Simplified stochastic calculus via semimartingale representations

We develop a stochastic calculus that makes it easy to capture a variety of predictable transformations of semimartingales such as changes of variables, stochastic integrals, and their compositions. The framework offers a unified treatment of real-valued and complex-valued semimartingales. The proposed calculus is a blueprint for the derivation of new relationships among stochastic processes with specific examples provided below.


Introduction
"Because in mathematics we pile inferences upon inferences, it is a good thing whenever we can subsume as many of them as possible under one symbol. For once we have understood the true significance of an operation, just the sensible apprehension of its symbol will suffice to obviate the whole reasoning process that earlier we had to engage anew each time the operation was encountered." -Carl Jacobi (1804-1851) [31, p. 67] We study the following concept. A semimartingale Y is said to be represented by a semimartingale X if, roughly speaking, there is a predictable function ξ acting on the increments of X such that the increments of Y satisfy dY t = ξ t (dX t ), where ξ t (dX t ) is given some "natural" meaning. Such representation of Y in terms of X, if it exists, is measure-invariant. One hopes that common operations on Y yield processes that are again X-representable, for example, (i) a stochastic integral ζ t dY t "ought to" yield ζ t dY t = ζ t ξ t (dX t ); (1.1) (ii) for a change of variables by means of some smooth function f it should be true that (1.2) (iii) for a new process Z such that dZ t = ψ t (dY t ) one would like to obtain the composition rule dZ t = ψ t (ξ t (dX t )).  Table 1: Useful identities involving the power function, stochastic exponential E , and stochastic logarithm L.

P4.3 analogous
|E (X (1) )| α |E (X (2) (1) ), L(X (2) )) When E (X) is signed, one can evaluate |E (X)| α and sgn(E (X))|E (X)| α separately to (1.7) E[ sgn(E (X))|E (X) t | α ] = E (B (sgn(1+id)|1+id| α −1)•X ) t ; (1.8) see [3,Examples 4.4 and 4.5]. We refer the reader also to the introductory paperČerný and Ruf [5], where other concrete illustrations of the calculus are given. 1 On the theoretical side, the paper introduces the class U of universal representing functions that are well-behaved with respect to operations (1.1)-(1.3); if one uses only locally bounded integration, change of variables, and composition, one is guaranteed never to leave U, which makes the calculus completely straightforward. For example, the representing functions in (1.5)-(1.8) and also those in Table 1 are all in U. The most important results pertaining to the class U are highlighted in Table 2. Furthermore, we develop a coherent theory for a wider class I(X) of representing functions specific to X, in which U appears as a special case. Here the "natural" composition rules (1.1) and (1.3) sometimes fail. We study sufficient conditions for their validity and offer counterexamples when such conditions are not met. The proposed framework does deliver closedness under composition for general stochastic integrals without further assumptions; this and other important properties of the class I(X) are collected in Table 3.

C3.20
ξ ∈ I(X); ψ ∈ I(ξ • X); ψ (0) locally bounded ψ(ξ) ∈ I(X); ψ • (ξ • X) = ψ(ξ) • X algebraically, as compositions of functions (indeed, ξ • X can be interpreted in some cases as the ξ-variation of X; see Remark 3.10). The benefits of doing so are significant, especially in the context of measure changes. One might expect the operations (1.1)-(1.3) to always work when the representing process X is a pure-jump process of finite variation. Using only standard techniques, this intuition is false, however, because an integral of a finite variation semimartingale need not itself be of finite variation. We do obtain universal validity of rules (1.1)-(1.3) for pure-jump processes after suitably extending the standard integrals with respect to random measures. This universality then applies to all representing processes X that belong sigma-locally to the class of finite-variation pure-jump semimartingales; see Subsection 2.5. The paper is organized as follows. Section 2 introduces notation and reviews important concepts such as integration with respect to a complex-valued semimartingale. Section 3 defines representation of a semimartingale and derives important properties thereof, such as (1.1)-(1.3). Here one gets to see an explicit formula for the object ξ • X, which is formulated in terms of real derivatives of the complex function ξ. This formula looks quite natural in the special case when both ξ and X are real-valued; such simplicity is also preserved when ξ is analytic at 0 but this is much harder to see in the original definition. Subsection 3.3 provides an alternative form of the most general representation formula in terms of so-called Wirtinger derivatives, where the simplification in the analytic case is plainly visible. Section 4 lists and proves a number of useful representations, among them generalizations of the Yor formula, thereby illustrating the strength of the proposed calculus. This section also provides counterexamples that document tightness of the results obtained in Section 3. Section 5 summarizes the computation of predictable characteristics of a represented semimartingale. Finally, Section 6 discusses additional benefits of the proposed calculus and directions for future research.

Setup and notation
This section provides background on complex numbers and the probabilistic setup. It furthermore reviews stochastic integration for complex-valued semimartingales, the notion of predictable functions, and sigma-localized integrals with respect to random measures.

