Pierre Dolbeault

Complex Plateau problem: old and new results and prospects

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1. Introduction 2. Volume minimality of complex analytic subvarieties and of Levi-flat hypersurfaces in Kähler manifolds 3. Possible origin: holomorphic extension; polynomial envelope of a real curve 4. Solutions of the complex Plateau problem (or boundary problem) in different spaces 5. Extension to real parametric problems 6. Levi-flat hypersurfaces with prescribed boundary: preliminaries 7. Levi-flat hypersurfaces with prescribed boundary: particular cases 1. Introduction.

Given a Hermitian manifold X, the complex Plateau problem is the research of an even dimensional subvariety with negligible singularities, with given boundary, and of minimal volume in X. We will call mixed Plateau problem the research of a real hypersurface with given boundary, and of minimal volume in X. More briefly, both problems will be called complex Plateau problem.

First we shall recall or show that complex analytic subvarieties, resp. Levi-flat hypersurfaces are solutions of the Plateau problem when X is Kähler (section 2). Then we will consider the complex Plateau problem as the research of the extension of an odd dimensional, compact, oriented, connected submanifold into a complex analytic subvariety, and recall known solutions (section 3, 4).

To solve the mixed Plateau problem as the research of the extension of an oriented, compact, connected, 2-codimensional submanifold into a Levi-flat hypersurface, we will need solutions of the complex Plateau problem with real parameter, in C n and CP n ; in CP n , it is an open problem to explicit satisfactory conditions on the boundary. In this way, we get very peculiar solutions of mixed Plateau problems (section 5).

Finally, the mixed Plateau problem is solved in C n , in particular cases, as a projection of a Levi-flat variety, and set up in CP n (sections 6,7): known solutions are recalled in C n when the complex points of the boundary are elliptic; special elliptic and hyperbolic points of the boundary are defined, and a solution when the boundary is a "horned sphere" is described; this will be the opportunity to precise and complete results announced in ([D 08], section 4). Problems when the boundary has general hyperbolic points are still open.

Proofs of the results in sections 6 and 7 will appear in detail elsewhere [D 09]. Acknowledgments. I thank G. Tomassini and D. Zaitsev for discussions, corrections and remarks about several parts of this paper .

2. Volume minimality of complex analytic subvarieties and of Levi-flat hypersurfaces in Kähler manifolds.

Wirtinger's inequality (1936) [H 77].

In C n ,with complex coordinates (z 1 , . . . , z n ), we have the Hermitian metric H = From the real vector space IR 2n ∼ = C n , we consider the real vector space Λ 2p IR 2n of the 2p-vectors with the associated norm |.|; every decomposable vector (exterior product of elements of IR 2n ) defines a real 2p-plane of C n i.e. an element of the Grassmannian G 2p 2n . We define the norm

|| ζ ||= inf j | ζ j | where ζ = j ζ j , ζ j is decomposable. Let P pp = { N j=1 λ j ζ j ; ζ j decomposable defining a complex p-plane of C n ; λ j ≥ 0; N ∈ IN * }.
2.1.1. Theorem. For every ζ ∈ Λ 2p C n , we have: Let ϕ ∈ Λ r IR 2n , the comass of ϕ is defined as

1 p! ω p (ζ)
|| ϕ || * = sup{ϕ(ζ) : ζ ∈ G r 2n ⊂ Λ 2p IR 2n }
Let Ω be an open subset of C n , for every differential form ϕ of degree r on Ω, let

|| ϕ || * = sup{|| ϕ(z) || * : z ∈ Ω} where || ϕ(z) || * is the comass of ϕ(z).
Let T be a current with measure coefficients on Ω, K be any compact subset of Ω and

χ K the charac- teristic function of K, M K (T ) = sup ||ϕ|| * ≤1 |χ K T (ϕ)|
is, by definition, the mass of T on K.

The measure which assigns the number M K (T ) to each compact set K ⊂ Ω is called the mass or volume measure of T and denoted || T ||, so that M K (T ) =|| T || (K). 2.3. Complex Plateau problem.

