Brownian excursions, stochastic integrals, and representation of Wiener functionals

A stochastic calculus similar to Malliavin’s calculus is worked out for Brownian excursions. The analogue of the Malliavin derivative in this calculus is not a diﬀerential operator, but its adjoint is (like the Skorohod integral) an extension of the Itˆo integral. As an application, we obtain an expression for the integrand in the stochastic integral representation of square integrable Wiener functionals; this expression is an alternative to the classical Clark-Ocone formula. Moreover, this calculus enables to construct stochastic integrals of predictable or anticipating processes (forward, backward and symmetric integrals are considered).


Introduction
Consider the Wiener space Ω = C(R + , R d ) of continuous paths ω : R + → R d , with the Wiener measure P, the standard Wiener process W t (ω) = ω(t), and the right continuous filtration (F t ) generated by W . The two main questions which are addressed in this article are: • the representation of a variable F defined on Ω as a stochastic Itô integral with respect to W ; • the construction of stochastic integrals of predictable or anticipating processes with respect to W .
These two questions have been dealt with for a long time by means of basic stochastic calculus and of Malliavin's calculus; here, we want to describe an alternative approach based on Brownian excursions.
Consider first the representation problem. Let F be a square integrable random variable defined on Ω. It is well known that the martingale E[F |F t ] can be written as where Z t is a R d -valued predictable process such that E |Z t | 2 dt is finite; moreover Z t (ω) is unique to a P(dω) dt negligible set. This formula has important theoretical consequences (this is the predictable representation property for the Wiener process), and it is also useful in some applications (mathematical finance is a recent example). Consequently, finding an expression of Z in terms of F is a natural question.
For instance, in the particular case F = f (W 1 ), let denote the heat semigroup. Then the martingale E[F |F t ], 0 ≤ t ≤ 1, can be written as P 1−t f (W t ), and an easy application of the Itô formula shows that for t ≤ 1, A classical extension of this particular case is based on the Malliavin calculus. If F is in a suitable functional space, one can consider its Malliavin derivative D t F , and Z can be written by means of the Clark-Ocone formula ( [7], Proposition 1.3.5 of [6]) which expresses Z as the predictable projection of DF . However, in this formula, we have to assume that F is differentiable in the sense of Malliavin and this is a restriction (consider for instance variables related to the first exit time of an interval, or to some local time). In order to bypass this difficulty, we can write a similar formula for distributions on the Wiener space (in this case the derivative D t F is considered in distribution sense, see [12]), but such a formula is not always very tractable; some other extensions have also been worked out in view of mathematical finance applications.
Our aim is to obtain another formula for Z which does not use any differentiation and which is valid in many cases. Notice that (1.2) follows from the fundamental formula of Malliavin's calculus (Definition 1.3.1 of [6]) This formula expresses the duality between the Malliavin derivative D and a divergence operator δ (the Skorohod integral); then (1.2) is a consequence of the fact that δ(ρ) coincides with the Itô integral for predictable processes ρ. Actually, any operator D satisfying the same property will provide a representation formula similar to (1.2). Here, we are going to replace D by an operator D which modifies the excursions of the Brownian path. It is not a differential operator; nevertheless, by working out a stochastic analysis of these excursions, we will prove a formula expressing the duality between D and an operator Φ which is again an extension of the Itô integration. Thus we will deduce that More explicitly, it will appear that the predictable projection in the right-hand side is an integral with respect to a measure which is strongly related to the Itô measure for Brownian excursions.
The operator Φ(ρ) coincides with Itô integrals for predictable processes ρ, so its study is related to our second question, namely the construction of stochastic integrals by means of the Brownian excursions. We will construct integrals with respect to a fixed component of the Wiener process, so we will write the Wiener process as (W t , V t ) ∈ R×R d−1 and look for integrals with respect to the first component W . If ω is a one-dimensional continuous path with finite variation, the Lebesgue-Stieltjes integral with respect to ω can be obtained by counting the points of increase and decrease of ω (this will be explained in Section 3). If now ω is a Brownian path, it has no point of increase or decrease, but one can consider the excursions of ω above or below any level, and the beginning (respectively the end) of an excursion can be viewed as a point of increase or decrease on the right (respectively the left). This procedure provides generalised Lebesgue-Stieltjes measures; notice that we have not one measure, but four measures because we have the choice of considering beginnings or ends of excursions below or above any level. Predictable processes are not directly integrable with respect to these measures, but an approximation procedure leads to a construction of Itô, Stratonovich and backward integrals (depending on the choice of the measure). This approximation is equivalent to a regularisation of the Wiener path by means of a path with finite variation; in this sense, this procedure is similar to the approach of [11], though the approximation and the results are different. If now we integrate anticipating processes, we obtain forward, backward and symmetric integrals, and we can study the adjoint Φ of D.
Technically, the study of these measures relies on classical formulas for excursions and more generally exit systems, see [2,1]; it is related to a calculus which consists in appending and removing excursions to the Brownian path, and which is similar to appending and removing jumps to Poisson processes as in [8]; this calculus has features which are similar to the Malliavin calculus, but it is not a differential calculus.
Let us outline the contents of this article. The representation formula (1.4) of a variable F is stated in Section 2, and an elementary proof is given for simple functionals F = f (W t 1 , . . . , W tn ). The general case relies on the duality formula (1.3) between D and Φ, the proof of which is postponed; some examples are discussed.
In Section 3, the family of generalised Lebesgue-Stieltjes measures associated to a onedimensional continuous path ω are introduced, and their stochastic analysis is worked out.
In Section 4, we deduce a construction of Itô, Stratonovich and backward integrals of predictable processes, and we complete the proof of the representation formula (1.4).
In Section 5, we construct the associated anticipating integrals, and explore the duality (1.3) between D and Φ.
We finish this introduction with a warning related to the general theory of processes. We will often consider predictable processes. Since we use the Wiener filtration, predictable and optional can be viewed as synonyms, but most of the results stated for predictable processes will not be true for processes which are only progressively measurable; this is usual in excursion theory and is due to the fact that the set of beginnings of excursions is progressively measurable but not predictable.

Representation of Wiener functionals
In this section, we first state in §2.1 the result, give some hints about the proof in §2.2 (it will be completed in Section 4), and describe some applications in §2.3.