The lift from C to R 2
Below, we explicitly allow quantities to be complex-valued in order to allow for a consistent treatment of complex integrals, exponentials, etc., and in particular characteristic functions. The reader interested only in real-valued calculus can easily skip this subsection and always replace the general 'C-valued' by the special case 'R-valued' in their mind. Throughout this section, let m ∈ N denote an integer. To simplify notation later on, we write C m = C m ∪ {NaN} for some 'non-number' NaN / ∈ k∈N C k . We introduce the function id : The definitions below now hinge on the identification mapîd : , v ∈ C;îd(NaN) = NaN, and its appropriate multidimensional extension, again denoted byîd : C m → R 2m ∪{NaN} given byî , v ∈ C m ;îd(NaN) = NaN.
Observe thatîd(v) ∈ R 2m for v ∈ C m contains the values of Re v and Im v, interlaced. At times we silently use matrix-valued versions of these canonical maps, which are taken to double the row dimension but which we do not introduce formally to avoid excessive notation.
So as not to obscure the main ideas with notation, we will highlight the key properties of the liftîd for m = 1. To this end, the inverse map toîd isîd −1 : The following two properties ofîd are of importance: •îd andîd −1 are linear, when restricted to C and R 2 ; • for u, v ∈ C one obtainŝ (2.1)

Probabilistic quantities
We fix a probability space (Ω, F , P) with a right-continuous filtration F. We shall assume, without loss of generality, that all semimartingales are right-continuous, and have left limits almost surely. For a brief review of standard results without the assumption that the filtration is augmented by null sets, see Perkowski and Ruf [29,Appendix A]. We follow mostly the notation of Jacod and Shiryaev [18].
For a C m -valued stochastic process V we shall writeV =îd(V ) for the corresponding R 2m -valued process. Definition 2.1 (Complex-valued process properties). A C m -valued stochastic process V is said to have a certain property, for example to be a semimartingale (respectively, martingale; local martingale; special semimartingale; process of finite variation; process with independent increments; predictable; locally bounded; etc.) if the R 2m -valued processV =îd(V ) has that same property, i.e., ifV is a semimartingale (respectively, martingale, etc.).
We denote the left-limit process of a (complex-valued) semimartingale V by V − and use the convention V 0− = V 0 . We also set ∆V = V − V − ; in particular we have ∆V 0 = 0.
For complex-valued processes, the quadratic variation process is defined to be bilinear. 2 That is, for C-valued semimartingales V and U we set denotes the corresponding C m×m -valued quadratic variation, formally given by  [18, I.4.27]. Note that V c (P) depends on the underlying measure P. To wit, for two equivalent measures Q ∼ P, we usually have V c (Q) = V c (P) if Q = P. Nevertheless, we always have see also Dellacherie and Meyer [7, Theorem VIII.27] and Protter [30, p. 70].
Let µ V denote the jump measure of a semimartingale V and ν V its predictable compensator (under a fixed probability measure P). Then for a C-valued bounded predictable function ξ (a precise definition is provided in Subsection 2.4) with ξ(0) = 0 If V is special, we let the triplet (B V , [V,V ] c , ν V ) denote the corresponding semimartingale characteristics of V under a fixed probability measure P. 3 In particular, the drift B V , i.e., the predictable finite-variation part of the Doob-Meyer decomposition of V , is always assumed to start in zero, i.e., We can then define the 'clock' (or 'activity') process where TV denotes total variation. Then A V is non-decreasing and locally bounded. Thanks to [18,II.2.9], there exists an appropriate transition kernel F V such that

Stochastic integration
In this subsection we discuss stochastic integrals of predictable processes with respect to complex-valued semimartingales. To begin, consider a C 1×m -valued process ζ and a C m -valued semimartingale V . Here ζ is explicitly allowed to take the value NaN, but needs to be C 1×m -valued, (P × A V )-a.e., for the integral to be defined. If V is real-valued, then we write ζ ∈ L(V ) if both Re ζ and Im ζ are integrable with respect to V (in the standard sense). We then set ζ · V = (Re ζ) the Kronecker product; recall also (2.1). We then write for the stochastic integral of ζ with respect to V . For real-valued V the class L(V ) is defined twice but it is clear that the two definitions are consistent and ζ · V is well defined. For m = 1 one has ζ ∈ L(V ) if and only if [ζ iζ] ∈ L(V ). It is clear how to extend this definition to a C n×m -valued process ζ, where n ∈ N.

Remark 2.3 (Caveat of complex-valued integration).
Complex-valued stochastic integrals appear in the literature in a very limited context such as stochastic differential equations (e.g., [18, I.4.60]) or the Itô formula (e.g., [32,Proposition V.2.3]). In those circumstances the integrands are locally bounded, meaning that vector-valued integration is not required and integrability itself is not an issue. Our definition coincides with these special cases when ζ is locally bounded but in general the (real) stochastic integrals on the right-hand side of (2.2) cannot be computed component-wise.
Finally, for a C n×m×m -valued process ζ and a C m×m -valued semimartingale V (usually a quadratic variation process), let vec r (ζ) and vec c (V ) denote the row-wise and column-wise flattening of ζ and V , respectively. Then vec r (ζ) is (n × m 2 )-dimensional and vec c (V ) is m 2 -dimensional. We then write ζ ∈ L(V ) if vec r (ζ) ∈ L(vec c (V )) and ζ · V = vec r (ζ) · vec c (V ).