[H 77]

On Ω ⊂ C n , or more generally, on a Hermitian manifold (X, ω), let B be a d-closed current of dimension 2p -1 with compact support, and let T be a (2p)-current with compact support and measure coefficients such that dT = B. The complex Plateau problem is to find such a T with minimal mass, i.e. for every compactly supported current S, with measure coefficients such that dS = B, to have M (T ) ≤ M (S), or equivalently, for every compactly supported, d-closed (2p)-current with measure coefficients R,

M (T ) ≤ M (T + R) Such a T is said absolutely volume minimizing on X.
Let T be a d-closed (2p)-current with measure coefficients on X. If, for each compact subset K of X,

M K (T ) ≤ M (χ K T + R)
for all compactly supported d-closed (2p)-current R with measure coefficients on X, then T is said to be absolutely volume minimizing on X.

Theorem. [H 77]

Let T be a 2p-current with measure coefficients on a Hermitian manifold (X, ω) and K be a compact subset of X. Then

(χ K T )(ω p /p!) ≤ M K (T ).
and equality holds iff χ K T is strongly positive.

⊔ ⊓

[H 77]

Volume minimality of complex analytic sets in a Kähler manifold. 2.3.4. Corollary to Theorem 2.3.2. Assume that X = (X, ω) is a Käkler manifold and does not contain compact p-dimensional complex subvarieties. Let V be a p-dimensional complex subvatiety, and T = [V ], then T is absolutely volume minimizing on X. Proof. T is strongly positive. Let K be a compact subset of X and R be a compactly supported d-closed (2p)-current with measure coefficients. From Theorem 2.3.2, M K (T ) = (χ K T )(ω p /p!). But locally ω = dd c ψ, then ω p = ω p-1 ∧dd c ψ = d(ω p-1 ∧d c ψ), so in the neighborhood of any point of X, R(ω p ) = R(d(ω p-1 ∧d c ψ)).

Let (α j ) j∈J be a partition C ∞ of unity subordinate to a locally finite open covering (U j ) j∈J of X such that for every j, ω| Uj = dd c ψ j . Then

R(ω p ) = j α j R(d(ω p-1 ∧ d c ψ j )) = ± j d(α j R)(ω p-1 ∧ d c ψ j ) = 0, because: j d(α j R) = j dα j ∧ R + j α j ∧ dR = 0
and, as in the proof of ([H 77], Corollary 1.25), in an open set of the Hermitian C n ,

M K (T ) = (χ K T )(ω p /p!) = (χ K T + R)(ω p /p!) ≤ M K (T + R). ⊔ ⊓ 2.3.5. Remark. If X contains a compact p-dimensional complex subvariety W , d[V ] = 0, but M K ([V ] > 0;
then T is relatively volume minimizing on X.

2.4. Volume minimality of Levi-flat hypersurfaces in Kähler manifolds. We suppose to be in the category of currents with measure coefficients. Recall the definition: A Levi-flat subvariety (with negligible singularities), of odd dimension, is, outside of the singularities, a submanifold with Levi form ≡ 0, or, equivalently, is foliated by complex analytic hypersurfaces.

Let M be a C ∞ Levi-flat hypersurface of a C ∞ Kähler manifold X = (X, ω) bearing a foliation L by complex hypersurfaces M l and let L be the space of the foliation L assumed to be a C ∞ real curve.

Let M ′ be a C ∞ hypersurface of X bearing a foliation L ′ with the same space L; the leaves of L ′ being C ∞ subvarieties with negligible singularities.

Let S be a C ∞ compact submanifold of codimension 2 of X. We denote by the same notation the hypersurfaces and submanifold and the integration currents they define. 2.5. Mixed Plateau problem.

Given S to find a C ∞ hypersurface in X \ S whose boundary is S in the category H of foliated hypersurfaces with the same space of foliation, a real curve. If M ′ is such a hypersurface whose space of foliation is L and the leaves (M ′ l , l ∈ L), then vol(M ′ ) = L vol(M ′ l )dl. From section 2, for every l ∈ L, vol(M ′ l ) ≥ vol(M l ) then M is relatively volume minimizing in the category H and, by definition, M is solution of the mixed Plateau problem.

Research of solutions of the complex Plateau problem.

The present method of resolution consists in finding complex analytic, resp. Levi-flat subvarieties, in X \ S, whose boundary S (in the sense of currents) is a submanifold of X with convenient properties. .

Possible origin: holomorphic extension; polynomial envelope of a real curve.

