Stochastic differential equations driven by processes generated by divergence form operators I : a Wong-Zakai theorem

September 2006, Vol. 10, p. 356–379 www.edpsciences.org/ps

DOI: 10.1051/ps:2006015

STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY PROCESSES GENERATED BY DIVERGENCE FORM OPERATORS I: A WONG-ZAKAI

THEOREM

^∗

Antoine Lejay

¹

Abstract. We show in this article how the theory of “rough paths” allows us to construct solutions of differential equations (SDEs) driven by processes generated by divergence-form operators. For that, we use approximations of the trajectories of the stochastic process by piecewise smooth paths. A result of type Wong-Zakai follows immediately.

Mathematics Subject Classification. 60H10, 60J60.

Received April 05, 2005. Revised May 12, 2006.

1. Introduction

The goal of this article is to define a stochastic differential equation of the type

dY_t=f(Y_t) dX_t, (1)

for a stochastic processX generated by the divergence form operator L=

N i=1,j

1 2

∂

∂x_i

a_i,j ∂

∂x_j

+ N i=1

b_i ∂

∂x_i,

wherea(x) = (a_i,j(x))_i,j=1,...,N is symmetric, measurable, uniformly elliptic and bounded, andb is measurable and bounded. The coefficientsaandb are not assumed to be regular.

Ifais smooth, thenLmay be transformed into a non-divergence form operator N

i,j=1

1

2a_i,j ∂²

∂x_i∂x_j + N i,j=1

1 2

∂a_i,j

∂x_i

∂

∂x_j + N i=1

b_i ∂

∂x_i,

Keywords and phrases.Rough paths, stochastic differential equations, stochastic process generated by divergence-form operators, Dirichlet process, approximation of trajectories.

∗This work has been partially supported by European Union’s TMR Stochastic Analysis Network (grant reference 960075).

1 Projet OMEGA (INRIA Lorraine), IECN, Campus scientifique, BP 239, 54506 Vandœuvre-l`es-Nancy Cedex, France;

Antoine.Lejay@iecn.u-nancy.fr

c EDP Sciences, SMAI 2006

Article published by EDP Sciences and available at http://www.edpsciences.org/psor http://dx.doi.org/10.1051/ps:2006015

and in this case, X is a semi-martingale. So, Y may be defined as the solution of some SDE in the Itˆo or Stratonovich sense. But if ais discontinuous, thenX is no longer a semi-martingale, but a Dirichlet process, that is the sum of a local martingale and a term of zero quadratic variation. The Itˆo theory of integration is no longer valid in this case.

However, using time-reversal techniques, stochastic integrals of the form_t

0f(X_s)◦dϕ(X_s) were defined by Rozkosz in [30] and Lyons and Stoica in [23]. There are also other approaches to define some integrals with respect to a Dirichlet process: see [8, 11] for example. But passing from integration of one forms to integration of (stochastic) differential equations has, at the best of our knowledge, not been treated.

The theory of rough paths developed by Lyons [24] (see also [17, 22]) gives a pathwise view of stochastic differential equations of type (1) by viewing them as ordinary differential equations controlled by irregular paths. The price to pay, when one deals with a processX of finiteα-variation, is to use the topology induced by the α-variation norm, and to know not only the process X, but also its iterated integrals up to orderα. Thus, we need to define a pathX, called arough path, that lives in a space of dimensionN+N²+. . .+N^α. This theory has been applied for a wide class of processes: infinite dimensional Gaussian processes [21], L´evy processes [34], fractional Brownian motion [5], Brownian Motion on fractals [13], ...

In [2], Bass, Hambly and Lyons have adapted the theory of rough paths to reversible Markov processes. This case includes processes generated by a divergence form operator when b = 0, but their proof requires the use of the invariant measure

m( dx)Px. Dealing with the measurePx for an arbitrary starting pointx∈R^N was left as an open problem: See Point 3 in Section 6.3 in [2]. As processes generated by divergence form operators are of finiteα-variation as soon asα >2, we need to consider their second-order iterated integrals. The results of [23, 30] may be used to construct second-order iterated integrals of processes generated by divergence form operator under Px for any x∈ R^d, yet their α/2-variations have never been studied. Moreover, the method of [2] cannot be transposed in our case, since it strongly relies on the fact that one works under the invariant measure.