Predictable functions
For this subsection, let m, n ∈ N. As in [18,II.1.4], we consider the notion of a predictable function on Ω m = Ω×[0, ∞)×C m . For two predictable functions ξ : Ω m → C n 3 We use the real-valued lift of V to describe the continuous part of the quadratic variation in the characteristic triplet. This is necessary to capture the full dynamics of V . For example, let V and W denote two independent R-valued Brownian motions and set Z = and ψ : Ω n → C we shall write ψ(ξ) to denote the function (ω, t, x) → ψ(ω, t, ξ(ω, t, x)) with the convention ψ(ω, t, NaN) = NaN. If ψ and ξ are predictable, then so is ψ(ξ). For a predictable function ξ : Ω m → C n we shall writeξ =îd(ξ) and ξ (k) for the k-th component of ξ, where k ∈ {1, · · · , n}. We also writeDξ andD 2 ξ for the real derivatives of ξ, i.e.,D i ξ (k) is the composition of the i-th element of the gradient of ξ (k) (î d −1 ) and the liftîd andD 2 i,j ξ (k) is the composition of the (i, j)-th element of the Hessian of ξ (k) (î d −1 ) and the liftîd, for i, j ∈ {1, · · · , 2m}. Note thatDξ has dimension n × (2m),D 2 ξ has dimension n × (2m) × (2m), and the domains ofDξ,D 2 ξ equal Ω m , i.e., they coincide with the domain of ξ. We want to allow for predictable functions such as ξ = log(1 + id) whose effective domain is not the entire C. For this reason, we define, for a given predictable function ξ : Ω m → C n , the set of semimartingales whose jumps are compatible with ξ, i.e., If for another predictable function ψ : Ω n → C m we have ψ(ξ(∆V )) = ∆V for all V ∈ Dom(ξ), we say ξ allows for a left inverse. If ξ(ψ(∆V )) = ∆V for all V ∈ Dom(ψ) we say that ξ allows for a right inverse. If ψ represents both left and right inverse we shall use the notation ξ −1 = ψ.

Sigma-localized integrals with respect to random measures
We next recall fromČerný and Ruf [4] relevant results about the sigma-localized version of the * integral of a predictable function with respect to ν V and µ V for a semimartingale V , which we fix from now on to the end of this section. The following is adapted from [4, Definition 3.1].
Definition 2.4 (Extended integral with respect to random measure). Denote by L(µ V ) the set of predictable functions that are absolutely integrable with respect to µ V . We say that a predictable function ξ belongs to L σ (µ V ), the sigma-localized class of L(µ V ), if there is a sequence (C k ) k∈N of predictable sets increasing to Ω × [0, ∞) and a semimartingale Y such that 1 C k ξ ∈ L(µ V ) for each k ∈ N and In such case the semimartingale Y is denoted by ξ µ V . Similarly, we define L σ (ν V ) and ξ ν V .
(ii) The following two conditions hold.
Next we recall a composition property for stochastic integrals. Such result does not hold if the integral were to be replaced by the * integral.

Proposition 2.8 ([4]
, Proposition 3.9). For ξ ∈ L σ (µ V ) taking values in C n for some n ∈ N and a C 1,n -valued predictable process ζ the following statements are equivalent. We next denote by V the set of semimartingales with finite variation on compact time intervals and by V d the subset of finite variation pure-jump processes, i.e., those semimartingales V ∈ V that satisfy V = V 0 +id * µ V . The statements in this subsection can also be expressed in terms of a special class of semimartingales V d σ , i.e., the σ-localized class of finite variation pure-jump processes. The key connection is the following.
We conclude this section with a natural decomposition of V into jumps at predictable times and a quasi-left-continuous process.
Hence V is special if and only if both V qc and V dp are special. Let T V denote a countable family of stopping times that exhausts the jumps of V dp . 4 For each V there may be many ways to choose T V . The following statement holds for any such T V .

Proposition 2.11 (Drift of a pure-jump process jumping only at predictable times).
Assume that V dp is special. Then we have Proof. Thanks to (2.3), we have B V dp = id ν V dp . Moreover, B V dp is of sigma-finite variation and T V exhausts its jumps. Proposition 4.6 in [4] applied to B V dp then yields the result.

Semimartingale representation 3.1 Definition and basic properties
From now on we shall fix some d, n ∈ N and consider a C d -valued semimartingale X. We shall then study a variety of predictable transformations of X. Of course, an R d -valued semimartingale can always be considered a special case.

