## Abstract

We provide cubature formulae for the calculation of derivatives of expected values in the spirit of Terry Lyons and Nicolas Victoir. In financial mathematics derivatives of option prices with respect to initial values, so called Greeks, are of particular importance as hedging parameters. The proof of existence of cubature formulae for Greeks is based on universal formulae, which lead to the calculation of Greeks in an asymptotic sense—even without Hörmander's condition. Cubature formulae then allow to calculate these quantities very quickly. Simple examples are added to the theoretical exposition.

## 1. Introduction

Cubature formulae provide approximative values for integrals with respect to a given measure. The well-developed theory of cubature formulae in finite dimensions was recently applied to provide cubature formulae on Wiener space by Lyons & Victoir (2004) in their seminal article. Cubature formulae are based on the basic observation that any diffusion can be constructed—up to given high-order time asymptotics—by a superposition of iterated Stratonovich integrals in well-specified directions. Hence, it is sufficient to obtain cubature results for those iterated Stratonovich integrals in order to obtain them for any diffusion. It is of crucial importance to realize then that the weighted points respect the geometry of the problem to obtain a deterministic, pathwise interpretation of the result. In the same spirit, we construct the well-known Malliavin weights—up to given (high) order of time asymptotics—through superposition of universal weights.

We briefly outline in this section the main results and possible applications. In the sequel, we shall always work with *C*^{∞}-bounded vector fields *V*_{i} and *C*^{∞}-bounded functions *f*. Given a stochastic differential equation in of the typeon a stochastic basis with *d*-dimensional Brownian motion, then cubature formulae provide a method to approximate . More precisely, choosing a degree *m*≥1 of accuracy, there is a number *r*≥1 and there are *H*^{1}-trajectories and weights *λ*_{j}>0, *j*=1, …, *r*, such thatwhere denotes the point in , which is obtained by solving the ordinary, non-autonomous differential equation:

The constant in the order estimate depends in general on derivatives of *f* up to order *m*+1. Note also the important fact that the number *r* of trajectories can be bounded by a number depending on *d* and *m*, but not on *N* (see Lyons & Victoir 2004 for the precise estimate). Obviously, this procedure can be iterated for small time steps by the Markov property, which yields then a high-order numerical approximation scheme for the calculation of expected values .

In the hypo-elliptic case, we can calculate the derivatives of with respect to the initial value by weight formulae, i.e. for any direction there is a random variable *π* such that(1.1)for bounded measurable *f*. Explicit formulae for *π* can be given by Malliavin Calculus (see for instance Malliavin & Thalmaier 2004; Gobet & Munos 2005) and are in principle well known for a long time (see for instance Bismut 1984).

If we fix an order of approximation *m*≥1 and a (homogenous) direction *v*, such that (for the notations see §2)holds with some 1≤*k*≤*m*, then we can construct a universal weight (calculated in a *universal way* from the coefficients *v*_{I} through a universally given random linear operator, see remark 3.7), such thatholds true, where the constant in the order estimate depends only on the first derivative of *f*. Theorem 4.5 asserts that we are able to find weights *μ*_{j}≠0 and *H*^{1}-trajectories such that(1.2)holds true. Also, the constant in the order estimate depends on derivatives of *f* up to order *m*+1. The number *r* can be estimated similarly as in Lyons & Victoir (2004).

We have the property(1.3)(1.4)

A calculation of the inner expectation by the original cubature formula (or any other numerical scheme), and of the outer expectation by formula (1.2) lead to the desired iterative procedure: in the uniformly hypo-elliptic case we, consequently, obtain the following recipe.