In this article, we study the regularity of the second-order iterated integrals of a process generated by a divergence form operator under Px for any x∈ R^d. Although a time-reversal technique plays a fundamental role here as in [2], we use it for the underlying process and not for defining the area. Thus, our proof is not an extension of the one in [2]. The regularity of the second-order iterated integrals is mainly all we need to apply the rough paths theory to this class of processes. It means that one can construct a rough pathXlying aboveX with α ∈ (2,3), and this construction is performed by approximating X with piecewise linear functions X^δ. Thus, the solutionsY^δ of the ordinary differential equations

dY_t^δ

dt =f(Y_t^δ)dX_t^δ

dt , Y₀^δ =y

converge in probability underPxfor anyxto the solutionY of (1) defined by the rough paths theory, which is denoted by

dY_t=f(Y_t)dX_t, Y₀=y.

A similar result occurs when one considers stochastic integrals of type _t

0g(X_s)dX_s. This result is then an extension of the Wong-Zakai theorem [35].

Moreover, in our proof, we may use any deterministic family of partitions whose meshes decrease to 0 in order to constructX^δ, while many equivalent results regarding rough paths have been proved using dyadic partitions.

In a companion article [19], we study the convergence as rough paths, or multiplicative functionals, of a family (X^ε)ε>0 of processes generated by the divergence-form operators ¹_2∂x^∂

i

a^ε_i,j∂x^∂

j

+b^{ε ∂}_∂x

i, when thea^ε’s are uniformly elliptic with the same constant, and the (a^ε, b^ε)’s are uniformly bounded. In particular, it will be shown that the condition UTD introduced in [6] (see also [28]) is a sufficient condition to assert that the convergence as rough paths holds and that the limit corresponds to the rough path “canonically” constructed above the limiting processX.

The coefficientsaandbmay be approximated by some smooth coefficientsa^εandb^εand the corresponding processesX^εare semi-martingales. The integrals and the stochastic differential equations, with a right choice of the rough pathsX^ε, correspond then to the usual Stratonovich or Itˆo integrals [4, 32]. We will then prove that the integrals we have constructed for divergence form operators may then be approximated by usual stochastic integrals of Stratonovich or Itˆo type, and also correspond to the ones constructed using the time-reversal techniques in [23, 30].

With [19], this article provides a natural way to defined stochastic differential equations driven by processes generated by divergence form operators. In addition, the SDEs so defined possess also some natural properties:

they may be approximated both by solutions of ODEs driven by piecewise smooth approximations of the processes or by semi-martingales when the coefficients ofLare approximated by smooth coefficients. Moreover, using a result of Wong-Zakai type, one gets immediately a change of variable formula for these kind of stochastic integrals or solutions of SDEs. And regarding stochastic integrals, we recover the objects constructed in [23,30] as a natural extension of stochastic integrals. Finally, our results may be extended to time-inhomogeneous processes or to the process (ϕ1(X), . . . , ϕ_m(X)) whereϕ_i∈W¹_loc^,∞(R^N) with∇ϕ_i ∈L^∞(R^N;R^m) fori= 1, . . . , m.

Outline.In Section 2, we present the results we use on stochastic processes generated by divergence-form operators. Our main theorem is stated in Section 3. Section 3.3 contains the core of the proof, which is that piecewise linear approximations of the trajectories of X and their second-order iterated integrals converge in α-variation. In Section 4, we show how to construct different rough paths lying above the same trajectories, and their effects.

Convention.Throughout this article, we use the Einstein convention, which means that an index variable that appears twice in an expression is implicitly summed over all its possible values.

2. On processes generated by divergence form operators 2.1. Construction and main properties

We fix two constantsλ and Λ with 0< λ <Λ. Leta be a measurable function fromR^N into the space of symmetric matrices which is uniformly elliptic with respect toλand bounded by Λ,i.e.,

λ|ξ|²≤a(x)ξ·ξ≤Λ|ξ|²

for any ξ in R^N and almost every x∈R^N, where |ξ| is the Euclidean norm of ξ. Let also b be a measurable function fromR^N toR^N which is bounded by Λ. Under these conditions, the second-order differential operator

L=1 2

∂

∂x_i

a_i,j ∂

∂x_j

+b_i ∂

∂x_i with domain Dom(L) =

f ∈H¹(R^N) Lf ∈L²(R^N)

is the infinitesimal generator of a conservative and continuous stochastic process X with a density which is absolutely continuous with respect to the Lebesgue measure [16, 33].