3.1. The extension theorem of Hartogs, obtained at the beginning of the 20 th century, has been completely proved by Bochner and Martinelli, independently, in 1943. The simplest version is:

Let

Ω be a bounded open set of C n , n ≥ 2. Suppose that ∂Ω be of class C k (1 ≤ k ≤ ∞) or of class C ω (i.e. real analytic). Let f be a function in C l (∂Ω), 1 ≤ l ≤ k.
Then the two conditions are equivalent: (i) f is a CR function, i.e. the differential of f restricted to the complex subspaces of the tangent space to ∂Ω, at every point, is C-linear;

(ii) there exists F ∈ C l (Ω) ∩ O(Ω) such that F | ∂Ω = f .
Then the graph of f is the boundary of the complex analytic submanifold defined by the graph of F in C n+1 .

Let M be a compact submanifold of dimension 1 of C

n , we call polynomial envelope of M , the compact set {z ∈ C n ; | P (z) |≤ max ζ∈M | P (ζ) |; P ∈ C[z]
, the polynomial ring with complex coefficients }.

Then (J. [START_REF] Wermer | The hull of a curve in C n[END_REF]), the polynomial envelope of M is either M , or the union of M with the support of a complex analytic variety T , of complex dimension 1, whose boundary is M [We 58]. 4. Solutions of the complex Plateau problem (or boundary problem) in different spaces.

4.1. The first result has been obtained in 1958, by J. Wermer, in C n , for p = 1 and M holomorphic image of the unit circle in C [We 58]; this result has been generalized to the case where M is a union of C 1 real connected curves by Bishop, Stolzenberg (1966), looking for the polynomial envelope of M according to section 3.2.

In C n , after preliminary results by [START_REF] Rothstein | Bemerkungen zur Theorie komplexer Räume[END_REF] [Rs 59] , the boundary problem has been solved by [START_REF] Harvey | On boundaries of complex analytic varieties, I[END_REF], for p ≥ 2, under the necessary and sufficient condition: M is compact, maximally complex and, for p = 1, under the moment condition: M ϕ = 0, for every holomorphic 1-form ϕ on C n [ HL 75]. For n = p + 1, the method, inspired by the Hartogs' theorem consists in building T as the divisor of a meromorphic function the defining function R; this function itself is constructed, step by step, from solutions of ∂-problems with compact support. T can also be viewed as graph (with multiplicities on the irreducible components) of an analytic function with a finite number of determinations. For any p, we come back to the particular case using projections.

In CP n \ CP n-r , 1 ≤ r ≤ n, for compact M , the problem has a une solution if and only if, for p ≥ r + 1, M is maximally complex and if, for p = r, M satisfies the moment condition: M ϕ = 0, for every ∂-closed (p, p -1)-forme ϕ . The method consists in solving the boundary problem, in C n+1 \ C n-r+1 , for the inverse image of M by the canonical projection [HL 77].

In both cases, the solution is unique.

Harvey et Lawson assume the given M to be, except for a closed set of Hausdorff (2p -1)-dimensional measure zero, an oriented manifold of class C 1 ; we will say: M is a variety C 1 with negligible singularities.

The boundary problem in CP n has been set up, for the first time, by J. King [ Ki 79]; uniqueness of the solution is no more possible, since two solutions differ by an algebraic p-chain.

4.2. In CP n , a solution of the boundary problem has been obtained by P. Dolbeault et G. Henkin for p = 1, (1994), then for every p (1997) and more generally, in a q-linearly concave domain X of CP n , i.e. a union of projective subspaces of dimension q [DH 97 ].

The necessary and sufficient condition for the existence of a solution is an extension of the moment condition: it uses a Cauchy residue formula in one variable and a non linear differential condition which appears in many questions of Geometry or mathematical Physics. In the simplest case: p = 1, n = 2, this is the shock wave equation for a local holomorphic function in 2 variables ξ, η, f ∂f ∂ξ = ∂f ∂η .

From a local condition, the above relation allows to construct, by extension ot the coefficients, a meromorphic function playing, in C n , the same part as the Harvey-Lawson defining function described above; it defines a holomorphic p-chain extendable to CP n using the classical Bishop-Stoll theorem. 5. Extension to real parametric problems.

5.1.

In a real hyperplane of C n .