Choose a small

*s*_{0}>0 and relatively large*t*−*s*_{0}. Subdivide with for*i*=1, …,*l*.Calculate by (1.2) with the function for a certain degree of accuracy

*m*. The order estimate yields (by careful choice of homogenous) directions for some 1≤*k*≤*m*. The constant in the order estimate depends on the (*m*+1)st derivative of , which—by methods of Malliavin Calculus—can be reduced to dependence on the first derivative of*f*, i.e. there is an absolute bound given bysince*t*−*s*_{0}is bounded away from 0.Calculate by cubature formulae with certain degree of accuracy

*m*′ (see Lyons & Victoir 2004). The order estimate, which again—by methods from Malliavin Calculus (see Kusuoka 2001)—can be reduced to dependence on the first derivative of*f*, i.e. there is an absolute bound given byFinally, we obtain a high-order approximation scheme for through solving ordinary differential equations with estimate of the absolute error through(1.5)

The procedure only involves ordinary differential equations and knowledge on the cubature trajectories. By choosing *m* big enough, one can bound the difference *m*−*k* from below.

A direct application of formula (1.1) is often very slow numerically in the purely hypo-elliptic case (see for instance Gobet & Munos 2005). In the elliptic case, there are very efficient ways to simulate *π* (see Fournie *et al*. 1999). Hence, in particular the purely hypo-elliptic case constitutes an interesting field of applications, since one does not need to worry about invertibility of the covariance matrix.

For all cubature formulae, there are algebraic methods to determine the trajectories *ω _{j}* and the weights

*λ*(or

_{j}*μ*) respectively. These methods are based on calculus in nilpotent Lie groups, which constitute the local model for hypo-elliptic diffusions in the sense of Gromov (see for instance the very well written monograph by Montgomery (2002) for a general outline, and Lyons (1998) and Baudoin (2004) for applications in the theory of—stochastic—differential equations).

_{j}## 2. Cubature on Wiener space

Cubature formulae on Wiener space (see the seminal article by Lyons & Victoir 2004) rely on the analysis of hypo-elliptic diffusions on free, nilpotent groups on the one hand (see the recently published book by Baudoin 2004), on the other hand on stochastic Taylor expansion (see for instance Ben-Arous 1989). Furthermore, the proof of a cubature formula on Wiener space appears as an application of Chakalov's Theorem on finite dimensional cubature formulae (see Bayer & Teichmann in press). We state these three main ingredients and show—in order to explain the mathematical background of cubature formulae—a sketch of the proof (following Lyons & Victoir 2004).

We fix a probability space together with a *d*-dimensional Brownian motion . We need the following notations for convenience.

We abbreviate by the set of all finite sequences ,

*k*≥0 and we define a degree function on bywhich simply means that the zeros appearing in (*i*_{1}, …,*i*_{k}) are counted twice. Additionally, we define deg()=0. We define a semi-group structure on via:We denote by the free, nilpotent algebra with

*d*+1 generators*e*_{0}, …,*e*_{d}, i.e. the set of all non-commutative polynomials in those variables, such that the following nilpotency relations hold: if deg(*I*)>*m*, then for all*I*∈. Hence, is a finite dimensional, non-commutative, real algebra with unit element 1 and we are given a grading via the degree function, i.e. a monomial is said to have degree*n*if deg(*i*_{1}, …,*i*_{k})=*n*. Denote by*W*_{n}the linear span of all monomials of degree*n*, then we obtainfurthermore, (where we define*W*_{p}=0 for*p*>*m*due to the given relations), , so is a graded algebra. We denote the canonical projections of*x*on the subspaces*W*_{n}by*x*_{n}for*n*≥0.On the finite dimensional algebra , we define the exponential serieswhere the series converges everywhere due to the nilpotency relations. We define the logarithm on elements

*x*with*x*_{0}≠0. We identify*x*_{0}with a real number, so the seriesis well-defined, since (i.e. the series is finite).We define the Lie algebra generated by

*e*_{0}, …,*e*_{d}with respect to the Lie bracket . The Lie algebra inherits the grading from the algebra via , for*n*≥1. Henceis a graded Lie algebra. In fact, the Lie algebra is free, nilpotent of step*m*, with*d*generators of degree 1 and one generator of degree 2.We denote the exponential image of by and call it the free, nilpotent Lie group. is indeed a Lie group as a closed subgroup of the Lie group . The tangent space at is spanned by the left (or right) translations