Ifais of classC¹, thenLmay be transformed into the non-divergence form operator¹₂a_i,j∂x^∂2

i∂xj+¹₂^∂a_∂x^i,j

i

∂

∂xj+ b_i∂x^∂

i. If follows thatX is a semi-martingale and solution to some SDE. This is no longer true if the derivative ofais not defined. The theory of Dirichlet forms [10, 26] is helpful to study the properties of such a process.

The most useful property on the transition density pofX is probably the Aronson estimates [1, 33]: there exists a constant M depending only on λ, Λ and the dimension N such that for any t ∈ (0,1] and any (x, y)∈R^N×R^N,

1 Mt^N/2exp

−M|x−y|² t

≤p(t, x, y)≤ M t^N/2exp

−|x−y|² Mt

· (2)

In other words, one may compare the transition density with the Gaussian kernel. In particular, there exists a constantC that depends only onλ, Λ andN such that

1

0 R^N|p(s, x, y)|^αdy _β/α

ds≤C,

whereαandβare the conjugate exponents ofα∈(1,∞] andβ ∈(1,∞] (i.e.,α =α/(α−1) andβ=β/(β−1)) satisfying the relationN/2α+ 1/β <1.

In fact,palso satisfies [1], Theorem 5, p. 656

1

0 R^N

|p(s, x, y)|^α+|∇p(s, x, y)|^α dy

_β/α

ds≤C, (3)

for some constant C depending only on λ, Λ andN, where α and β are the conjugates exponents of real numbersα∈(2,∞] andβ ∈(2,∞] satisfyingN/2α+ 1/β <1/2.

Let us denote by (X_t,Px,Ft;t≥0;x∈R^N) the Hunt process (and then a strong Markov process) generated by (L,Dom(L))viathe probability transition densityp, on a probability space (Ω,F,P). The filtration (Ft)_t≥0

is the minimal admissible filtration satisfying the usual hypotheses.

Many of the constants that appear in the computations depend only onλ, Λ,N, andα. This motivates the following notation.

Convention 1. An expression of typeabmeans in fact thata≤C×b, whereCis a constant that depends only on λ, Λ, the dimension N, the choice of αand the time interval, which is [0,1] here. For example, the convexity inequality (a+b)^α≤2^α−1(a^α+b^α) is then written (a+b)^αa^α+b^α.

An upper bound of exponential type on the probability thatX stays in a given ball between times 0 and t follows from both the upper and lower bound in (2).

Lemma 1(See [33], Lem. II.1.2). There exists a positive constant C depending only onλ,Λ andN such that for any t∈[0,1]and anyR≥0,

Px

0sup≤s≤t|X_s−x| ≥R

exp

−CR² t

·

This lemma is useful when one has to proceed by localization. If the first-order differential termbis equal to 0, then the conclusion of Lemma 1 is true for anyt∈(0,∞).

As said above, X is in general not a semi-martingale, but belongs to the more general class of Dirichlet processes (the reader has to be warned that there exists several definitions of Dirichlet processes with slight differences).

We denote by (Π^δ)_δ>0 a family of deterministic partitions of [0,1], whose meshes decrease to 0 with δ.

Definition 1 (Dirichlet process along Π^δ [9]). A stochastic process (X,P,(Ft)_t≥0) is called aDirichlet process (along Π^δ) if it admits the decompositionX_t=M_t+A_t, t∈[0,1], where M is a (P,(Ft)_t≥0)-local martingale andAis a (Ft)_t≥0-adapted process such that

Q(A) =

k^δ−1 i=0

|A_tδ

i+1−A_tδ i|²

converges to 0 in probability asδ→0, where Π^δ is the partition 0≤t^δ₀≤t^δ₁≤ · · · ≤t^δ_kδ ≤1.