The result
We first consider the one-dimensional case d = 1 (the extension to the multidimensional case is given at the end of the subsection). In order to introduce our result, we need some notation. If ω and θ are two paths indexed by R + , and if t ≥ 0, we can consider the path ω|t|θ defined by If θ is a path, we can consider the hitting time and define the path θ by The path θ is the reflection of θ up to ζ.
Let I be the Itô measure of Brownian excursions (see Chapter XII of [10]); this is a σ-finite measure on Ω supported by It can be characterised by the following property; under the restriction of I to {ζ > t} (which is a finite measure), conditionally on F t , the process (θ(t + s); s ≥ 0) is a Wiener process stopped at 0, and its entrance measure is Excursions are either positive or negative, and this leads to a decomposition Θ = Θ ↑ ∪ Θ ↓ and I = I ↑ + I ↓ . Notice that θ → θ transforms I ↑ and I ↓ into each other.
Under I, the canonical process is stopped at ζ; it is useful to also consider the unstopped process, that is an Itô excursion followed by a Wiener path. More precisely we let J be the push forward of the measure I(dθ) P(dω) by the concatenation map It is a measure on the set of paths θ ∈ Ω such that θ(0) = 0 and ζ(θ) > 0. It has a characterisation similar to I; on {ζ > t}, conditionally on F t , the process (θ(t + s); s ≥ 0) is a Wiener process, and Notice that under J , the map θ → θ consists of reflecting the first excursion without modifying the remaining of the path. Like I, the measure J has a decomposition J = J ↑ + J ↓ involving the sign of the first excursion, and θ → θ transforms J ↑ and J ↓ into each other. We are now ready to state the main result.
Then the stochastic integral representation (1.1) of E[F |F t ] satisfies the relation almost everywhere on A.
Remark 2.1.7. Variables F are often only defined almost surely. On the other hand, for t fixed, the law of ω|t|θ under P ⊗ J ↑ is singular with respect to P, so it seems that J t F is not well defined; however, it will appear that this law is absolutely continuous after a time integration, so J t F is actually well defined for almost any (t, ω) of A.
be the height of the excursion. Then (2.1.9) and the law I ↑ conditioned on {η = a} is described as follows; if T a (θ) is the first hitting time of a, then (θ(t); 0 ≤ t ≤ T a (θ)) and (a − θ(T a (θ) + t); 0 ≤ t ≤ ζ(θ) − T a (θ)) are two independent Bessel processes of dimension 3, up to their first hitting time of a. One deduces a description of J ↑ (dθ) by adding a Wiener path after ζ(θ). This decomposition can be useful for the numerical simulation of Brownian excursions, and therefore for the computation of Z t by means of a Monte-Carlo method.
Before entering the proof of the theorem, let us give its multidimensional extension. Similarly to J t F in (2.1.5), we define and

About the proof
We first prove Theorems 2.1.4 and 2.1.12 for a particular class of variables; the proof is indeed much simpler in this case, and is an application of the reflection principle for the Brownian motion. We then give some hints about the general case (the proof will be completed in Section 4).
is a bounded simple functional. Then, with the notations of Theorem 2.1.4 (onedimensional case), the set A has full measure and Z t (ω) = J t F (ω) almost everywhere. Similarly, with the notations of Theorem 2.1.12 (multidimensional case), the set A k has full measure and Z k t (ω) = J k t F (ω) almost everywhere.
Proof in the one-dimensional case. Fix t, and notice that for t j < t < t j+1 , one has F (ω|t|θ) = F (ω|t|θ ) on {θ; ζ(θ) ≤ t j+1 − t}; the complement of this event has finite measure for J ↑ . Thus A has full measure. Moreover, the martingale has the form for the function which is smooth with respect to the last variable. If g denotes the derivative with respect to this last variable, then For the last line, notice that the paths ω|t|θ ± ε are not continuous at time t and are therefore not in Ω, but F (ω|t|θ ± ε) can still be defined for simple functionals F . The integral involves the paths θ ± ε separately, so we can use another coupling of these two processes. If we choose the coupling based on the reflection principle, we get, with the notation (2.1.1), Define δ = t j+1 − t and let ζ ε (θ) = inf t ≥ 0; θ(t) = −ε be the coupling time of θ + ε and its reflected path; the difference in the integral of (2.2.4) is 0 on {ζ ε (θ) ≤ δ}, so In order to compute this expression, we need the law of (θ(δ + s) + ε; s ≥ 0) under 1 {ζ ε (θ)>δ} P(dθ)/(2ε). This process has Brownian increments, and its initial measure is obtained from the reflection principle; more precisely By studying the limit as ε ↓ 0, it appears that this process converges in law (for the convergence on all bounded Borel functions) to the process with Brownian increments and initial measure But (recall (2.1.3)) this is also the law of (θ(δ + s); s ≥ 0) under the measure 1 {ζ(θ)>δ} J ↑ (dθ), so Since the indicator can be removed, we deduce that Z t = J t F .
Proof in the multidimensional case. Proving that A k has full measure is similar. On the other hand, Z k t is given by a formula similar to (2.2.2), but involving the derivative of g(w 1 , . . . , w j , x) with respect to the k th component of x.
Let us now consider the case of a general square integrable variable F . We can approximate it in L 2 by simple functionals F n , so Z t is the limit of J t F n . However, it is not evident in the general case to prove the convergence of J t F n to J t F , so we work out another method. The main tool is the following duality formula which will be proved in Section 4 (see after Theorem 4.1.11).
Lemma 2.2.5. Let ρ t be a real-valued bounded predictable process. Suppose 1 ≤ k ≤ d and let F be a bounded F 1 -measurable variable such that so that ρ t J k t F is well defined almost everywhere. Then Taking for granted this lemma, let us explain how the representation formula is deduced.
This is a predictable process. For any bounded predictable process ρ, we can apply Lemma 2.2.5 to ρ1 {Z ≤µ} for µ > 0, and deduce Thus Z t = J t F holds on {Z t ≤ µ} and therefore on A by letting µ tend to infinity. If F is not bounded but only square integrable, we approach it by Let Z M t be the integrand involved in (1.1) for F M . Then J t F M converges to J t F on A, and Z M t converges to Z t in L 2 , so we can take the limit in the formula for F M and conclude.