*xw*for .We need the canonical dilatations on Lie algebra and Lie group. We define for an algebra homomorphism

*Δ*_{t}viafor*x*_{n}∈*W*_{n},*n*≥1 and*t*>0. The homomorphism is well defined and restricts to a Lie algebra homomorphism*Δ*_{t}on and a Lie group homomorphism*Δ*_{t}for*t*>0.We apply the notation . We abbreviate the left invariant vector field associated to

*e*_{i}by*D*_{i}, i.e.*D*_{i}(*x*)=*xe*_{i}for . We define a stochastic process on via(2.1)(2.2)(2.3)and see immediately that for 0≤*t*≤*T*almost surely.Note here and in the sequel that vector fields on a vector space are used with double meaning: either as tangent directions on a smooth geometric object, or as first-order differential operators on smooth functions for . For smooth maps , we apply the notion of the

*tangent map or Jacobian*(2.4)for . A vector field is called*C*^{∞}-bounded if all derivatives of order greater than 0 are bounded.

In order to see the relation between free, nilpotent Lie groups on the one hand and asymptotic analysis as *t*↓0 on the other hand, we formulate stochastic Taylor expansion (see Ben-Arous 1989; Baudoin 2004), which simply results from iterating the defining equations for a stochastic differential equation.

(stochastic Taylor expansion). *Given C ^{∞}-bounded vector fields*

*on*

*, then the diffusion process*

*for*

*admits the following series expansion: for n≥*1

*and*

*the expansion*

*holds true for 0≤t≤T. Additionally*,

*where the constant in the order estimate depends on the (m+*1

*)st derivative of f*.

The above procedure works for ‘any’ differential equation (see the literature on rough paths, for instance the seminal work of Lyons 1998), in particular, if we consider the non-autonomous, ordinary differential equationfor a *H*^{1}-trajectory , we have that for *n*≥1 and the expansionholds true for 0≤*t*≤*T*. Notice here that in order to obtain an asymptotic expansion (for a given fixed trajectory *ω*) of certain order in time *t*, one has to change the degree function.

*If we apply this series expansion to the process* *in the vector space* *, we obtain for any linear function* *for* *and* 0*≤t≤T, since**for* *, hence* . *Consequently, we obtain in* *that**for* 0*≤t≤T, almost surely, by duality. This formula provides a nice stochastic representation of the solution of equation (**2.1**), where we apply that* *is embedded in the algebra* .

Stochastic differential equations provide solutions for the associated heat equation via expectations, so we obtain for for all and *t*≥0. The heat equation itself admits an iterative construction similar to stochastic Taylor expansion, namely any solution *u*(*t*, *x*) with initial value *u*(0, *x*)=*f*(*x*), where *f* is *C*^{∞}-bounded, can be written as

In the case of *f*=*λ* for a linear function , this leads for the process in , to the nice formulaeand hence(2.5)for 0≤*t*≤*T*, which is one basic formula for the construction of cubature formulae (see Lyons & Victoir 2004; Baudoin 2004).

In order to obtain a cubature formula, we need a version of Chakalov's theorem (see Bayer & Teichmann in press for a short proof).

*Fix a number m≥1. Let μ be a probability measure on* *such that moments up to order m exist, then there are points x _{1}, …, x_{r}∈*supp(

*μ*)

*and weights λ*0

_{j}>*for j=*1

*, …, r, such that*

*for all polynomials on*

*up to order m. Furthermore,*

*, where*

*is the vector space of polynomials on*

*with degree less or equal m*.

On the basis of these preparations, we can define cubature formulae on Wiener space.

*Fix m≥*1 *and* 0*≤t≤T, a cubature formula on Wiener space is given by a finite number of points x _{1}, …, x_{r}*

*and finitely many weights λ*

_{1}

*, …, λ*0

_{r}>*, such that*

*or equivalently due to formula*

*(2.5)*

*Furthermore,* .