ReversingX in time under Px for a given starting point gives us more information about the term of zero quadratic variation ofX. For that, let us denote by (Ft)_t∈[0,1] the time-reversed filtration defined by

Ft=σ(X_s;t≤s≤1).

Proposition 1 ([30], Th. 2.2, p. 97; [23]). Letϕbe a function in W^1,p(R^N)with p >max(2, N). Then, Px-a.s.

for every pointx,(ϕ(X_t))0≤t≤1 is a Dirichlet process with decomposition:

ϕ(X_t) =ϕ(X0) +M_t^ϕ+A^ϕ_t withA^ϕ_t =−1

2M_t^ϕ+1

2(M^ϕ_1−t−M^ϕ₁) +V_t^ϕ (4) whereM^ϕ is the martingale part ofϕ(X),M^ϕ is the martingale part of the reversed processϕ(X1−t)(this is a (Ft)0≤t≤1-martingale) and

V_t^ϕ=

t 0

b_i(X_s)∂ϕ

∂x_i(X_s)−1

2p⁻¹(s, x, X_s)a_i,j(X_s) ∂p

∂x_j(s, x, X_s))∂ϕ

∂x_i(X_s)

ds

is a term of integrable variation. The martingales M^ϕ andM^ϕ are square-integrable with M^ϕt=

t 0

a_i,j∂ϕ

∂x_i

∂ϕ

∂x_j(X_s) dsandM^ϕt=

t 0

a_i,j∂ϕ

∂x_i

∂ϕ

∂x_j(X1−s) ds.

Remark 1. In [10], a process generated by a Dirichlet form may be decomposed under

R^N dxPxas a sum of a local martingale and a process locally of zero-energy. Definition 1 of a Dirichlet process is a more stringent.

Remark 2. If N = 1, then Proposition 1 is also true forp= 2: See [29]. This proposition is also true when the coefficients aandb are time-inhomogeneous, as proved in [18].

Remark 3. A decomposition of type (4), which is called a Lyons-Zheng decomposition, also holds under the infinite measure

R^N dxPx, in which caseV_t^ϕ=_t

0b(X_s)∇ϕ(X_s) ds.

Remark 4. Ifϕ∈W¹_loc^,∞(R^N), then (4) is still true, butM^ϕandM^ϕare only local martingales, andV^ϕis not necessarily of integrable variation [30], Theorem 2.2, p. 97. Yet if∇ϕin L^∞(R^N;R^N) then it is easily checked that M^ϕ and M^ϕ are still martingales, since a is bounded. Besides, although p(t, x, y) is singular at t = 0, (3) implies that

Ex

1 0 |dV_t^ϕ|

1. (5)

2.2. Regularity of the trajectories

In the theory of rough paths, the results strongly depend on the regularity of the path which is considered.

In our case, the regularity of a path x : [0,1] → R^N if characterized by its α-variation for α ≥ 1, which is defined by

Var_αx= sup

partition{ti}i=1,...,kof [0,1]

_k−1

i=0

|x_ti+1−x_ti|^α 1/α

. (6)

Let us start by recalling the Kolmogorov criterion.

Lemma 2 (Kolmogorov criterion [31], Th. 2.1 p. 25). If (X,P)is a stochastic process on [0,1]such that for someα, ε >0,

E[|X_t−X_s|^α]≤C|t−s|^1+ε for all0≤s≤t≤1,

then for any β in [0, ε),

E

0≤s<t≤sup 1

|X_t−X_s|^α

|t−s|^β

≤C,

where C is a constant that depends only onα, β andC. Besides, there exists a modification of X which is β/α-H¨older continuous.

In particular, for anyγ > α/ε,E[ Var_γX]≤E[ Var_γ(X)^γ]^1/γ ≤C^1/γ.

The Kolmogorov criterion will be used to prove that the trajectories of the process X generated by a divergence-form operator are β-H¨older continuous for any β < 1/2, as for the trajectories of the Brownian motion. Then, almost every trajectory ofX is of finiteα-variation for anyα >2.

Lemma 3. Let X be a process generated by a divergence form operator as in Section 2.1. For any α≥2 and any 0≤s≤t≤1,

sup

x∈R^NEx

sup

r∈[s,t]|X_r−X_s|^α

|t−s|^α/2.