Examples
In the examples, we check that A has a negligible complement; this means for almost any (t, ω), or a similar relation with J ↑ k in the multidimensional case. Then the representation formula (1.1) holds with Z t = J t F almost everywhere. If θ is a path under J , then ζ(θ) and η(θ) will denote the length and height of the first excursion. Recall that J ↑ is the push forward of I ↑ ⊗ P by the concatenation map (θ, ω) → θ|ω; the law of η(θ) under I ↑ or J ↑ has already been given in (2.1.9), and the law of ζ(θ) is deduced from (2.1.2) and is given by on (0, ∞). In particular, ζ γ ∧ 1 is integrable for γ > 1/2. The first examples are relevant to the one-dimensional case d = 1.
be a bounded function of the first hitting time at time t = 1 and level x, or of the last hitting time Then F (ω|t|θ) and F (ω|t|θ ) are different only when the height η(θ) of the excursion exceeds |x − ω(t)|, so, from (2.1.9), almost everywhere; since we have assumed that F is bounded, we deduce (2.3.1).
Example 2.3.5. Let F = T W 1 be the first time t at which W t hits W 1 . Then because on the complement of the right hand side, one has for any x. This set has again finite measure for J ↑ , so we can conclude as in Example 2.3.3. The same argument can be applied for instance to the first hitting time of W 1 /2, or to the time of [0, 1] at which W is minimal.
where denotes the local time of θ (it can be defined like L from Tanaka's formula (2.3.4)), and (at most one of two local times ±ω(t) ζ(θ)∧δ is non zero). The right-hand side is zero except on {η(θ) ≥ |ω(t)|}. Conditionally on this event, after hitting |ω(t)|, the process θ is a Brownian motion, so almost everywhere, so (2.3.1) again holds.
Notice that in this example the martingale E[F |F t ] takes the form L 0 t + f t (W t ) and Z t = f t (W t ). The next example is devoted to a non Markovian example. Example 2.3.7. Consider the local time F = L W 1 1 at level W 1 . Then (2.3.1) again holds.
Proof. As in previous example, put δ = 1 − t and let denote the local time of θ. The difference F (ω|t|θ) − F (ω|t|θ ) again involves but at two random levels; moreover, if ζ(θ) > δ, one has W 1 (ω|t|θ) = W 1 (ω|t|θ ) and it also involves the local time L of ω at two random levels. Thus we obtain the estimate We also notice that the difference is zero if |θ(δ)| > η(θ), so An easy Gaussian estimate shows that For the second term of the product, we know that I ↑ {ζ > δ/2} is finite, and we have by applying Doob's inequality. For the first term of the product, by applying maximal inequalities on local times (Theorem XI.2.4 of [10]). This completes the proof of (2.3.1).
Example 2.3.8. In the multidimensional setting, consider the first exit time of some domain D 0, and let F = f (W τ ) for f a Borel bounded function (we do not need any smoothness assumption on f nor on the boundary of D). Then F (ω|t|θ) = F (ω|t|θ (k) ) as soon as the height η(θ) of the excursion is smaller than the distance between ω(t) and the complement of D, so (2.3.1) can be proved on {t < τ } as in Example 2.3.3. In this case, E[F |F t ] = h(W t ) is given on {t < τ } by the solution h of the Dirichlet problem with boundary condition f , and Z t = ∇h(W t ), so we obtain a stochastic representation Z t = J t F for the gradient of h. Notice that this representation can be implemented by means of a simple Monte-Carlo method, whereas the classical Clark-Ocone formula is not directly applicable in this case; another classical stochastic interpretation of harmonic functions in Malliavin's calculus is to view them in duality with densities of some excessive measures (see for instance [9]), but again such an interpretation is not easy to use from a practical point of view. We can also consider F = f (τ, W τ ). Example 2.3.9. We now check that our result can also be applied to solutions of stochastic differential equations with Lipschitz coefficients.
with b and σ Lipschitz and bounded (we write the proof for real processes but the multidimensional case is similar). Then (2.3.1) holds true almost everywhere as soon as f satisfies for some rate α > 1/2.

Proof.
For ω fixed, we have to compare the solutions of the equation starting at x = X t (ω) and driven respectively by the semimartingales θ and θ . More precisely, put δ = 1 − t, let ξ s = ξ s (θ) be the solution of and similarly for ξ s = ξ s (θ ); we have where the last line is obtained by conditioning on the excursion (θ(s); 0 ≤ s ≤ ζ(θ)) and by applying classical estimates on stochastic differential equations with Lipschitz coefficients. We want to prove that this term is finite. We have and similarly for ξ , so For the first term, we have from (2.1.2) and for the others, it is sufficient (from Burkholder's inequalities) to prove that Thus for q > α/2 ∨ 1. For the first term, we notice that the exponent of ζ ∧ δ is greater than 1/2 if q is chosen large enough (because α > 1/2), so this term is finite from (2.3.2). On the other hand, from classical estimates on Itô processes and by applying again (2.3.2), We can conclude that I is finite.

Generalised Stieltjes measures
In this section, considering a real-valued function, we construct in §3.1 a family of measures which generalise the classical Lebesgue-Stieltjes measure, and we show in §3.2 and §3.3 how the calculus for Brownian excursions can be applied to them. This will be the main tool for the construction of stochastic integrals in next section, and in particular for the proof of Lemma 2.2.5. Throughout this section, except in §3.4, we suppose that we are in the one-dimensional case (d = 1).

A family of measures
If ω is a real-valued continuous increasing path, then More generally, if ω is continuous and piecewise monotone, let S ω (t) be +1 (respectively −1) on intervals where ω is increasing (respectively decreasing). Then the Lebesgue-Stieltjes integral with respect to ω can be written as If now ω is a general continuous path (such as a typical Brownian path), this formula has no clear sense since the sum can be uncountable and the path may have no increase or decrease time (so S ω is not well defined). Instead of considering all the times t in ω −1 (x), we will only consider times t such that ω is decreasing or increasing on the left or on the right of t. In other words, we will be concerned with beginnings or ends of excursions above or below any level x, and S ω will be linked to the sign of these excursions. An end of excursion is obtained as a first hitting time , and a beginning of excursion is obtained as a last hitting time More precisely, consider which is almost surely finite for a Brownian path. We say that an excursion begins at time t if ζ t (ω) > 0. In this case, we define the excursion Υ t ω ∈ Θ beginning at t by The length and the height of this excursion are ζ t and η t = η(Υ t ) (see Figure 2). For other values of t, we let Υ t = 0 be the null excursion. We obtain a path with values in Θ ∪ {0}; we will write Υ t > 0 or Υ t < 0 depending on the sign of the excursion. Then we let ω x (dt) be the counting measure for the set of beginnings of excursions of the path ω above level x, that is for A ⊂ R + , and we define The measure ω can be obtained on [0, 1] as the increasing limit of measures ω ∆n for any refining sequence of subdivisions such that |∆ n | = max(t n j+1 −t n j ) ↓ 0. If ω has finite variation, then ω (dt) is the positive part ω + (dt) of the Lebesgue-Stieltjes measure associated to ω.
Proof. We first verify the description (3.1.5) of the measure ω ∆ . The set B ∆ ∩ [t j , t j+1 ] is the set of beginnings of positive excursions beginning in [t j , t j+1 ] and straddling time t j+1 , so that (with notation (3.1.1)) ω and (3.1.5) is proved. It is clear that ω is the limit of ω ∆n , and since ω ∆n is finite, this implies that ω is σ-finite. Let us now consider the case of a path ω with finite variation on [0, 1]. We deduce from (3.1.5) that At the limit, we obtain that ω ([0, 1]) ≥ ω + ([0, 1]); since ω ≤ ω + , the proof of ω = ω + on [0, 1] is complete.
Remark 3.1.6. The integral with respect to ω ∆ (dt) can be written as In the definition (3.1.3), the measure ω (dt) is decomposed according to the level x = ω(t); we now see that it can also be decomposed according to the height a = η t of the excursion starting at time t; this will be useful in Section 4 (see the proof of Theorem 4.4.13).
Lemma 3.1.8. The measure ω of (3.1.3) can be written as Proof. Following the notation of Lemma 3.1.4, it is sufficient to study the measure ω ∆ on [t j , t j+1 ]. Consider the restriction of ω:

This map is injective, its image is the interval
, and the pull back ω −1 j (dx) of the Lebesgue measure on I is the measure ω ∆ . On the other hand, for each level x ∈ I, let us consider the height α(x) = η g x t j+1 of the excursion starting at g x t j+1 (see Figure 3). Then is the time at which this excursion is maximal. Actually, x → τ (x) is piecewise constant; it changes its value when the end d x t j+1 of the excursion is a local minimum of ω, and the excursion starting at this minimum has a height greater than α(x); in this case one has α(x−) > α(x). Thus x → α(x) is decreasing, and is the sum of an affine function with slope −1 and of jumps. We deduce that the pull back α −1 (da) of the Lebesgue measure on α(I) is the Lebesgue measure on I; by composition, the pull back (α thus it holds for A ⊂ B ∆ , and therefore for any A by letting |∆| ↓ 0.
Similarly to ω x and ω , we can consider the measures associated to beginnings of negative excursions. In the finite variation case, ω is the negative part of the Lebesgue-Stieltjes measure of ω.
Let us now consider ends (instead of beginnings) of excursions; the excursion Υ † t ending at time t will be defined later; now let us simply denote Υ † t > 0 or Υ † t < 0 depending on the sign of this excursion (the sign is well defined even if the excursion is incomplete). Then we can define Thus we have four elementary measures associated to ω. Then we put which are 0 in the finite variation case. More precisely, we have All these definitions are summarised in the following table, where GLSM stands for generalised Lebesgue-Stieltjes measure.
ω beginnings of positive excursions ω beginnings of negative excursions ω ends of positive excursions ω ends of negative excursions Remark 3.1.12. It is known that positive excursions of a path can be viewed as a tree (see for instance [4,3]). Then ω and ω actually correspond to the Lebesgue measure on this tree. This point of view will be explored in another article.
If we apply this construction to the Wiener paths, we obtain the measures W x (dt), . . . and W (dt), . . . . Of course, a result for one of them can be transferred to the three others by time reversal and/or change of sign. Notice however that if we want to study predictable processes, there is a big difference between the measures W (dt) and W (dt) which are predictable, and the two others which are supported by a set (beginnings of excursions) intersecting no graph of stopping time but only graphs of honest times, see [2]. Notice also that all of these measures are supported by dense Lebesgue negligible subsets of R + .

Calculus for excursions
We need to define some transformations of Ω consisting in adding and removing excursions.
The transformation E + t,θ consists in inserting the excursion θ ∈ Θ at time t, so that On the other hand, E − t removes the excursion beginning at time t if an excursion indeed begins at this time, otherwise it does nothing; thus Notice that if φ t is a predictable process, then This property is evident for φ t = 1 {r<t≤s} F and F a F r -measurable variable, and it is extended to other predictable process with a monotone class theorem. Let us now study the effect of these transformations on non predictable processes.
In order to prove this theorem, we first state the following basic lemma about excursions at a given level.
Lemma 3.2.6. Fix some real x and let L x t be the local time at level x given by the Tanaka formula (2.3.4). Let φ t be a nonnegative measurable process. Then In particular, where ψ t is predictable. Then it is known (see Chapter XX of [2] or Chapter III of [1]) that On the other hand, if we look at the right-hand side of (3.2.7), we have Under P(dω)I(dθ), the concatenated path θ|Υ t (ω) is independent of F t and has law J , so Proof of Theorem 3.2.4. We integrate (3.2.8) with respect to x. In the left-hand side we obtain the measure W (dt), and in the right-hand side, integrating dL x t with respect to x gives the Lebesgue measure dt.
Of course, a similar formula holds for W (dt) and I ↓ (dθ).

Iterated formulas
We now notice that Theorem 3.2.4 can be iterated to study integrals of higher order. Theorem 3.3.1. Let φ t 1 ,t 2 be a nonnegative process indexed by R 2 + . Then Proof. We iterate (3.2.5) by applying it firstly with respect to t 2 , secondly with respect to t 1 . Since we are on {t 1 < t 2 }, we notice that and obtain (3.3.2). The other cases are similar.
In the right-hand side of (3.3.2), we first append an excursion at time t 1 , then another one at time t 2 . This may result in two different figures. If t 2 > t 1 + ζ(θ 1 ), then we have appended two disjoint excursions. On the other hand, if the reverse inequality holds, we insert θ 2 inside θ 1 and obtain an augmented path E + t 2 −t 1 ,θ 2 θ 1 containing θ 2 as an excursion; this augmented path always begins with an excursion; however it is not an excursion if θ 1 and θ 2 have opposite signs and η(θ 2 ) ≥ θ 1 (t 2 − t 1 ) . We denote by θ the first excursion of this augmented path (see Figure 4), so that We are going to see that θ plays an important role in our estimations. Theorem 3.3.1 can be applied to processes of type φ t 1 ,t 2 = φ t 1 φ t 2 , in order to deduce the L 2 norm or the variance of φ t W (dt) or φ t W → (dt). Let us explain this calculation in the simple case φ t = 1 B (t, Υ t ) for a B ⊂ R + × Θ of finite measure for dt I(dθ). Relation (3.2.5) is written as On the other hand, for t 1 < t 2 , By comparing (3.3.4) and (3.3.5), we get the variance Notice that in this integral we only have to consider times t 2 ≤ t 1 + ζ(θ 1 ) (otherwise θ = θ 1 ). We can handle similarly negative excursions, and by considering opposite signs, we get the covariance We also deduce if θ 1 and θ 2 have the same sign, and it is non zero when θ 1 have an height less than a, but θ 2 has height greater than a and becomes a sub-excursion of θ 1 . Thus If X t is a Bessel process (of dimension 3), then X 2 t −3t is a martingale, so the expected time of the first hitting time of a is a 2 /3; consequently, the Williams decomposition recalled in Remark 2.1.8 shows that the expected length of an excursion of height a is 2a 2 /3. We deduce that the conditional expectation under I ↑ of ζ(θ) given η(θ) is equal to 2η(θ) 2 /3, and that 1 We also check with the Williams decomposition that the conditional expectation of ζ(θ) 2 given η(θ) is finite and proportional to η(θ) 4 , so the difference between (3.3.10) and (3.3.11) is bounded by On the other hand, the covariance is given by (3.3.7) and if θ 1 and θ 2 have opposite signs, and is non zero when excursions θ 1 and θ 2 have height greater than a, but the height of θ is less than a; this happens when t 2 − t 1 is less than the first hitting time T a (θ 1 ) of a (or −a) by θ 1 . The contributions of I ↑ (dθ 1 )I ↓ (dθ 2 ) and I ↓ (dθ 1 )I ↑ (dθ 2 ) are equal, and Conditionally on {η(θ) ≥ a}, T a (θ) is the hitting time of a by a Bessel process, so it has expectation a 2 /3 and The difference between (3.3.15) and (3.3.16) is bounded by By proceeding as in previous example, we can prove that this variance is dominated by its limit and similarly for W . One can also write the formula involving mixed excursions for the covariance, but calculations are more complicated. Subsequently, we will only use the boundedness of the covariance, which follows from the boundedness of the variances.