Applying Chakalov's theorem to the law of the process in , 0<*t*≤*T*, which is supported in , yields that we find points and weights *λ*_{j}>0, such thathence, the existence of cubature formulae for any *m*≥1.

Finally, we apply Chow's theorem of sub-Riemannian geometry (see Montgomery 2002; Baudoin 2004), which tells that every point can be reached by a *H*^{1}-horizontal curve, i.e. for every we find a *H*^{1}-curve , such that the solution of the non-autonomous ordinary differential equationreaches *x* at time *t*>0, i.e. .

Taking for each of the *x*_{j} appropriate curves *ω*_{j} with the previous property, we obtain—by Taylor expansion of the solution of the non-autonomous equation as in remark 2.2—the familiar version of cubature formulae on Wiener space, namely

This leads to a redefinition of cubature formulae, where we replace the points *x*_{i} by endpoints of evolutions of ordinary differential equations for *i*=1, …, *r*. This is the announced pathwise interpretation of cubature formulae, which respects the geometry of the problem.

As proposed in Lyons & Victoir (2004), we could also find for any point *x* in the support of a *H*^{1}-trajectory , such that for *t*∈[0,*T*] satisfies . Hence, we could equally work with this simpler class of curves, which we do not point out in this article, but which remains true in any statement.

*Given C ^{∞}-bounded vector fields V*

_{0}

*, …, V*

_{d}on*, then the diffusion process*

*for*

*admits the following cubature formula of degree m≥*1

*: for*0

*<t≤T we find H*

^{1}-curves*and weights λ*

_{1}

*, …, λ*0

_{r}>*, such that*

*for*.

*The constant for the order estimate depends in general on derivatives of f up to order m+1. The curve*

*is understood as the solution of*

Since cubature formulae define moment-generating families, we can apply the impressive results of Kusuoka (2001) on iterative numerical schemes for high-order Taylor methods yielding the estimate (1.5). However, Kusuoka's method does not respect the geometry of the problem, since only the moments of iterated Stratonovich integrals up to a certain degree are used.

## 3. Calculation of the Greeks

For the calculation of Greeks, we proceed by methods from Malliavin Calculus (see Nualart 1995; Malliavin 1997 for all details). We shall consider stochastic differential equations of the typeon a stochastic basis (*Ω*, , *P*), where we are given a *d*-dimensional Brownian motion in its natural filtration, up to a finite time horizon *T*>0. For the vector fields *V*_{i}, we shall assume the following regularity assertion.

The vector fieldsare

*C*^{∞}-bounded.

If the distribution(3.1)has constant rank *N* at a fixed point , we say that *Hörmander's condition* holds at . Here, and in the sequel, we apply the notion of Lie brackets of two vector fields , for .

We shall apply usual notions of Malliavin calculus (see Nualart 1995; Malliavin 1997). The first variation denotes the derivative of with respect to *y*; hencewhere we apply the Jacobian d*V*_{i} of the vector field *V*_{i} as defined in (2.4). This is an almost surely invertible process and we obtain the representationfor 0≤*s*≤*T*, 0≤*t*≤*T*, 0≤*k*≤*d* of the Malliavin derivative. We have the fundamental partial integration formulawhere is a Skorohod integrable process. *δ*(*a*) is real valued random variable. Notice that if *a* is predictable and square-integrable, thenhence, the Skorohod integral coincides with the Ito integral.

*Fix t>*0*,* *and* . *If condition* *(3.1)* *holds, there is a weight π*∈^{∞}*, such that for all bounded, measurable functions* *the equation**holds true depending on t, v, y and the whole stochastic process* .

For the proof see also Gobet & Munos (2005). Assume *f* is , thenwhere d*f* denotes the differential of *f*. If we are able to solve the equationwith a Skorohod integrable strategy such that *δ*(*a*)∈^{∞}, then we have proved the assertionhence the result.