5.1.1. Let E ∼ = R×C n-1 , and k : R×C n-1 → R be the projection. Let N ⊂ E be a compact, (oriented) CR subvariety of C n of real dimension 2n -4 and CR dimension n -3, (n ≥ 4), of class C ∞ , with negligible singularities (i.e. there exists a closed subset τ ⊂ N of (2n -4)-dimensional Hausdorff measure 0 such that N \ τ is a CR submanifold). Let τ ′ be the set of all points z ∈ N such that either z ∈ τ or z ∈ N \ τ and N is not transversal to the complex hyperplane k -1 (k(z)) at z. Assume that N , as a current of integration, is d-closed and satisfies:

(H) there exists a closed subset L ⊂ R x1 with H 1 (L) = 0 such that for every x ∈ k(N ) \ L, the fiber k -1 (x) ∩ N is connected and does not intersect τ ′ . 5.1.2. Theorem [DTZ 09] (see also [DTZ 05]). Let N satisfy (H) with L chosen accordingly. Then, there exists, in E ′ = E \ k -1 (L), a unique C ∞ Levi-flat (2n -3)-subvariety M with negligible singularities in E ′ \ N , foliated by complex (n -2)-subvarieties, with the properties that M simply (or trivially) extends to E ′ as a (2n -3)-current (still denoted M ) such that dM = N in E ′ . The leaves are the sections by the hyperplanes E x 0 1 , x 0 1 ∈ k(N ) \ L, and are the solutions of the "Harvey-Lawson problem" for finding a holomorphic subvariety in

E x 0 1 ∼ = C n-1 with prescribed boundary N ∩ E x 0 1 . 5.2.
In a real hyperplane of CP n+1 . 5.2.1. The simplest significant case is the boundary problem in CP 3 . For the boundary problem with real parameter in C 3 , we considered a boundary problem in IR × C 3 , i.e. in the subspace of C 4 , in which the first coordinate is real. In the same way, we will consider in CP 4 , with homogeneous coordinates (w 0 , w 1 , . . . , w 4 ), a boundary problem in the subspace E defined by w 1 = λw 0 , with λ ∈ IR. Then, for personal convenience, we will follow, step by step, the known construction in CP 3 in the oldest version [DH 97].

Particularly, the coefficients C m of the defining function R of the solution are estimated as for the problem in CP 3 .

The end of the proof of the main theorem seems analogous to the known case in IR × C 3 . 5.2.2. The projective space CP 3 has homogeneous coordinates w ′ = (w 0 , w 2 , . . . , w 4 ); denote Q = {w 0 = 0} the hyperplane at infinity of CP 3 .

For w 0 = 0, let k be the projection: E → IR λ , (w 0 , w 1 = λw 0 , w 2 , w 3 , w 4 ) → λ; for w 0 = 0, λ is indeterminate.. We also have the projection:

π : E → CP 3 , (w 0 , λw 0 , w 2 , w 3 , w 4 ) → (w 0 , w 2 , w 3 , w 4 ). In the same way, (E \ {w 0 = 0}) ∼ = IR × C 3 . 5.2.3. Let N ⊂ E ⊂ CP 4 be a submanifold of class C ∞ , CR, oriented, compact of E, of dimension 4, of CR dimension 1, with negligible singularities. N being compact in E, k(N ) is compact in IR, i.e. in N , the parameter λ varies in a closed, bounded interval Λ of IR.

Assume that N satisfies the same properties as in subsection 5.1.1. 5.2.4. Consider the complex hyperplanes of CP 4 , whose equation is h(w) = w 4ξ 2 w 0η ′ 2 w 1η 2 w 2 = 0 and, in E, the subspaces