The multidimensional case
In this short subsection, we describe how results of §3.2 and §3.3 are extended to the multidimensional case Ω = C(R + , R d ). Excursions of each component have to be considered separately, so let us suppose that we want to study the first component; then we write the Wiener process as (W t , V t ) ∈ R × R d−1 . We let Θ 1 be the space of d-dimensional continuous paths such that the first component θ 1 is an excursion with length ζ(θ), and the other components are paths stopped at time ζ(θ). On this space, we consider the measures I ↑ 1 and I ↓ 1 under which the first component θ 1 is a Brownian positive or negative excursion, and the other components are independent Wiener paths up to time ζ(θ); the main difference with respect to the one-dimensional case is that now θ(ζ(θ)) = 0. Then E + t,θ again consists in inserting θ at t; the difference with respect to (3.2.1) is that for s > t. The process Υ t is defined as in (3.1.2) (its first component is an excursion). Then Theorems 3.2.4 and 3.3.1 hold with the measures W (dt) and I ↑ 1 (dθ) (one first consider processes φ t which are the product of a functional of W and of a functional of V , then the general case follows from a standard monotone class theorem).

Stochastic integration
In this section, we use the generalised Lebesgue-Stieltjes measures (3.1.10) in order to construct stochastic integrals of predictable processes. As in §3.4, we write the Wiener process as (W t , V t ) ∈ R×R d−1 , and we want to construct integrals with respect to the first component W . More precisely, we check that the measure W → (dt) leads to Itô integrals, that the measures W ↑ (dt) and W ↓ (dt) lead to Stratonovich integrals, and that W ← (dt) leads to so-called backward integrals. Each type of integral is studied by means of an approximation procedure; more precisely, the approximation consists in restricting the generalised Lebesgue-Stieltjes measures to subsets of the time interval in order to obtain finite signed measures. As an application, the construction of Itô integrals is closely related to the duality formula of Lemma 2.2.5, so we are able to complete the proof of the representation theorem for Wiener functionals (Theorems 2.1.4 and 2.1.12). Results of this section are summarised in §4.5.
Notice however that this is a nonlinear approach to integration; we can separately construct an integral with respect to each component of the Wiener process, but if now we consider the integral with respect to the vector-valued process as the sum of these integrals, then the result depends on the choice of the frame on R d . For instance, the measures W → 1 + W → 2 and (W 1 + W 2 ) → are not equal (a beginning of excursion of W 1 is not a beginning of excursion of W 1 + W 2 ), though they would be equal if W 1 and W 2 were of finite variation (since they would coincide with Lebesgue-Stieltjes measures).
It is sometimes easier for notational convenience to consider processes indexed by R. In this case, (W t , V t ; t ≥ 0) and (W −t V −t ; t ≥ 0) are independent standard Wiener processes. However, we always construct integrals on the time interval [0, 1]; this means that we integrate a process ρ t , 0 ≤ t ≤ 1, which is extended by ρ ≡ 0 out of this interval.

Itô integrals
It can be seen from results of previous section that predictable processes cannot be directly integrated with respect to W → (dt) (consider for instance a non zero constant process). However (with the notations of §3.4), if B is a subset of [0, 1]×Θ 1 which has finite measure for dt I 1 (dθ), we have and similarly for W (dt), so we can consider for any bounded process ρ, where is an anticipating process with finite variation. Since we want positive and negative excursions to compensate, we assume that B is symmetric, that is with θ (t) = (−θ 1 (t), θ 2 (t), . . . , θ d (t)) (it was denoted by θ (1) in (2.1.10) but we prefer a lighter notation for this section). Then we can define Ψ → (ρ) as the limit in some sense of Ψ → B (ρ) when B grows to [0, 1] × Θ 1 , if this limit exists. In this subsection, we suppose that ρ is predictable; we check that the convergence holds for a weak topology and that the limit Ψ → (ρ) coincides with the Itô integral. We also check that the convergence may be stronger under some assumptions.

4.1a The general case
We now introduce the analogue of the Malliavin derivative in our calculus; this is the difference operator which is defined for (t, ω) such that the integrand of the right-hand side is integrable. Notice in particular that the operator J 1 t defined in (2.1.11) can be written as the predictable projection We have checked in Lemma 2.2.1 that for simple bounded functionals F , J 1 t F is the process Z 1 t involved in the integral representation formula (1.1), so we deduce that if F is simple, bounded, and ρ t is predictable, bounded, then We can now state the general convergence result of Ψ → B (ρ) to the Itô integral of ρ.
be a bounded simple variable, and let (ρ t ; 0 ≤ t ≤ 1) be a bounded predictable process. Then for any sequence B n of symmetric sets with finite measure increasing to [0, 1] × Θ 1 , one has This theorem states that Ψ → B (ρ) converges to the Itô integral of ρ for the weak topology induced by bounded simple variables. The basic result used in the proof is the following lemma.
Lemma 4.1.7. Let ρ t be a bounded predictable process, and let F be a bounded variable such that Then, for any (B n ) as in Theorem 4.1.5, Proof. First notice as in (3.2.3) that since ρ t is predictable, one has ρ t • E + t,θ = ρ t . Then Theorem 3.2.4 (generalised in §3.4) implies that and similarly for the W -integral, so, from the symmetry of B n , Assumption (4.1.8) enables to take the limit in the right-hand side and to deduce the lemma.
Proof of Theorem 4.1.5. If (t j ) are the times associated to the simple variable F , one has On the other hand, from (2.3.2), < ∞ so (4.1.8) holds. We can apply Lemma 4.1.7, and the conjunction of (4.1.4) and (4.1.9) enables to conclude.