The above equation can be solved directly by the reduced covariance matrix *C*^{t}: take any scalar product 〈.,.〉 on , then we define the symmetric random operatorwhich is under Hörmander's condition (3.1) almost surely invertible with (see Nualart 1995). Hencefor 0≤*s*≤*t* solves the above problem with an anticipative strategy, where the resulting Skorohod integral lies in ^{∞} (see Nualart 1995). Furthermore, the outcome *δ*(*a*) depends smoothly on *y*. ▪

For the pull-back of the stochastic flow, we can provide a similar stochastic Taylor expansion on the space of vector fields as in §2, or—in the case of a Lie group—on the Lie algebra. We apply the following notation:for *I*=(*i*_{1}, …, *i*_{k})∈.

*Given C ^{∞}-bounded vector fields V*

_{0}

*, …, V*

_{d}on*, then the diffusion process*

*for*

*, admits the following series expansion for the pull-back*

*on smooth vector fields*

*for*0

*≤t≤T. For the remainder term we obtain*

*for*0

*≤t≤T.*

This Taylor expansion leads on the Lie group to an action on the Lie algebra , namelyby inserting the vector fields *D*_{i} for *V*_{i} and evaluating at *x*=1. Notice the decrease in order from *m* to *m*−1, since in Lie brackets only up to order *m*−1 can be non-zero. This stochastic process on the Lie algebra will be applied in the sequel in order to construct approximative, universal weights for the calculation of the Greeks. First, we need a Lemma on scaling invariance.

*The following identities in law for the stochastic process* *hold:**for t>0 and w∈W _{n}*.

By scaling invariance, we know that for *t*>0 and *i*=1,…,*d*. This extends to all iterated stochastic integrals. Notice the importance of the degree function, which associates degree 2 to *e*_{0}, and hence the correct scaling property. Notice furthermore that for the element is an element of the Lie algebra , but not necessarily of *W*_{n}! ▪

Notice that the stochastic process is not hypo-elliptic, since the direction *e*_{0} always points into directions, where the density does not admit a logarithmic derivative. Therefore, we only calculate with directions , which means directions with vanishing *e*_{0} component. In all those directions, we are able to conclude a result on differentiability.

In the sequel, we shall construct Skorohod integrable processes by defining them only on [0,*t*] for 0<*t*≤*T*. It is understood that these processes are defined on [0,*T*], but vanish for *s*>*t*, hence for 0≤*s*≤*T* and *i*=1, …, *d*.

*Fix* *(no e*_{0} *component). For t>*0 *there are Skorohod-integrable processes* *(the t-dependence is not visible in our notation) with* *for* 0*≤s≤T, such that**and*

*Furthermore, we obtain that for any bounded measurable function* *the equation**holds true, where* *denotes the Skorohod integral of the strategies a. Additionally,* .

We can define the processes by Malliavin Calculus: we choose a scalar product 〈,〉 on , which respects the grading, i.e. *W*_{k} is orthogonal to *W*_{l} for *k*≠*l*, and define the reduced covariance operator *C*^{t} as symmetric operator by

This operator is almost surely invertible on with *p*-integrable inverse (see Nualart 1995). Obviously, the image of *C*^{t} lies in , since there is no *e*_{0}-component in , the kernel of *C*^{t} can be calculated by the classical method from Nualart (1995) and vanishes on ; hence is invertible with *p*-integrable inverse, since the Norris Lemma applies.

Furthermore, we observe the following scaling property: notice that is a self-adjoint operator on for *t*>0. We can compare with *C*^{t} in law via lemma 3.3

So we have thatfor *t*>0. We define the (non-adapted) strategies via(3.2)for 0≤*s*≤*t*, *i*=1,…,*d*, and obtainfor . Hence, the strategies satisfy the desired equation by duality on . Concerning the order estimate for the strategies we observe thatso consequently

In a similar way, the second-order estimate is calculated and we obtain by well-known formulae on the covariance between Skorohod integrals (see Nualart 1995, p. 39) the result on . One can shortcut this calculation by applying that the Malliavin derivative reduces the asymptotic behaviour of iterated Stratonovich integrals by a factor . ▪

The strategies are *universal* in the sense that they depend only on the Brownian motion, the choice of a scalar product on , and—in a linear way—on the ‘input’ direction *w*. We denote the Skorohod integrals of those strategies by and call them *universal weights*. The time dependence is achieved by calculating the strategies with respect to the direction instead of *w*. We omit the dependence of on *w* in our notation.