Example 3.1 (A motivational example)
. Let X denote an R-valued semimartingale and let f : R → R denote a twice continuously differentiable function. Then it is well known that also the process Y = f (X) is a semimartingale. More precisely, the Itô-Meyer change of variables formula, [18, I.4.57], provides the representation Note that the derivatives Dξ f,X and D 2 ξ f,X exist. The representation in (3.1) then can be written in the more compact form Observe that ∆Y = ξ(∆X) and that Y is fully determined by X and the predictable function ξ f,X .
The connection between (3.1) and (3.2) motivates the key concept of this paper, Definition 3.8 below. Recall from Subsection 2.5 the predictable set H X , on which X dp has no 'activity.' Definition 3.2 (Representing functions for a given semimartingale X). Let I n (X) denote the set of all predictable functions ξ : Ω d → C n such that the following properties hold.
We write I(X) = k∈N I k (X). To see this, we only need to argue (6). This follows from observing that we have 1 H\H XD ξ(0) ∈ L(X), yielding 1 H\H XD ξ(0)îd ∈ L σ (µX ) by Proposition 2.8. Example 4.6 below provides an instance where X = X dp , ξ ∈ I(X), ξ is twice differentiable at zero, but Dξ(0) / ∈ L(X). Thus, allowing for the existence of an appropriate predictable set H X such that only 1 H X Dξ(0) ∈ L(X) is required, indeed allows for a bigger class I(X).
As Propositions 3.6 and 3.15 and Theorem 3.17 below argue, the following class U enjoys closedness with respect to common operations and universality in the sense that a representing function ξ ∈ U satisfies ξ ∈ I(X) for any semimartingale X provided that ξ(∆X) is finite. Definition 3.4 (Universal representing functions). Let U n denote the set of all predictable functions ξ : Ω d → C n such that the following properties hold, P-almost surely.
(4) There is a predictable locally bounded process K > 0 such that We write U = n∈N U n . Remark 3.5 (A special case: real-valued semimartingales). If X is real-valued then we may consider ξ as a predictable function with real domain. In this case, it can be easily checked that in Definitions 3.2 and 3.4 we may omit the hats on top of D, id, and X, with D and D 2 being the standard gradient and Hessian, respectively. Proposition 3.6 (Universality of U). Fix some ξ ∈ U such that X ∈ Dom(ξ). We then have ξ ∈ I n (X), (ξ −Dξ(0)îd) ∈ L(µ X ), and Proof. The first claim follows from Remark 3.3. For the second claim it suffices to observe that |ξ −Dξ(0)îd| * µ X < ∞ by localization.
Proposition 3.7 (Properties of I(X)). The following statements hold.

Definition 3.8 (Semimartingale representation).
For a predictable function ξ ∈ I(X) we use the notation we say that the semimartingale Y is represented in terms of the semimartingale X. see also Proposition 3.6. In this case the integral can be replaced by the standard * integral.
Remark 3.10 (Interpretation of ξ • X as ξ-variation). The object ξ • X with time-constant deterministic ξ, most often a power function, resurfaces several times in the literature under the name ξ-variation, see Doléans [8], Monroe [27,28], Lépingle [24], Jacod [17], and Carr and Lee [2]. The terminology and Émery's [13] notation · 0 ξ(dX s ) originate from the fact that, for suitably regular time-constant deterministic ξ, the partial sums n∈N ξ tn−1 (X tn − X tn−1 ) converge uniformly on compact time intervals in probability to ξ • X as the time partition (t n ) n∈N becomes finer; see [  For example, one could abstain from requesting that x → 1 H X ξ(x) is real-differentiable in the i-th component for times when dA X (i) = 0. Moreover, one could assume that the second real derivative of x → 1 H X ξ(x) only needs to exist (P × trace[X,X] c )-a.e.
However, such generalisations would come with more complicated notation and would obscure the main results, hence we do not pursue them here. Remark 3.12 (Measure invariance of representations). Note that I(X) is invariant under equivalent changes of measures. More precisely, if Q is a probability measure absolutely continuous with respect to P and if ξ ∈ I(X) under P, then also ξ ∈ I(X) under Q (recall Remark 2.7 to see this). Moreover, if we define Y = ξ • X under P, then we also have Y = ξ • X under Q. Hence, ξ • X is measure-invariant in the sense that (3.3) only depends on the null sets. A similar statement holds for U. This is in contrast to the common (and frequently also very useful) representation of Y in terms of predictable characteristics.
For (6), note that ξ(·, ∆Y ) ∈ I(X) yields that Now, the result follows by comparing the jumps on the left and right hand side of (3.5), for example by using (1). Proposition 3.14 (Representation of stochastic integrals). Let ζ be a C 1×d -valued predictable process in L(X). Then ζid ∈ I 1 (X) and Hence, ξ belongs to I(X) as per Definition 3.2, and (3.3) together with (2.2) yield the claim.