4.1b The weak L 2 convergence
The convergence of Theorem 4.1.5 holds for a large class of sequences (B n ). By choosing particular sequences, we can have a better (but still weak) convergence, namely for the weak topology of L 2 . In the same time, we are able to complete the proof of Lemma 2.2.5, and therefore of the integral representation formula (Theorems 2.1.4 and 2.1.12). Bounded sequences of L 2 are relatively compact for the weak L 2 topology, so it is easy to see that the statement of Lemma 4.1.10 holds true as soon as Ψ → Bn (ρ) is bounded in L 2 for any bounded predictable ρ; these sequences (B n ) are subsequently called good approximating sequences. We give below three examples of good approximating sequences (B n ), so this will complete the proof of the lemma.
An important consequence of Lemma 4.1.10 is the following duality formula.
Theorem 4.1.11. The duality formula holds for any bounded F and any bounded predictable process ρ satisfying (4.1.8).
Proof. The left-hand side of (4.1.12) is expressed in (4.1.9) as a limit, and the fact that this limit is the also the right-hand side of (4.  We have already studied in Example 3.3.8 the case ρ ≡ 1, and check that Ψ → B (1) is centred with bounded variance. We now have to consider other bounded predictable processes ρ; the approximation Ψ → B (ρ) is again centred and we have to compute the variance. First notice that if φ t is a bounded symmetric process in the sense that then, from (3.2.5), Consequently, because this is the integral with respect to W → (dt 2 ) of a symmetric process. Thus by applying Theorem 3.3.1, and with the notation θ given in (3.3.3). By applying the decomposition I = I ↑ + I ↓ , the sign of 1 B (t 1 , θ) − 1 B (t 1 , θ 1 ) given in (3.3.9) and (3.3.14), and formulas (3.3.6) and (3.3.7), we check that the quadruple integral is half of the variance of Ψ → B (1). It is dominated by the variances of Ψ B (1) and Ψ B (1) which have been proved to be bounded in Example 3.3.8. Thus Ψ → B (ρ) is bounded in L 2 , and it is sufficient to choose a sequence B n = B 0 (a n ) corresponding to a sequence a n ↓ 0 to obtain a good approximating sequence. This means that we consider all excursions straddling time 1, independently of their height. The approximation Ψ → B (ρ) for B = B 1 (a) is now F 1 -measurable, and it converges like previous one, because the difference between this approximation and previous one is dominated by a. Subsequently, as mentioned in the beginning of the section, it will be useful to consider paths indexed by R; in this case, we also add to B 1 (a) excursions beginning before time 0 and ending in [0, 1]; this means that we put We are again reduced to the boundedness of the variances corresponding to ρ ≡ 1 (Example 3.3.19). These approximations will be called length-based approximations.
Example 4.1.18. Consider a subdivision ∆ = (t j ) of [0, 1], and let B = B(∆) be the set of (t, θ) such that the interval [t, t + ζ(θ)] contains a point of the subdivision. With the notation ω ∆ introduced in (3.1.5), a similar notation ω ∆ and ω → ∆ = ω ∆ − ω ∆ , we have from (3.1.7) where ω[r, s] is the range of ω on [r, s]. Thus An application of the duality formula (3.2.5) shows that We check that Y [r,s] is F s -measurable, and an application of (4. so the variance of Y [t j ,t j+1 ] is of order t j+1 − t j , and the sum Y [t j ,t j+1 ] is bounded in L 2 . Thus Theorem 4.1.11 can be applied to B n = B(∆ n ) for any sequence ∆ n = (t n j ) of subdivisions of [0, 1] such that max j (t n j+1 − t n j ) → 0.

4.1c The strong L 2 convergence
Notice that the approximations Ψ → B (ρ) generally do not converge strongly in L 2 . For instance, for ρ ≡ 1, consider the approximation based on time discretization of Example 4.1.18. In this case we have From the invariance of the Wiener measure by time reversal, we check that so Y [r,s] is the sum of W s − W r and of a non trivial orthogonal variable and Thus does not converge to E[W 2 1 ] = 1, and the convergence lim Ψ → B (1) = W 1 does not hold strongly in L 2 .
Nevertheless, we now prove the strong convergence for a particular approximation when ρ is continuous. Proof. We already have the weak convergence, so it is sufficient to prove that for B = B(a) and a ↓ 0. This has been checked in (3.3.18) for ρ ≡ 1, and we have to write the formulas in the general case. The variance has already been estimated in Example 4.1.13; we again use (4.1.14), but we handle it differently, taking advantage of the continuity of ρ. We have The first term is studied from Theorem 3.3.1 (notice that ρ t 1 is not affected by E + t 1 ,θ 1 and E + t 2 ,θ 2 ), and we get By proceeding as in Example 3.3.8, we check For I 2 , we have enumerated in Example 3.3.8 the cases where the difference between the indicators is non zero; this can happen only when t 2 belongs to the excursion beginning at t 1 , and before this excursion hits the level ±a, so we must have a . The first condition shows that and the second one shows that for t 2 fixed, This term is shown to converge to 0 from the Cauchy-Schwarz inequality because R a converges almost surely to 0 and 1 B (Υ t )|W → (dt)| is of order 1/a in L 2 . Adding the asymptotic behaviours of I 1 and I 2 yields (4.1.21).
Remark 4.1.22. In §4.4, in Remark 4.4.10, we will see that almost sure convergence holds in some cases.
Remark 4.1.23. We do not know whether the same result holds for length-based approximations of Example 4.1.17, but we doubt it. From the proof which is given here, the strong convergence for height-based approximations may seem to come from the miraculous formula (3.3.18). Actually, the particularity of height-based approximations with respect to others will appear more clearly in §4.3.

Backward integrals
In this subsection we consider the measure W ← (dt) based on ends of excursions. More precisely, we suppose that (W t , V t ; t ≥ 0) and (W −t , V −t ; t ≥ 0) are independent standard Wiener processes. We can define the length of the excursion of W ending at t by and the excursion ending at t by Then, similarly to (4.1.1) and (4.1.2), we consider the operator . The process W ← t (B) also has finite variation, but an important difference with respect to W → t (B) is that it is predictable. If we use for instance height-based approximations (4.1.16), then Ψ ← B(a) (ρ) is measurable with respect to (W t , V t ; 0 ≤ t ≤ 1) (see the discussion of Remark 4.1.15). We are going to check that when ρ is a semimartingale, then Ψ ← B (ρ) is an approximation of what is usually called the backward integral of ρ. for bounded predictable processes α t , b t and β t . Let B = (B n ) be a sequence such that W ← t (B n ) is bounded in L 2 . Then Ψ ← B (ρ) converges for the weak topology of L 2 to Proof. Theorem 4.1.5 and a time reversal imply that for any bounded simple F ; from our assumption about the L 2 boundedness of W ← t (B n ), this convergence also holds for any F ∈ L 2 . On the other hand, Itô's formula applied to (4.2.1) and (4.2.4) yields and If now F is a square integrable variable, it can be written from (1.1) as a stochastic integral of a predictable square integrable process Z t = (Z 1 t , Z t ) with respect to (W t , V t ), and we have and From (4.2.5), each term of the right-hand side of (4.2.7) converges to the corresponding term of (4.2.8), so the first statement of the theorem is proved. For height based approximations, one proves as in Theorem 4.1.20 that W ← t (B(a)) converges strongly to W t , so one can take the limit in (4.2.6) and obtain the strong convergence of Ψ ← B(a) (ρ). Remark 4.2.9. As for Itô integrals, we will see in Remark 4.4.10 that the convergence can be almost sure.
is a weak approximation of the quadratic covariation ρ, W 1 . Next subsection is devoted to another approximation of this covariation.