So,X isβ-H¨older continuous for anyβ <1/2, and E[ Var_αX]1 for anyα >2.

Proof. With the Markov property,

sup

x∈R^NEx

sup

r∈[s,t]|X_r−X_s|^α

= sup

x∈R^NEx

sup

r∈[0,t−s]|X_r−x|^α

.

WhentandRare fixed, let us denote byτ the first timeX exits from the ball of radiusRwith centerx. Then, with the help of the strong Markov property,

sup

x∈R^NEx

r∈sup[0,t]|X_r−x|^α

≤R^α+ sup

x∈R^NEx

r∈sup[0,t]|X_r−x|^α;t > τ

≤R^α+ 2^α−1

R^α+ sup

x∈R^NEx

sup

r∈[τ,t]|X_r−X_τ|^α;t > τ

≤(2^α−1+ 1)R^α+ 2^α−1 sup

x∈R^NEx

sup

r∈[0,t]|X_r−x|^α

sup

x∈R^NPx[t > τ]. (7) But we have seen in Lemma 1 that sup_x∈RNP_x[t > τ]≤Cexp(−CR²/t), whereC andC are some constants depending only on λ, Λ and N. Hence, the lemma is proved ifR is chosen to be equal to κ√

t with κlarge

enough so that 2^α−1Cexp(−Cκ²)≤1/2.

2.3. Stochastic integrals driven by processes generated by divergence-form operators

If X denotes a process generated by a divergence form operator as in Section 2.1, Proposition 1 gives a decomposition of X in term of forward and backward martingales, and a term of finite variation. In [28]

(however, see Rem. 2.6 in [30]) and in [23], the decomposition (4) is used to prove existence of stochastic integrals of f(X_t) againstϕ(X_t) for two functions f andϕ.

Let (Π^δ)δ>0 be a family of deterministic partitions whose mesh decreases to 0 as δgoes to 0. For a δ >0, we set Π^δ ={0≤t^δ₁≤ · · · ≤t^δ_kδ ≤1}and fort∈[0,1],M^δ(t) =iwitht^δ_i ≤t < t^δ_i+1.

Theorem 1 [23, 28]. Let ϕandf be two functions inW^1,p_loc(R^N)for some p >2∨N. Then, for any starting point x∈R^N and any t∈[0,1], the limits in probability underPx of

M^δ(t) i=1

f(X_tδ

i)(ϕ(X_tδ

i+1)−ϕ(X_tδ

i)) (8)

and

M^δ(t) i=1

1 2(f(X_tδ

i+1) +f(X_tδ

i))(ϕ(X_tδ

i+1)−ϕ(X_tδ

i)) (9)

exist whenδdecreases to0. These limits are denoted by_t

0f(X_s) dϕ(X_s)for (8)and_t

0f(X_s)◦dϕ(X_s)for (9).

The theory of rough paths requires one to construct the integral ofX against itself, which can be done using this Theorem. Then, one may consider both stochastic differential equations controlled by X and integrals driven by X. In the companion paper [18], we will identify the latter integrals with the one constructed in Theorem 1.

Using Theorem 1, we define fori, j= 1, . . . , N andt∈[0,1],

t

0 X_sⁱdX_s^j = lim

δ→0 M^δ(t)

k=1

X_tⁱδ k(X_t^j_δ

k+1−X_t^j_δ

k), and

t 0

X_sⁱ◦dX_s^j = lim

δ→0

1 2

M^δ(t) k=1

(X_tⁱ_δ

k+X_tⁱ_δ

k+1)(X^j

t^δ_k+1−X^j

t^δ_k), where the convergences hold in probability underPx,x∈R^N.