P ν ′ λ whose equation is h1 (w ′ , λ) = w 4 -ξ 2 w 0 -η ′ 2 λw 0 -η 2 w 2 = w 4 -(ξ 2 + η ′ 2 λ)w 0 -η 2 w 2 = 0, of real dimension 5. Their restrictions to (E \ IR × Q) ∼ = IR × C 3 are real affine sub-spaces of dimension 5. We note ν ′ λ the 1 × 2-matrix (ξ 2 + η ′ 2 λ) η 2 . Generically, Γ ν ′ λ = N ∩ P ν ′ λ is of dimension 2. For z ∈ N, λ = k(z). let E λ = k -1 k(z); for λ / ∈ L, N ∩ E λ is of dimension 3 and is contained in E λ ∼ = CP 3 .
Consider the linear forms

h(w) = w 3 -ξ 1 w 0 -η ′ 1 w 1 -η 1 w 2 h0 (w ′ , λ) = w 3 -ξ 1 w 0 -η ′ 1 λw 0 -η 1 w 2 = w 3 -(ξ 1 + η ′ 1 λ)w 0 -η 1 w 2 Denote ν λ = (ξ λ , η) the 2 × 2-matrix (ξ 1 + η ′ 1 λ) η 1 (ξ 2 + η ′ 2 λ) η 2
For fixed λ, ν λ is a coordinate system of a chart of the Grassmannian G C (2, 4), i.e. ν λ is a coordinate system of a chart of G C (2, 4) × IR. and we identify ν λ with the point of G C (2, 4) × IR having these coordinates.

Let ξ λ = t (ξ 1 + η ′ 1 λ) (ξ 2 + η ′ 2 λ) ; η = t (η 1 η 2 ).
Remark that ξ λ depends on (ξ 1 , ξ 2 , η ′ 1 , η ′ 2 ); to get effective dependance on the parameter λ, it suffices to fix η ′ 1 = 0, .η ′ 2 = 0. Recall: ξ λ = t (ξ λ1 ξ λ2 ), ξ λl = ξ l + η ′ l λ, l = 1, 2, η = t (η 1 η 2 ). Let z j = w j /w 0 , j = 2, . . . , 4, be the non homogeneous coordinates ; h0 defines the affine function:

h = z 3 -(ξ 1 + η ′ 1 λ) -η 1 z 2
The two forms h0 et h1 are linearly independent, then the set of their common zeros D ν λ is of real dimension 3, is contained in P ν ′ λ ; in general, D ν λ ∩ N is a finite set Z ν λ ; then, for general enough fixed λ and ν λ , Z ν λ = ∅. For every fixed real number λ / ∈ L, the situation in E λ is the classical situation in CP 3 . 5.2.5. Boundary problem. Given N , find a complex analytic subvariety M depending on the real parameter λ such that dM = N in the sense of currents, under a necessary and sufficient condition on N .

To do this, we can check, step by step, the solution of the boundary problem in CP 3 [HL 97], introducing the parameter λ.

For

λ / ∈ L, γ ν ′ λ = N ∩ P ν ′ λ ∩ E λ is of dimension 1.
Under the notations of the sub-section 5.2.4, consider the function

(1) G(ν λ ) = 1 2πi γ ν ′ λ z 2 dh h
5.2.6. Tentative statement. The following two conditions are equivalent: (i) There exists, in E ′ = E \ k -1 L, a C ∞ Levi-flat subvariety M , (with negligible singularities), of dimension 5, foliated by complex analytic subvarieties M λ of complex dimension 2, such that M extends simply (or trivially) to E ′ as a current of dimension 5 (still denoted M ) such that dM = N in E ′ . The leaves are the sections by the subspaces E λ , λ ∈ k(N ) \ L, and are the solutions of the boundary problem for finding complex analytic subvarieties in E λ ∼ = CP 3 with given boundary N ∩ E λ .

(ii) N is a submanifold CR, oriented, of CR dimension 1 outside a closed set of 4-dimensional Hausdorff measure 0.

There exists a matrix ν * λ * in the neighborhood of which

D 2 ξ λ G(ν λ ) = D 2 ξ λ N j=1 f j (ν λ )
where f j , j = 1, . . . , N , is a holomorphic function in ν λ , C ∞ en λ, and satisfies the system of P.D.E.

(2) f j ∂f j ∂ξ λl = ∂f j ∂η l , l = 2, 3 5.2.7. Remark. This result is not satisfactory because the relation of the analytic conditions with the geometry of the submanifold N is not explicit. 5.3. Boundary problem in a real hyperplane of C n+1 or CP n+1 . C n+1 and CP n+1 are both Kähler. The solutions of the above boundary problems are both Levi flat, hence, from a plain extension of section 2.5, volume minimal, i.e. solution, of codimension 3, of mixed Plateau problems.

6. Levi-flat hypersurfaces with prescribed boundary: preliminaries.

Introduction.