Quadratic covariation
We now study the correction measure W (dt) given by (3.1.11) (one can of course study similarly W (dt)). We consider like previously paths indexed by R and put As for Ψ → B and Ψ ← B , we are interested in the asymptotic behaviour of this expression. However, we only consider height-based approximations B = B(a) defined in (4.1.16). We suppose that ρ is a smooth Itô process, that is a process given by (4.2.3) for bounded α, b, and β, such in the spaces L p , 1 ≤ p < ∞.
The particularity of height-based approximations is that positive and negative excursions of height greater than a are in bijection with each other, and actually, this explains the particular behaviour of these approximations which was already noticed in Theorems 4.1.20 and 4.2.2. Let us define If t is the beginning of a positive excursion of height greater than a, then σ t is the end of a negative excursion of height greater than a, and t → σ t is a bijection between these two sets. This remark is important for the following result. We first prove the following lemma. Proof. Let us first consider the case b ≡ 1. The difference between the two sides is of order a in L 2 (recall the calculation of Example 3.3.8). We can deduce from the Borel-Cantelli lemma that the almost sure convergence holds for the sequence a n = 1/n. Moreover, the monotonicity of a → B(a) implies that for a n+1 ≤ a ≤ a n . We deduce the whole almost sure convergence as a ↓ 0. The case b t = 1 [0,r] (t) is studied similarly, so almost surely, the measure a1 B (t, Υ t )W (dt) converges weakly to dt/2, and the lemma is proved for any continuous b. But the continuity of b follows from (4.3.1) and the Kolmogorov lemma (Theorem I.1.8 of [10]).
for any 0 ≤ r ≤ t ≤ 1, and for some finite variable K.
Remark 4.3.8. If b were assumed to be a semimartingale, this type of result can be used to check that the process (W, b) with its Lévy area has finite p-variation for p > 2 (see [5]). The multiplicative property satisfied by rough paths is here written as for r ≤ s ≤ t.
Proof of Lemma 4.3.7. The first estimate is classical from the Kolmogorov lemma and relies on (4.3.1). In particular, from (4.3.9), for r ≤ s ≤ t and for some finite K 0 . On the other hand, the variable We apply this inequality to r = t n k = k2 −n , t = t n k+1 , and obtain if p is chosen large enough. Thus the Borel-Cantelli lemma shows that for some finite K 1 . Now, for 0 < r < t < 1, let n = n(r, t) be the maximal integer such that for some k. Then because if it is smaller, then n + 1 satisfies (4.3.12). The almost additive property (4.3.10) shows that for 0 < γ < 1. Thus The integral in the left-hand side is close to the integral of the left-hand side of (4.3.5). For the right-hand side, notice that (σ t − t) γ ∧ 1 converges almost surely to 0 as a ↓ 0, and is bounded by ζ γ t ∧ 1. Moreover, almost surely. Thus we deduce from the dominated convergence theorem that the righthand side of (4.3.15) converges to 0. This completes the proof of the almost sure convergence of Ψ B (ρ).
Proof of Theorem 4.3.3 in the general case. We have already considered the case β ≡ 0, so we can now suppose α ≡ b ≡ 0. We use the filtration F t generated by (W s ; 0 ≤ s ≤ 1) and (V s ; 0 ≤ s ≤ t).
where the stochastic integral is understood in the filtration (F t ). Similarly to the previous case, we can prove that W t (B) = Ψ B (1 [0,t] ) converges almost surely to 0, uniformly in t, so we deduce the convergence in probability of Ψ B (ρ) to 0.

Stratonovich integrals
In this subsection, we consider the measures W ↑ (dt) and W ↓ (dt) on R, and define (4.4.2) and similarly Actually, we focus our study on Ψ ↑ B since the other one is similar. Notice that so we deduce from Theorems 4.2.2 and 4.3.3 that for a smooth Itô process and for heightbased approximations, it converges in probability to the Stratonovich integral. Our aim is to prove that the convergence in probability actually holds for any approximation and that it is almost sure when β ≡ 0 or for height-based approximations. Notice that if Ψ ↑ B (ρ) converges in probability as B ↑ Θ ↑ 1 , the limit is necessarily the Stratonovich integral (since this is the limit on a subsequence).
The definition (4.4.1) of Ψ ↑ B involves the beginnings and ends of the same excursions, so we can write almost surely, then Ψ ↑ B (ρ) converges almost surely to Proof. Since ρ is bounded, in order to prove (4.4.5), we can neglect the first and last incomplete excursions and only prove that We deduce from (4.3.14) that ρ t+ζt − ρ t ≤ K(ζ γ t ∧ 1), so (4.3.16) implies (4.4.8). The other statements of the theorem follow from the above discussion.
Remark 4.4.9. The approximation (4.4.1) of the Stratonovich integral is written as the integral with respect to the anticipating process with finite variation W ↑ t (B). For instance, for the height-based approximations of (4.1.16), the path W ↑ t (B(a)) is depicted in Figure  5, and one has W t − a ≤ W ↑ t (B(a)) ≤ W t . For approximations associated to subdivisions ∆ = (t j ) of [0, 1] (Example 4.1.18), one has W .  Proof. The case β ≡ 0 has already been studied in Theorem 4.4.7, so we now suppose α ≡ b ≡ 0 (in particular, Itô, backward and Stratonovich integrals coincide). We use as in Theorem 4.3.3 the enlarged filtration F t generated by (W s , 0 ≤ s < ∞; V s , 0 ≤ s ≤ t).
The process ρ t is a semimartingale for this filtration, and the process W ↑ t (B) given by (4.4.2) is F 0 -measurable. We can write This estimation is sufficient to deduce the convergence in probability of Ψ ↑ B (ρ) in the form (4.4.12).
We now show that in the case of height-based approximations, almost sure convergence to the Stratonovich integral holds even when β ≡ 0. Proof. The result has already been proved for β ≡ 0 in Theorem 4.4.7, so we now suppose α ≡ b ≡ 0, and consider the enlarged filtration F t . We use Lemma 3.1.8 to write (4.4.4) in the form where the 'O(a)' involves the small excursions straddling time 0 or 1. Thus the theorem will proved if we check that is bounded in L 1 as a → 0 because the sum in the right-hand side is 0 or 1, so (4.4.14) holds true.

Summary
We summarise in a table the results obtained in this section about convergence to the stochastic integrals and the quadratic covariation in various frameworks, when ρ is a smooth semimartingale satisfying (4.2.3) and (4.3.1), and either for any approximating sequence, or for the height-based approximations B(a) of (4.1.16).
The weak L 2 convergence ((w) in the table) holds if Ψ → B (ρ) is bounded in L 2 (for the Itô integral) or when Ψ ← B (1) is bounded in L 2 (for the backward integral).

Anticipating calculus
In this section, we explain how techniques of previous section can be applied to study Ψ → B (ρ), . . . for some anticipating processes ρ and to construct anticipating integrals Ψ → (ρ), . . . . We also obtain a duality between the transformation D of the Wiener path and an operator Φ; this duality is similar to the duality between the Malliavin derivative and the Skorohod integral (see for instance [6]).