*We consider the simplest non-trivial example: m=*2 *and d=*2*. Hence, the nilpotent algebra is generated by**and the Lie algebra by e*_{0}*, e*_{1}*, e*_{2}*, *[*e*_{1}*, e*_{2}]*. The dimensions of* *and* *are six and four, respectively. We can solve explicitly**by*

*The pull-back yields a shorter expression, namely**for* 0*≤t≤T, which can also be calculated directly. Consequently, the above equation reduces to find a Skorohod integrable strategy such that**for* . *This can be done by introducing a scalar product on* *such that e*_{1}*, e*_{2}*,* [*e*_{1}*, e _{2}*]

*, e*

_{0}

*are an orthonormal basis. We then obtain the symmetric matrix*

*with respect to the given basis, and we obtain the strategies via equation (3.2). So C*

^{t}is almost surely invertible on 〈e_{1}

*, e*

_{2}

*,*[

*e*

_{1}

*, e*

_{2}]

*〉—as claimed in*

*proposition 3.6*

*, indeed the determinant is given through*

*which is positive if and only if one of the two Brownian motions is not vanishing identically on*[0

*,t*]

*. The order assertion in t is also nicely visible*.

We consider the first derivative of the function for *t*>0. By Hörmander's theorem, we know that this function is smooth for all bounded measurable functions (see for instance Nualart 1995 or Malliavin 1997). We generalize this assertion on the one hand, since we leave out Hörmander's condition. On the other hand, we only prove an approximative result with certain time asymptotics.

*Fix t>0,* *and* . *Furthermore, we fix an order m≥*1 *of approximation. We assume that the vector v is a linear combination of Lie brackets up to order m*−1 *(except the direction V*_{0}*), i.e.**and define*

*Then, there is universal weight* *associated to* *, such that for all C ^{∞}-bounded functions*

*the equation*

*holds true, where the constant in the order estimate depends only on the first derivative of f*.

Assume *f* is , thenwhere d*f* denotes the differential of the function *f*. Instead of solving the equationwe take the universal weight of proposition 3.6 and solve the equationby the above universal construction. We choose real numbers *w*_{I} for such thatand define

Then, we take the universal weight associated to *w* and solve consequently the previous equation. This leads tohence the result, since due to proposition 3.6 we obtain—by the Cauchy–Schwartz inequality and integration with respect to *t*—an order estimate of the type (*t*^{(m+1)/2}): indeed, the remainder satisfies ▪

## 4. Cubature formulae for Greeks

First, we have to get expressions for derivatives of heat equation evolutions on free, nilpotent Lie groups, which is an easy task for linear functions by the considerations of §2.

*Fix a linear function λ on* *, then**for* . *Consequently*(4.1)*for* .

We know that the function solves the associated heat equation. The solution of the heat equation on with initial value *λ* is given byhence the derivative in direction *w*_{x} can be calculated and yieldswherefrom the result follows by duality and theorem 3.9. ▪

Next, we apply Chakalov's theorem twice in order to obtain the appropriate cubature result.

*Fix a free, nilpotent Lie group* *and* *, then there are points* *and weights μ*_{1}*, …, μ _{r}≠*0

*such that*

*Furthermore,* .

We write . We define two probability measures on Wiener space, absolutely continuous with respect to Wiener measure *P*:if the respective positive and negative parts have non-vanishing expectation. Then

There is a null set *N* on Wiener space, such that on *N*^{c} the process ; hence also almost surely with respect to *Q*_{±}. Consequently by Chakalov's theorem, we find points and weights such thatwhich yields the desired formula by proposition 4.1. ▪

We can additionally require that , and |*μ*_{i}|≤*E*(|*π*|) for *i*=1, …, *r*.