Proposition 3.15 (Representation of a change of variables).
Let U ⊂ C d be an open set such that X − , X ∈ U and let f : U → C n be twice continuously real-differentiable. Then the predictable function ξ f,X : Ω d → C n defined by belongs to U n , X ∈ Dom(ξ f,X ), and Proof. Denote by R > 0 the distance from X − to the boundary of U, by R * its running infimum, and by τ > 0 the first time R * hits zero. The left-continuity of R now yields τ = ∞ and R * > 0. Therefore, (τ n ) n∈N given by τ n = inf{t : R * t ≤ 1/n}, is a localizing sequence of stopping times that makes both K = 2 /R and sup |x|≤ 1 /K |D 2 ξ(x)| locally bounded, yielding ξ f,X ∈ U. Since X ∈ Dom(ξ f,X ), Proposition 3.6 now yields that is the Itô-Meyer change of variables formula for the real-valued functionîd(f (î d −1 )) applied to the real-valued processX; see [18, I.4.57].
Remark 3.16 (Itô's formula requires smoothness). It is possible to exhibit a function f : R → R and a semimartingale X such that ξ f,X ∈ I(X), in the notation of Proposition 3. 15, and such that f (X) is a semimartingale, but For example, choose X equal to Brownian motion started at 0 and f = |id|. Here f is not twice differentiable but ξ f,X ∈ I(X) anyway as it is Lebesgue-a.e. twice differentiable.
to the tower property in Carr and Lee [2, Proposition 2.4].
Here, we have used the associativity of the stochastic integrals with respect toX and [X,X] c as well as the associativity of the jump-measure integral.
Example 4.11 below shows that without the assumption that ψ is twice real-differentiable at zero, (P × A X )-a.e., the conclusion of Theorem 3.18 does not necessarily hold.
Proof. Sinceξ(îd −1 ) is continuously differentiable at zero on H X ,ψ(îd −1 ) is actually an inverse ofξ(îd −1 ) in a neighbourhood of zero on H X . Thus 1 H X ψ is twice realdifferentiable at zero withDψ(0) = (Dξ(0)) −1 on H X . If now the smallest singular value ofDξ(0) is locally bounded away from zero, then the largest singular value ofDψ(0) is locally bounded and by equivalence of the Schatten and maximum matrix norms each element ofDψ(0) is locally bounded. The assertion follows from Remark 3.22.
If the assumption that the smallest singular value ofDξ(0) is locally bounded away from zero is replaced by the weaker assumption that is is merely positive, then Corollary 3.23 is wrong as Examples 4.9 and 4.10 below illustrate, even if ψ is an inverse of ξ and d = 1.
One then has to collect all terms manually in order to simplify this expression and eventually recast it in terms of µ X . In contrast, the notation of (3. Only the function ξ f,Y (ξ) needs to be computed and then the corresponding representation applies. This is pedagogically pleasing because ξ f,Y (ξ) describes the jumps of f (Y ) in terms of the jumps of X, i.e.,

Alternative Émery formula
In the non-analytic case, which too is of practical importance, it can be helpful to rephrase the Émery formula (3.3) in terms of the C 2d -valued process (X, X * ). Here X * denotes the complex conjugate of X. This allows the use of Wirtinger partial derivatives (see [35]), given by This turns out to be convenient in some applications; see Proposition 3.26 and Example 3.28. Observe, however, that the proposed calculus allows one to write simply ξ • X and operate on the level of ξ, where the specific physical implementation of ξ • X is immaterial.
To arrive at the alternative Émery formula, we introduce the functionǐd : where ⊗ again denotes the Kronecker product. This allows us to introduce the procesš X =ǐd(X).

(3.11)
Observe thatX is the R 2d -valued process containing the values of Re X and Im X, interlaced, whileX is the C 2d -valued process containing X and its conjugate X * , interlaced.
Next, we denote byĎξ the row vector of Wirtinger derivatives, given by and byĎ 2 ξ the corresponding 'Wirtinger Hessian,' given by The following technical observation will be very useful in the subsequent proposition.

Lemma 3.25 (Invertible linear transformations in a stochastic integral). Fix m ∈ N.
Let Λ 1 , Λ 2 be arbitrary invertible matrices in C m×m . Let ζ denote a C m×m -valued predictable process and let V denote a C m×m -valued semimartingale. Then the following are equivalent.
If one (hence both) of these conditions holds, then Proof. Note that it suffices to argue the implication from (i) to (ii) and to show (3.13).
We now provide two examples of complex-valued representations where the representing functions are not assumed analytic at 0. Recall the notation for Wirtinger derivatives in (3.10).

Example 3.27 (Quadratic covariation of represented semimartingales)
. Let X be a Cvalued semimartingale. Then by Proposition 3.13(3) and Corollary 3.20, for ξ, ψ ∈ I 1 (X), we have [ξ • X, ψ • X] = ξψ • X. In the explicit form (3.14), this is written as This formula seems very intuitive. The first three terms capture the continuous covariation of ξ • X and ψ • X. The last term is the pure-jump component which multiplies together the jumps in ξ • X and ψ • X. Consider the predictable function ξ = |1 + id| α − 1, which on a sufficiently small neighbourhood of zero satisfies On this neighbourhood, apply formal Wirtinger calculus to obtain Next, ξ ∈ U, hence ξ ∈ I(X) for any C-valued semimartingale X with X ∈ Dom(ξ), in particular for any X with ∆X = −1. Formula (3.14) now yields We continue discussing this setup in Example 5.