Anticipating integrals
We again consider the space generated by the Wiener process (W t , V t ), and we want to integrate with respect to W processes ρ t , 0 ≤ t ≤ 1, which are not predictable for the Wiener filtration. We say that the bounded process ρ t is in the domain of Ψ → with integral Ψ → (ρ) if Ψ → B (ρ) defined in (4.1.1) converges in probability to Ψ → (ρ) for heightbased approximations B = B(a) of (4.1.16), a ↓ 0. We can give a similar definition for Ψ ← , Ψ ↑ , Ψ ↓ . In view of the predictable case, we can say that Ψ → (ρ) is a forward integral, that Ψ ← (ρ) is a backward integral, and that Ψ ↑ (ρ) and Ψ ↓ (ρ) are symmetric integrals; these two last integrals were equal in the predictable case and are also equal in the examples below.
Remark 5.1.1. One can consider other definitions, by requiring for instance the weak L 2 convergence instead of the convergence in probability.
Example 5.1.2. In the case dρ t = α t dt, it is not necessary to assume that α is adapted. Even in the anticipative case, one can write and similarly for the other integrals, so the four types of integrals exist and coincide with A similar result holds in the case dρ t = β t dV t , where β t is non anticipating with respect to V but may be anticipating with respect to W ; this means that we use (like previously) the filtration F t generated by (W s ; s ≥ 0) and (V s ; 0 ≤ s ≤ t).
Example 5.1.3. Suppose that ρ t = ρ t+ε , where ρ t is an Itô process (4.2.3). Then we divide the time interval into intervals of length ε. On each of these intervals, ρ is an integral with respect to the Wiener increments on the next interval, and these increments can be viewed as an independent process. Thus we are reduced to Example 5.1.2, and all the integrals coincide with where the integral is computed for the filtration F t+ε .
for some finite variable K and some 1/2 < γ ≤ 1, then ρ t+ζt − ρ t is integrable with respect to W ((4.4.5) holds true), and Ψ ↑ (ρ) can be defined by the right-hand side of (4.4.6). We can approach it by for a subdivision ∆ = (t j ) of [0, 1]. If we compare with Riemann sums, it appears that Estimates on the modulus of continuity of W and ρ show that this expression converges to 0, so Ψ ↑ (ρ) is the limit of Riemann sums and therefore coincides with the Young integral converges to 0, so Ψ B (ρ) (and also Ψ B (ρ)) converges to 0, and the four types of integrals coincide.

A duality property
We now describe for the multidimensional Wiener process (W 1 t , . . . , W d t ) the relation between our integrals and the adjoint Φ of D = (D 1 , . . . , D d ) in L 2 , where each operator D k is defined similarly to (4.1.3). If we compare this calculus with standard anticipating calculus, the operator Φ can be viewed as an analogue of the Skorohod integral, so our problem is similar to the description of some anticipating integrals (which are constructed as some limits) by means of the Skorohod integral, as in §3.1 of [6].
Let S be the class of simple functionals F = f (W t 1 , . . . , W tn ) with f bounded and Lipschitz; in particular, for any 1 ≤ k ≤ d and for t j < t < t j+1 , so D t F is bounded for these functionals. We define the domain of Φ to be the set of processes ρ t in L 2 (R + × Ω; R d ) such that for any F ∈ S, and for some C ≥ 0. Then Φ(ρ) is defined in L 2 by for any F ∈ S.
In particular, Φ(ρ) is centred. Since S is dense in L 2 , it is classical to verify that Φ(ρ) is uniquely defined by this relation. Moreover, Φ is a closed operator.
We use the notation ρ − k,t = ρ k,t • E − k,t where ρ k,t is the k th component of ρ t , and E − k,t removes the excursion of the k th Wiener component starting at time t (see §3.4 for k = 1); this excursion is denoted by Υ k,t . In particular, if ρ is predictable, then ρ − = ρ. converges for the weak topology of L 2 induced by S, and Φ(ρ) is its limit.
Proof. We deduce from Theorem 3.2.4 that If B is symmetric, we have a similar relation with W k (dt) and E + t,θ replaced by W k (dt) and E + t,θ (k) , so by taking the difference between these two relations, we obtain for F ∈ S Thus lim a↓0 E F Ψ → B(a) (ρ − ) = E (D t F, ρ t ) dt and the theorem is proved.
Proof. We already know Ψ → (ρ) so we have to study Ψ → (ρ − ρ − ). We have If we separate the last excursion from the others, we get The first term is easily studied as B ↑ R + × Θ, and leads to the last excursion in Γ x . Let us write the second term as Γ B x dx. Then Γ B x is a sum of orthogonal variables because the signs of the excursions are independent, and independent from the absolute excursions and the future (W t ; t ≥ 1). Thus where in the second line we apply a conditional expectation given F 1 , and in the third line we notice that the total length ζ t of excursions is bounded by 1, and that there is no term if the process does not hit x before time 1. Moreover, if we consider B = B(a n ) for a n ↓ 0 and if Γ n x are the associated variables, then (Γ n+1 x − Γ n x ; n ≥ 0) are orthogonal (for the above reason of independent signs), so Γ n x converges strongly in L 2 , and it is not difficult to deduce that Γ n x dx also converges ( Γ n x − Γ ∞ x 2 converges to 0 for any x, and is dominated by P[T x < 1] which is integrable, so we apply the dominated convergence theorem). We deduce the desired result.
Example 5.2.5. Returning to the multidimensional case, let P be the set of processes ρ t = (ρ k,t ; 1 ≤ k ≤ d) of the form ρ k,t = n−1 j=0 g k,j (W k t 0 , . . . , W k tn ; W 1 , . . . , W k−1 , W k+1 , . . . , where (t 0 , . . . , t n ) is a subdivision of [0, 1] and g k,j is bounded and Lipschitz with respect to its (n + 1) first arguments. By proceeding as in previous example, we can prove that P is included in the domain of Φ. In particular, the domain of Φ is dense in the space of square integrable processes.
We can define the adjoint D of Φ as follows. The domain of D is the set of variables F ∈ L 2 such that for any ρ in the domain of Φ, and for some C ≥ 0. If F is in this domain, DF is defined in L 2 by E (D t F, ρ t )dt = E F Φ(ρ) (5.2.6) for any ρ in the domain of Φ. Then D is a closed operator. Let us compare D and D; it is clear that they coincide on S.
Theorem 5.2.7. The operator D is the closure in L 2 of the restriction of D to S.
Proof. The domain of Φ is dense in L 2 (Example 5.2.5), so D is closed, and it is sufficient to prove that S is dense in the domain of D endowed with the norm Let G be in the domain of D and orthogonal to S; we want to check that G = 0. For any F ∈ S, we have