Let us end this section by showing the relationship between the two integrals. Fort∈(0,1], let us also set Q^δ,i,j_t = 1

2

t−t^δ_Mδ(t) t^δ_Mδ(t)+1−t^δ_Mδ(t)

X_tⁱδ

Mδ(t)+1−X_tⁱδ Mδ(t)

X^j

t^δ_Mδ(t)+1−X^j

t^δ_Mδ(t)

+1 2

M^δ(t)−1 k=0

(X_tⁱδ

k+1−X_tⁱδ k)(X^j

t^δ_k+1−X^j

t^δ_k). (10) Proposition 2 [28]. For any x ∈R^N and t ∈[0,1], Q^δ,i,j_t converges in probability under Px to ¹₂Mⁱ, M^jt

whenδ→0, whereM is the martingale part ofX. It follows easily that

t 0

X_sⁱ ◦dX_s^j=

t 0

X_sⁱdX_s^j+1

2Mⁱ, M^jt=

t 0

X_sⁱdX_s^j+1 2

t 0

a_i,j(X_s) ds

as for the semi-martingale case. Note that becauseX is a Dirichlet process, one may define the brackets ofX as equal to the brackets of its martingale partM.

3. Rough paths and processes generated by divergence form operators

The theory of rough paths allows us to construct integrals and solutions of differential equations driven by an irregular path, provided that this path is “enhanced” with another one that plays the role of the iterated integrals. In this section, we show how to construct a rough pathXlying aboveX by proving that the second- order iterated integrals ofX (sinceX is of finiteα-variation withα∈(2,3]) have the required regularity.

Before stating our main result in Theorem 2, we recall a few definitions. The reader is referred to [17, 22, 24]

for some expositions of this theory.

3.1. A few definitions from the theory of rough paths

We recall that from the rough paths terminology, arough path, or amultiplicative functional, (of order 2) is a function X= (X¹,X²) defined from ∆+ ={(s, t) 0≤s≤t≤1} to R^N ×(R^N ⊗R^N) such that for some functionX: [0, T]→R^N,

X^i,_s,t¹=X_tⁱ−X_sⁱ andX^i,j,_s,t²=X^i,j,_s,r²+X^i,j,_r,t²+X^i,_s,r¹ ×X^j,_r,t¹ (11) for all 0≤s≤r≤t≤1.

The functionXis asmooth rough path ift→X0,t is also smooth.

The functionXis ageometric rough path if there exists a functionY from ∆+ intoR^N ⊗R^N such that Y_s,t^i,j=−Y^j,iifi=j andY^i,i= 0, (12) Y^i,j_s,t=Y_s,r^i,j +Y^i,j_r,t+1

2(X^i,_s,r¹×X^j,_r,t²−X^j,_s,r¹×X^i,_r,t²), X^i,j,2_s,t =Y^i,j_s,t+1

2X^i,1_s,t×X^j,1_s,t.

We have to note that ifX is a geometric rough path and ifZ: [0,1]→R^N ⊗R^N satisfies (12), then (X_s,t+ Z_t−Z_s)(s,t)∈∆₊ is also a geometric rough path: see Section 4.1. Of course, any geometric rough path is also a rough path, but the converse is not true: see [25] for more details. This distinction will be important here, since we will construct two integrals, one of Stratonovich type and the other of Itˆo type. The first construction uses indeed geometric rough paths while constructing integrals of Itˆo types requires one to control also the brackets of the process.

We will not use the algebraic characterization of a geometric rough path, but another one that identifies them with the limit of a sequence of smooth rough paths. But first, we introduce theα-variation ofX.

IfX : ∆+ →V, where V is a Banach space with a norm| · |, then theα-variation ofX is

Var_αX = sup

partition{ti}i=1,...,kof [0,1]

_k−1

i=0

|X_ti,ti+1|^α 1/α

, (13)

provided that this quantity is finite. If X : [0,1]→ V, then itsα-variation is defined by (13), whereX_s,t is replaced byX_t−X_s, so that we recover (6).

Forα≥2, we denote byV^αthe space of rough pathsX= (X¹,X²) such thatXis continuous,X¹is of finite α-variation, andX²is of finite α/2-variation. This space is equipped with the norm

X_V^α=X_∞+ Var_α(X¹) + Var_α/2(X²).

One can easily show thatV^α⊂ V^α for allα≤α.

Proposition 3 [12, 22, 24]. A rough path X : ∆+ → R^N ×(R^N ⊗R^N) in V^α, α≥ 2 is a geometric one if there exists a sequence of functionsX^δ: [0,1]→R^N such that X^δ converges uniformly toX and inV^α for all α> α, where

X¹_s,t=X_t^δ−X_s^δ and X^i,j,_s,t^2,δ =

t s

(X_r^i,δ−X_s^i,δ) dX_r^j,δ.