Generic applications
If X is a C-valued semimartingale, then by [9, Théorème 1] (see also [18, I.4.60]) the stochastic exponential E (X) of X is the unique solution to the stochastic differential equation E (X) = 1 + E (X) − · X. The stochastic logarithm L(X) of a semimartingale X that can hit zero only by a jump (but not continuously) and is absorbed in zero is given by All representing functions shown in this subsection belong to the universal class U and can therefore be applied to any semimartingale whose jumps are compatible with the given function (Proposition 3.6). The simplified stochastic calculus yields many identities by straightforward computations. Using only the Itô-Meyer change of variables formula, those identities would involve convoluted arguments. As an example, we now establish a generalization of Yor's formula and its converse (see [18,).
(1) Assume that the following conditions hold.
Assume that these two conditions also hold with α and X (1) replaced by β and X (2) , respectively. We then have where complex powers with exponent in C \ Z are defined via the principal value logarithm. In particular, with α = β = 1 we have ä . (2) Assume next the following conditions.
• If α ∈ N, then X (1) does not reach zero continuously and is absorbed in zero.
The uniqueness of strong solutions to the stochastic differential equation (4.1) then yields (4.2).
Assume from now on that ∆X = −1. Observe that log(1 + id) is the right-inverse of the function e id − 1 over the domain C \ {−1} and that log(1 + id) ∈ U ∩ I(X). We may therefore define Y = log(1 + id) • X. From Finally, for a semimartingale Y satisfying Y > 0 and Y − > 0 one obtains by Proposition 3.15, the identity Y = Y − id • L(Y ), and Theorem 3.17 that again by composition.
Consider now X such that Re E (X) > 0, hence Re E (X) − ≥ 0 and E (X) − = 0. As in the previous step, by Proposition 3.15 and Theorem 3.17 one obtains where the last equality follows by comparing the respective Émery formulae.
(1) Assume that the following condition holds.
Assume that this condition also holds with α and X (1) replaced by β and X (2) , respectively. We then have (4.12) (2) Assume next the following conditions.

Example 4.4 (Iterated composition).
Let us now consider the following construction for a C-valued semimartingale X and for a constant α ∈ C. Define inductively the processes Y 0 = X; Then an induction argument, (4.6), and Theorem 3.17 yield that Y k = ξ k • X for all k ∈ N ∪ {0}, with ξ 0 = id; Explicitly, for each k ∈ N, ξ k is a nested function of the form Using the chain rule, one infers that ξ k is analytic at zero for each k ∈ N with Dξ k (0) = αDξ k−1 (0); where for α = 1 we interpret (α k − 1) /(α − 1) as k. We conclude that, for each k ∈ N, Note that this representation of Y k is the same for any starting process X, for each k ∈ N. For example, let X t = µt + σW t for all t ≥ 0, where W is Brownian motion with W 0 = 0. Here µ ∈ R denotes the drift rate and σ ∈ R the volatility. Then (4.15) yields for all k ∈ N. Classical calculus would yield the same result, of course. For each k ∈ N, one would repeatedly compute This is not too complicated but can easily become quite cumbersome, even in the case of drifted Brownian motion. To this end, consider an R n -valued semimartingale V and a predictable function ψ such that ψ(x, ·) ∈ I(V ) for each x ∈ R d . Define next a family of semimartingales (F (x)) x∈R d by setting One can now randomize the family F by allowing x to switch values stochastically in line with the R d -valued semimartingale X. Assuming F is sufficiently smooth, the randomized process F (X) defined by will again be a semimartingale. The observation then yields, under suitable technical conditions, that We leave the technical details to future work. For the moment, we only note that for R-valued continuous processes X and V and for ψ(x, v) = f (x)v, where x, v ∈ R and f : R → R is twice continuously differentiable, one formally obtains As the examples illustrate, the stochastic calculus introduced above is powerful and simple. Stochastic integration, Itô's formula, and the composition rule of Theorem 3.18 allow for a wide range of applications. Within the confines of their assumptions they show that it is enough to study jump transformations; i.e., to represent Y in terms of X it suffices to trace how the jump ∆X t is transformed into the jump ∆Y t at time t ≥ 0.

Counterexamples
This subsection illustrates the tightness of the results in Section 3 by providing several counterexamples. Example 4.6 (ξ ∈ I(X), but ξ (0) / ∈ L(X)). Here, we construct a process X ∈ V d σ and a predictable function ξ ∈ I(X), twice continuously differentiable at zero, such that ξ (0) / ∈ L(X). This illustrates the role of the predictable set H X in Definition 3.2. Let U ∈ V d σ denote a piecewise constant martingale that jumps at times 2 − 1 /n by ±1/n 2 . Let (Θ n ) n∈N denote an independent sequence of independent {0, 1}-valued random variables with P[Θ n = 1] = 1 /n 4 . Let (Ψ n ) n∈N denote a sequence, independent of U and (Θ n ) n∈N , of independent standard normally distributed random variables. Let now V ∈ V d σ denote a piecewise constant martingale that jumps at times 2 − 1 /n by Ψ n if Θ n = 1 and does not jump if Θ n = 0.
Set now and assume that the filtration be the right-continuous modification of the one generated by the finite-variation process X.
Consider now deterministic ξ and ψ given by ξ t = 1 id≤t 4 id and ψ t = id /t 2 1 t>0 for all t ≥ 0. Then ξ ∈ I(X) with However, it is clear that ψ(ξ) / ∈ I(X). An even stronger statement holds, namely that there exists no η ∈ I(X) such that ψ • Y = η • X. ∈ I(ξ • X)). Assume that X = W is standard Brownian motion. Let ξ denote some deterministic predictable function that satisfies ξ t (x) = tx + x 2 /2 for all t > 0 and x in a neighbourhood of zero (which may depend on t) and allows for an inverse. Then ξ t (0) = t and ξ t (0) = 1 for all t > 0. Hence ξ ∈ I 1 (X) and we can define Y = ξ • X, Moreover, with ψ = ξ −1 we have ψ(ξ) ∈ I 1 (X). Observe, however, that ψ (0) / ∈ L(Y ) and Thus ψ / ∈ I(Y ) but there is no contradiction to Remark 3.22 as (3.6) is not met. Because ψ (0) = 1 /ξ (0) is not locally bounded, this example does not contradict Corollary 3.23 either. Note that there exists no η ∈ I(Y ) such that X = η • Y . ∈ I(ξ • X); additionally ξ • X = X). Let (τ k ) k∈N be a sequence of independent random variables with τ k uniformly distributed on ( 1 /(k + 1), 1 /k). Let (U k ) k∈N be an independent sequence of independent and identically distributed {−1, 1}-valued random variables with P[U 1 and assume that the filtration be the right-continuous modification of the one generated by X.
Let ξ denote some deterministic predictable function that allows for an inverse and satisfies, for all t > 0, ξ t (x) = x for all x with |x| ≥ 1 /(t + 1) and ξ t (x) = tx for all x in a neighbourhood of zero (which may depend on t). Then ξ ∈ I(X) and X = ξ • X. However, since 1 H X = 1 and Dξ −1 (0) / ∈ L(X), we have ξ −1 / ∈ I(X), concluding the example.

