)) with L´evy triplet ( b, c , K ) and let ˜ f : R → R be a bounded, continuous, increasing directionally convex function. Thenwe have by Proposition 4.6 that for f ( ω ) = R T ˜ f ( ω t ) dt the functional G f is in C , b (Λ T ) .Further, G f is vertically directionally convex and vertically increasing by Example 4.8. Socondition ( i ) of Theorem 3.11 is fulfilled.We compare X to an Itˆo semimartingale Y with differential characteristics ( β, δ , η ) .Here β is an adapted process which is integrable with respect to the identity, δ is anadapted process which is integrable with respect to the Brownian motion and η is suchthat ν ( dt, dx ) := dtη t ( dx ) is the compensator of µ Y .Since X is a type C L´evy process we have that supp( P X t ) = R for all t . Hence, by thechoice of f we have that for all ω ∈ R [0 ,T ] ¯ U t G f ( t, ω t ) = 0 . It follows that the generalized Kolmogorov backwards equation ¯ U t G f ( t, Y t − ) = 0 holds forthe path of Y and consequently condition ( ii ) is fulfilled.If Assumptions ( iii ) and ( iv ) are imposed, we get from the dt × P almost sure ordering ofthe differential characteristics β t ≤ b,δ t ≤ c, Z R H G f ( t, Y t − , x ) η t ( dx ) ≤ Z R H G f ( t, Y t − , x ) K ( dx ) , that E (cid:2) f ( Y T ) (cid:3) ≤ E (cid:2) f ( X T ) (cid:3) , i.e. the comparison of the path-dependent function is valid. References
Ananova A., Cont R. (2017). Pathwise integration with respect to paths of finite quadraticvariation.
Journal de Mathematiques Pures et Appliquees, Volume 107, No 6 , 737–757.Arnaudon M., Breton J-C., Privault N.. (2008). Convex ordering for random vectors usingpredictable representation.
Potential Anal., 29, No 4 , 327–349.Bally V., Caramellino L., Cont R. (2016). Stochastic integration by parts and functionalItˆo calculus.
Birkh¨auser .Bellamy N., Jeanblanc M., Shreve S. (2000). Incompleteness of markets driven by a mixeddiffusion.
Finance Stoch., 4, no.2 , 209–222.31ergenthum J., R¨uschendorf L. (2006). Comparison of option prices in semimartingalemodels.
Finance and Stochastics, 10 , 222–249.Bergenthum J., R¨uschendorf L. (2007). Comparison of semimartingales and L´evy processes.
The Annals of Probability, 35 , 228–254.Bergenthum J., R¨uschendorf L. (2007). Some convex ordering criteria for L´evy processes.
Advances Data Analysis Classification 1 , 143–173.Bergenthum J., R¨uschendorf L. (2008). Comparison results for path-dependent options.
Statistics & Decisions, 26 , 53–72.Billingsley P. (1968). Convergence of probability measures.
John Wiley & sons.
Cont R., Fourni´e D.-A. (2010). Change of variable formulas for non-anticipative functionalson path space.
Journal of Functional Analysis, 259 , 1043–1072.Cont R., Fourni´e D.-A. (2010). A functional extension of the Itˆo formula.
C.R. Acad. Sci.Paris, Ser. I, 348 , 57–61.Criens D. (2019). Couplings for processes with independent increments.
Statist. Probab.Lett., 146 , 161–167.Dupire B. (2009). Functional Itˆo calculus.
Bloomberg .Ethier S.N., Kurtz T.G. (2005). Markov processes, characterization and convergence.
JohnWiley & Sons .El Karoui N., Jeanblanc-Piqu´e M., Shreve S. (1998). Robustness of the Black and Scholesformula.
Math. Finance, 8, no.2 , 93–126.El Karoui N., Peng S., Quenez M.C: (1997). Backwards stochastic differential equationsin finance.
Math. Finance, 7, no.1 , 1–71.F¨ollmer (1981). Calcul d’Itˆo sans probabilit´es.
S´eminaire de Probabilit´es de Strasbourg,Springer - Lecture Notes in Mathematics, 15 , 143–150.Geib Ch., Manthey R. (1994). Comparison theorem for stochastic differential equations infinite and infinite dimensions.
Stochastic Processes an their Applications, 53 , 23–35.Gushchin A.A., Mordecki E. (2002). Bounds on option prices for semimartingale marketmodels.
Proceeding of the Steklov Institute of Mathematics, 273 , 73–113.Hobson D. G. (1998). Volatility misspecification, option pricing and superreplication viacoupling.
The Annals of Applied Probability, 8 , 193–205.Jacod J., Shiryaev A.N. (2003). Limit theorems for stochastic processes.
Springer .32lein T., Ma Y., Privault N. (2017). Convex concentration inequalities via for-ward/backward stochastic calculus.
Electron. J. Probab., 11, No. 20 , 486–512.K¨opfer B., R¨uschendorf L. (2019). Comparison of path-independent functions of semi-martingales.
Preprint .K¨opfer B. (2019). Comparison of stochastic processes by Markov projection and functionalItˆo calculus.
Phd-Thesis, Albert-Ludwigs-Universit¨at Freiburg .Leventhal S., Schroder M., Sinha S. (2013). A simple proof of functional Itˆo’s Lemma forsemimartingales with an application.
Statistics and Probability Letters, 83 , 2019–2026.Ma L. X., Xu G. Q., Mastorakis N. E. (2010). A comparison theorem for stochasticdifferential equations and its applications in economics.
Newaspects of systems theoryand scientific computation, WSEAS Press , 34–40.Peng S., Zhou X. (2006). Necessary and sufficient condition for comparison theorem of1-dimensional stochastic differential equations.
Stochastic Process. Appl., Vol. 116, No.3 , 370–380.Protter P. (2005). Stochastic integration and differential equations.
Springer. .Riga C. (2015). Pathwise functional calculus and applications to continuous-time finance.
Phd thesis, Universit´e Pierre et Marie Curie and Scoula normale superiore di Pisa.
Sato K.-I. (2015). L´evy processes and infinitely divisible distributions.
Cambridge Univer-sity Press.
Shi Y., Gu Y., Liu K. (2005). Comparison theorems of backward doubly stochastic differ-ential equations and applications.
Stochastic Analysis and Applications, Vol. 23, No. 1 ,97–110.Wua Z., Xu M. Y. (2009). Comparison theorems for backward SDEs.
Statistics & Proba-bility Letters, Vol. 79, No. 4 , 426–435.Zhou S. W. (2004). Comparison theorem for multi-dimensional backward differential equa-tions.