In other words, the geometric rough paths are the one that can be approximated by a sequence of smooth rough paths X^δ whose second-order termX^2,δ is constructed using the iterated integrals ofX^δ. Of course, X^δ is a also a geometric rough path for anyδ >0.

3.2. The main result

With the help of Theorem 1, let us define fori, j∈ {1, . . . , N} and (s, t)∈∆+, K_s,t^i,j(X) =

t s

(X_rⁱ−X_sⁱ)◦dX_r^j andK_s,t^i,j,itˆo(X) =

t s

(X_rⁱ−X_sⁱ)dX_r^j.

We denote byK(X) andK^itˆo(X) the families (K_s,t^i,j(X))_{i,j∈{1,...,N},(s,t)∈∆+}and (K_s,t^i,j,itˆo(X))_i,j∈{1,...,N},(s,t)∈∆₊. We also set X_s,t =X_t−X_s. It is easily verified that (X_s,t, K_s,t(X))(s,t)∈∆₊ and (X_s,t, K_s,t^itˆo(X))(s,t)∈∆₊ are rough paths, in the sense they satisfy (11).

Givenωin the probability space Ω, letX^δ(ω) be the piecewise linear approximation ofX(ω) along Π^δ, that is X_t^δ(ω) =X_ti(ω) + t−t_i

t_i+1−t_i(X_si+1(ω)−X_ti(ω)) fort∈[t_i, t_i+1]. (14) Fori, j= 1, . . . , N and 0≤s≤t≤1, we also denote byK_s,t^i,j(X^δ) the ordinary integrals

K_s,t^i,j(X^δ) =

t s

(X_r^i,δ−X_s^i,δ) dX_r^j,δand K_s,t^i,j,itˆo(X^δ) =K_s,t(X^δ)−(Q^i,j,δ_t −Q^i,j,δ_s ),

whereQ^δhave been defined by (10). We denote byK(X^δ) andK^itˆo(X^δ) the families (K_s,t^i,j(X^δ))_i,j∈{1,...,N},(s,t)∈∆₊

and (K_s,t^itˆo,i,j(X^δ))_i,j∈{1,...,N},(s,t)∈∆₊. Our main result is the following.

Theorem 2.

(i) The rough paths(X, K(X))and(X, K^itˆo(X))are of finiteα-variation for anyα >2.

(ii) For anyα > 2, the sequences of smooth rough paths (X^δ, K(X^δ))_δ>0 and (X^δ, K^itˆo(X^δ))_δ>0 converge in probability in V^α respectively to(X, K(X))and(X, K^itˆo(X)).

(iii) The rough path(X, K(X))is a geometric one.

In this theorem, we give only a convergence in probability, while there are other types of processes (Brownian motion [21], fractional Brownian motion [5], semi-martingales [4], ...) for which an almost sure convergence is given. But we deal with a family of general, deterministic partitions, and not with dyadic partitions.

The following corollaries are then immediate using the results from the theory of rough paths [17, 22, 24].

We setX= (X_s,t, K_s,t(X))(s,t)∈∆₊ andX^itˆo= (X_s,t, K_s,t^itˆo(X))(s,t)∈∆₊.

Corollary 1. Let f = (f¹, . . . , f^N) be a function such that: For i = 1, . . . , N, fⁱ is a continuous function fromR^N toR^mwith a derivative which isγ-H¨older continuous for someγ >0.

Then the integrals Z = _·

0f(X_s) dX_s and Z^itˆo = _·

0f(X_s) dX^itˆo_s exist and define respectively a geometric rough path and a rough path, both of finite α-variation for anyα >2.

Let Z^δ andZ^δ,itˆo be the processes inR^m defined by the ordinary integrals Z_t^δ =z+

t 0

fⁱ(X_r^δ)dX_r^δ,i, andZ_t^δ,itˆo=z+

t 0

fⁱ(X_r^δ)dX_r^δ,i+

t 0

∂fⁱ

∂x_k(X_r^δ) dQ^δ,i,k_r

wherez∈R^mis fixed. Then, for any x∈R^N,Z^δ andZ^δ,itˆo converge in probability underPx to the stochastic processes Z_·=z+Z¹_0,· andZ_·^itˆo=z+Z^1,itˆo_0,· .