Predictable characteristics
Up to this point we have relied on a 'pathwise' perspective in the sense that the representation of the process Y by means of ξ • X depends on the probability measure only through the null sets; see also Remark 3.12. Now we will demonstrate the ability to convert an X-representation into predictable characteristics. In this section, we shall use generalized conditional expectation; see Shiryaev [33, pp. 475-476] and Jacod and Shiryaev [18, I.1.1].

Truncation functions
In [18,, a bounded function h : has bounded jumps and is therefore special. Below, it will be useful to not only control the jumps of X, but also those of a stochastic integral with respect to ' This leads to the following generalization of the classical truncation function where the boundedness and integrability requirements are relaxed. Moreover, h is no longer restricted to be a time-constant deterministic predictable function.
Definition 5.1 (Truncation function for X and its compatibility with ξ ∈ I(X)). We call a predictable function h : and if Moreover, for ξ ∈ I(X), we say that a truncation function h for X is ξ-

Remark 5.2 (Observations on truncation functions).
If h is a truncation function for X and ξ ∈ I(X), implying 1 H XD ξ(0) ∈ L(X), it does not follow that 1 H XD ξ(0) ∈ L( ' X[h]). Indeed, there exists an R-valued quasi-left-continuous process V ∈ V d σ with V 0 = 0 whose jumps are bounded and a predictable process ζ ∈ L(V − B V ) such that ζ / ∈ L(B V ) (see [4,Example 3.11]). Let now ). Consider now the process Y = ζ · X. We claim that h = 0 is not a truncation function for Y . Hence, this provides an example of a process Y such that Y [0] does not exist. Assume it did. Then id ∈ L σ (µ Y ), yielding ζid ∈ L σ (µ X ) = L σ (µ V ). In view of Proposition 2.8 and the fact that ζ / ∈ L(V ) this yields a contradiction.
As a final observation for the moment, note that the process 1 H XD ξ(0) · ' X[h] may not be special even if it is known that h is a truncation function for X, ξ ∈ I(X), and Furthermore, ifDξ(0) is locally bounded (in particular, if ξ ∈ U ∩ I(X)), then any truncation function for X is ξ-compatible.
Proof. By assumption, ς = 1 H XD ξ(0) is in L(X). We claim that h = id1 |id|≤1 and |ςî d|≤1 has the desired properties. Indeed, id − h ∈ L(µ X ) because both X and ς ·X have finitely many jumps larger than one in absolute value on any compact time interval. This also Observe that h(ω, t, x) = x on a (ω, t)-dependent neighbourhood of zero, (P × A X )-a.e. This yields that h is analytic at 0, Dh(0) is an identity matrix, and D 2 h(0) = 0. The representation formula (3.15) now gives The final claim follows by localization.

Remark 5.4 (Truncation at zero).
The previous lemma shows that sufficiently many truncation functions can be applied via the natural formula (5.1). We elect not to make (5.1) the only way to truncate because (5.1) does not hold for h = 0. Truncation at zero is convenient when X has jumps of finite variation; more generally, it can be applied whenever id ∈ L σ (µ X ).
The next proposition recognizes that the Émery formula (3.3) represents a whole spectrum of equivalent expressions where the jumps of X can be dialled down in the first term of (3.3) as long as they are equivalently modified in the last term of (3.3). In most applications, it is possible to choose as truncation one of the polar cases h = 0 or h = id; less frequently one may have to opt for an intermediate truncation such as h = id1 |id|≤1 ; in full generality it may be necessary to use the compatible truncation (5.2). Proposition 5.5 (Émery formula involving truncation). Fix ξ ∈ I n (X) and let g be a truncation function for ξ • X. Moreover, let h be a ξ-compatible truncation function for X. Then the following terms are well defined and we have If 1 H X ξ is analytic at 0, (P × A X )-a.e., the following terms are well defined and we have If g satisfies g(w) = w on an (ω, t)-dependent neighbourhood of 0, (P × A X )-a.e., then we also have g(ξ) ∈ I(X) and (ξ • X)[g] = g(ξ) • X.