Hence, we can formulate a cubature result for the calculation of Greeks.

*Fix m≥*1 *and* 0*≤t≤T, a cubature formula for the first derivative in direction* *is given by a finite number of points* *and finitely many weights μ*_{1}*, …, μ _{r}≠*0

*, such that*

*or equivalently due to formula*

*(4.1),*

*Again we use trajectories ω _{i} to represent the points x_{i} as endpoints of evolutions of ordinary differential equations Z_{t}*(

*ω*)

_{i}*=x*1

_{i}for i=*, …, r. Furthermore,*.

*Given C ^{∞}-bounded vector fields V*

_{0}

*, …, V*

_{d}on*, then the diffusion process*

*for*

*admits the following cubature formulae for derivatives in direction*.

*Fix m≥*1

*and assume that*(4.2)

*then we obtain*

*taking a cubature formula, which was derived in*

*and C*1

^{∞}-bounded f. The constant depends in general on derivatives of f up to order m+*. We can determine the weights μ*

_{j}and the trajectories ω_{j}bySince we shall in practise apply this procedure to smooth functions—as discussed in the recipe after formulae (1.3) and (1.4)—this already yields the interesting result for applications.

By the previous constructions we obtain

We then insert instead of *e*_{i}, the vector fields *V*_{i} and apply those vector fields to the function *f* at . Then, we obtain on the left-hand side—due to stochastic Taylor expansion—and on the right-hand side due to Taylor expansion of the non-autonomous equation

The left-hand side yields then the order estimate by partial integration and proposition 3.6, the right-hand side by the fact that iterated integrals with respect to *ω*_{j} behave like *t*^{(m+1)/2}. ▪

We additionally have an assertion on the construction of cubature paths due to the scaling properties. Assume that we found trajectories *ω*_{i} and weights *μ*_{i} for *m*≥1, *d*≥1 and *t*=1 fixed, then we can construct solutions for all *t*>0 with the same *m*, *d*: the equationholds; hence the trajectoriesfor *i*=1, …, *d*, *j*=1, …, *r* and 0≤*s*≤1 satisfy

## 5. Applications

### (a) Example for *m*=1 and *d*≥1

Fournie *et al*. (1999) provide the following expression for the Malliavin weight *π*: given *C*^{∞}-bounded vector fields *V*_{1}, …, *V*_{d} on such that a uniform ellipticity condition holds, i.e. there is *δ*>0 such thatfor all , with respect to a standard scalar product on , the formula(5.1)where we understand d*B*_{s} as column vector, where the random matrix acts on. *σ* is defined via

Notice that this formula provides adapted strategies, therefore, the Skorohod integral can be replaced by the Ito integral. We shall compare this formula to the formula obtained in our setting. We choose *m*=1, we can calculate explicitly, namelyfor *i*=1, …, *d* and *w* has *e*_{i}-component *w*_{i} for *i*=1, …, *d*. Hence, we approximatewhere *w*_{i} stem from the solution of the equationconsequently *w*=*σ*^{−1}(*y*).*v*. Hencewhich is obviously the expansion of the precise formula (5.1), sincefor *f* a -function.

### (b) Example for *m*=2 and *d*=2

In order to demonstrate the method we calculate one example of the above methodology. First, we fix *m*=*d*=2; hence we can include the calculations of example 3.8. We first fix a direction *w*=*e*_{1}. Then, we find one solution ofwhich can be given through *r*=2, *μ*_{1}=1/2, *μ*_{2}=−1/2 and , . Consequently, the trajectories are given bywhich leads to

## Acknowledgements

The author acknowledges the support from the RTN network HPRN-CT-2002-00281 and from the FWF grant Z-36.

## Footnotes

- Received October 27, 2004.
- Accepted September 13, 2005.

- © 2005 The Royal Society