In fact, it will be shown in [19] thatZ_t is equal to the Stratonovich integralz+_t

0fⁱ(X_r) ◦dX_rⁱ defined in Theorem 1, whileZ_t^itˆo is equal to the Itˆo integralz+_t

0fⁱ(X_r) dX_rⁱ.

Corollary 2. Let g= (g¹, . . . , g^N)be a function such that gⁱ is a continuous function fromR^m toR^N with a derivative which is γ-H¨older continuous for some γ >0.

Then there exist some solutionsYandY^itˆo to the stochastic differential equationsY_0,t=_t

0g(y+Y_0,s¹ ) dX_s andY^itˆo_0,t=_t

0g(y+Y₀^itˆo,1_,s ) dX^itˆo_s . These objects YandY^itˆo are rough paths of finiteα-variation for anyα >2 andY is a geometric rough path.

Ifgⁱ is twice differentiable and∂_x²_,xjgⁱ isγ-H¨older continuous for someγ >0, then YandY^itˆo are unique.

Moreover, let y be fixed and let Y^δ and Y^δ,itˆo be the processes in R^m whose trajectories are solutions to the ordinary differential equations

Y_t^δ =y+

t 0

gⁱ(Y_r^δ) dX_r^i,δ, andY_t ^,δ,itˆo=y+

t 0

gⁱ(Y_r^δ,itˆo) dX_r^i,δ+

t 0

g_kⁱ(Y_r^δ,itˆo)∂g^j

∂x_k(Y_r^δ,itˆo) dQ^δ,i,j_r

for = 1, . . . , m. Then, for any x∈R^N,Y^δ andY^δ,itˆo converge in probability under P_x as δ→0 respectively toY =y+Y¹_0,·andY^itˆo=y+Y¹_0,·^,itˆo.

Remark 5. Indeed, all the previous results are true if X is replaced by ϕ(X), where ϕ is a function in W_loc^1,∞(R^N;R^m) with∇ϕ∈L^∞.

Remark 6. Using some of the results in [18], these results are also be true for inhomogeneous diffusion processes.

Remark 7. Applying the Newton formula on the change of variables to a function of classC^2+γ withγ >0 to Z^δ, Z^δ,itˆo, Y^δ andY^δ,itˆo, it is immediately established that the Stratonovich and Itˆo formula hold for X and also forY, Y^itˆo, Z andZ^itˆo. The Itˆo formula for X could be proved under weaker assumptions forX (See for example [7, 10, 28] for different approaches).

3.3. Notations and useful results

We recall first a useful tightness criterion on the space of rough paths [17].

Lemma 4 (A tightness criterion). Let (X^δ)δ>0 be a family of rough paths of finite α-variation such that for any C >0 and for i= 1, . . . ,α,

lim sup

η→0 sup

δ>0P

|t−s|≤ηsup |X^i,δ_s,t|> C

= 0,

κ→∞lim sup

δ>0P

Var_α/i(X^i,δ)> κ

= 0. (15)

Then, there exists a rough path X of finiteα-variation such thatX^δ converges in distribution to Xin V^α for any α > α.

Furthermore, if X^δ lies above X^δ, and (X₀^δ)_δ>0 is tight, then the limit Xlies above X_t=X¹_0,t+X0, where X0 is a limit in distribution ofX₀^δ.

Remark 8. As the space of continuous functions of finite α-variation is not separable, the convergence in distribution inV^α ofX^δ to Xmay not imply that (15) is true.

Convention 2. In the sequel, we choose a family of deterministic partitions Π^δ whose meshes decrease to 0 withδ. Whenδis fixed, then the point of the partitions are denoted by{t_i}iand the reference toδis implicit.

Let Π ={0≤s₁≤ · · · ≤s_n≤1} be a partition of [0,1]. Whenδ is fixed, the expression A(s, t) =

i=0

B(t_i, t_i+1)

resp.

Π

i=0

B(s_j, s_j+1, t_i, t_i+1)