Characteristics under the measure P
Proposition 5.5 yields the next observation, which is the key step towards computing predictable characteristics of represented semimartingales. Proposition 5.6 (Drift of a truncated represented semimartingale). Fix ξ ∈ I(X) and let g be a truncation function for ξ •X. Moreover, let h be a ξ-compatible truncation function for X. Then the following terms are well defined and the predictable compensator of (ξ • X)[g] under P is given by If 1 H X ξ is analytic at 0, (P × A X )-a.e., the following terms are well defined and we have  10. Consider now a predictable function ξ ∈ I(X). Proposition 3.7(3) asserts (ξ • X) qc = ξ • X qc and (ξ • X) dp = ξ • X dp . Next, suppose ξ • X is special. By Propositions 2.11 and 3.13(1), the drift at predictable jump times then takes a particularly simple form, namely, (5.9) Observe that this formula is simpler than Proposition 5.6 applied to X dp in place of X. Therefore, in practice, Proposition 5.6 is used with X = X qc to obtain B ξ•X qc . One then has Finally, recall that X qc is quasi-left-continuous, hence B ξ•X qc is continuous, yielding ∆B ξ•X = ∆B ξ•X dp .
where the superscript * denotes again the complex conjugate. The statement in (5.12) follows from Proposition 5.6 on observing that ν Y (G) = B 1 G •(ξ•X) , where G = G 1 × G 2 with G 1 ⊂ [0, ∞) predictable and G 2 a closed set in C n not containing a neighbourhood of zero.
When ξ is of the form ξ = f (X − + id) − f (X − ) for a twice continuously differentiable real-valued function f and when X is real, then Corollary 5.8 reduces to the situation in Goll and Kallsen [16,Corollary A.6]. When ξ = R id for some R n×d -valued matrix R and X is real-valued, Corollary 5.8 yields the statement of Eberlein, Papapantoleon, and Shiryaev [12,Proposition 2.4]. Example 5.9 (Generalized Yor formula continued). We continue the discussion of Proposition 4.1. Consider α, β ∈ C and a C 2 -valued semimartingale X satisfying the assumptions of Proposition 4.1(1) and additionally X is stopped when ∆X (1) = −1 or ∆X (2) = −1.

Concluding remarks
Let us review the benefits of the proposed 'calculus of predictable variations.' Some of the advantages, such as universality of representations in U and the ease with which calculations can be performed in a very general class of complex-valued functions, have been showcased in the introduction and subsequently in the main body of the paper.
Here we want to mention several other benefits that are of a more philosophical kind or whose detailed treatment is beyond the scope of this paper and will be pursued in other work. The literature has a number of fragmented and specialized results that fit into the framework of semimartingale representations. On their own, these results are hard to generalize and do not suggest fruitful unification, hence are also difficult to recall and disseminate. The new calculus overcomes this barrier by providing a compact, systematic way of recording existing (and new) results. Let us mention two classical examples to illustrate these advantages.
• Recall that a C-valued continuous local martingale is called conformal if [X, X] c = 0.
Hence by (3.15), an analytic representation with respect to a continuous conformal local martingale is again a conformal local martingale. This not only covers a change of variables by means of an analytic function, as in Getoor and Sharpe [15,Proposition 5.4], but includes arbitrary representation analytic at the origin. For example, the stochastic logarithm of a natural exponential preserves continuous conformal local martingales as its representing function e id − 1 is analytic at 0.
As log(1 + id) is in U and analytic at 0, the Émery formula this time in full generality, because the jump to zero may be treated separately.
Further advantages of the new calculus emerge when one is tasked with computing the drift of a represented process under some new probability measure Q whose density Z with respect to P is also represented, say by L(Z) = ψ • X. It now suffices to observe that by Girsanov's theorem the Q-drift of X equals the P-drift of X + [X, L(Z)] = id(1 + ψ) • X.
We refer the reader toČerný and Ruf [3] for a detailed treatment of measure changes by means of non-negative, represented, multiplicatively compensated semimartingales and once again to [5] for specific applications.
The suggested calculus has one other benefit for applied stochastic modelling. In an applied setting it is impractical to work with the raw characteristics This issue can be addressed by decomposing the process X uniquely into a 'discrete-time' component X dp involving only jumps at predictable times and a 'continuous-time' part X qc , see Proposition 2.10. When it comes to computing drifts, the jumps at predictable times τ can be treated separately via the natural formula The remaining quasi-left-continuous part X qc is usually an Itô semimartingale in applications, i.e., the characteristics of X qc are assumed to be absolutely continuous with respect to time. One may then rephrase the drift computation for this component in terms of time rates, reverting to drift rates, quadratic variation rates (squared volatilities), and jump intensities (Lévy measures). Thus, the calculus naturally accommodates the two most common ways of specifying the underlying stochastic process X (discrete time vs. an Itô semimartingale) and even allows them to be combined in intricate ways, see [3,Example 4.5].
We shall close by mentioning possible directions for future research. As for extensions of the classes U and I(X), the most immediate generalization concerns the level of smoothness of the representing function at the origin. Lack of differentiability is associated with the need to consider local times in the Itô-Meyer formula; see Karatzas and Shreve [22,Theorem 3.6.22]. This suggests an appropriate modification of the Émery formula (3.3), for which the three key operations would have to be checked again. In Example 4.5, we have broached the subject of the Itô-Wentzell formula that we believe merits further investigation.