Let S ⊂ C n be a compact connected 2-codimensional submanifold. Find a Levi-flat hypersurface M ⊂ C n \ S such that dM = S (i.e. whose boundary is S, possibly as a current).

For n = 2, near an elliptic complex point p ∈ S, S \ {p} is foliated by smooth compact real curves which bound analytic discs (Bishop [Bi 65]). The family of these discs fills a smooth Levi-flat hypersurface.

In 1983, Bedford-Gaveau considered the case of a particular sphere with two elliptic complex points. If S is contained in the boundary of a strictly pseudoconvex bounded domain, then the families of analytic discs in the neighborhood of each elliptic point extend to a global family filling a 3-dimensional ball M bounded by S. In 1991, Bedford-Klingenberg [BeK 91] and Kruzhilin extended the result when there exist hyperbolic complex points on S with the same global condition.

Results of increasing generality have been obtained by Chirka, Shcherbina, Slodowski, G. Tomassini until 1999. The global sufficient condition of embedding of S in the boundary of a strictly pseudoconvex domain is still required in these papers.

A first result for n ≥ 3 (in the sense of currents), and for elliptic points only, has been obtained four years ago ([DTZ 05] and [DTZ 09] in detailed form); we got new results when S is homeomorphic to a sphere, with three elliptic and one hyperbolic special points (see [D 08] for a first draft), or a torus, with two elliptic and two hyperbolic special points and, more generally, a manifold which is obtained by gluing together elementary models.

A local condition is required because, in general, S is not locally the boundary of a Levi-flat hypersurface. The proof uses the construction of a foliation of S by CR orbits, Thurston's stability theorem for foliations on S, and a parametric version of the Harvey-Lawson theorem on boundaries of complex analytic varieties. There is no global condition. 6.2. Preliminaries and definitions. 6.2.1. A smooth, connected, CR submanifold M ⊂ C n is called minimal at a point p if there does not exist a submanifold N of M of lower dimension through p such that HN = HM | N . By a theorem of Sussman, all possible submanifolds N such that HN = HM | N contain, at p, one of the minimal possible dimension, called a CR orbit of p in M whose germ at p is uniquely determined. 6.2.2. S is said to be a locally flat boundary at a point p if it locally bounds a Levi-flat hypersurface near p. Assume that S is CR in a small enough neighborhood U of p ∈ S. If all CR orbits of S are 1-codimensional (which will appear as a necessary condition for our problem), the following two conditions are equivalent [DTZ 05]:

(i) S is a locally flat boundary on U ;

(ii) S is nowhere minimal on U .

Complex points of S [DTZ 05].

At such a point p ∈ S, T p S is a complex hyperplane in T p C n . In suitable holomorphic coordinates (z, w) ∈ C n-1 × C vanishing at p, S satisfies

(1) w = Q(z) + O(|z| 3 ), Q(z) = 1≤i,j≤n-1 (a ij z i z j + b ij z i z j + c ij z i z j )
S is said flat at a complex point p ∈ S if b ij z i z j ∈ λR, λ ∈ C. We also say that p is flat. Let S ⊂ C n be a locally flat boundary with a complex point p. Then p is flat. By making the change of coordinates (z, w) → (z, λ -1 w), we make b ij z i z j ∈ IR for all z. By a change of coordinates (z, w) → (z, w + a ′ ij z i z j ) we can choose the holomorphic term in (1) to be the conjugate of the antiholomorphic one and so make the whole form Q real-valued.

We say that S is in a flat normal form at p if the coordinates (z, w) as in (1) are chosen such that

Q(z) ∈ R for all z ∈ C n-1 . 6.2.4. Properties of Q.
Assume that S is in a flat normal form; then, the quadratic form Q is real valued. Only holomorphic linear changes of coordinates are allowed. If Q is positive definite or negative definite, the point p ∈ S is said to be elliptic; if the point p ∈ S is is not elliptic, and if Q is non degenerate, p is said to be hyperbolic. From section 6.4, we will only consider particular cases of the quadratic form Q.

From [Bi 65], for n = 2, in suitable holomorphic coordinates, Q(z) = (zz + λRe z 2 ), λ ≥ 0, under the notations of [BeK 91]; for 0 ≤ λ < 1, p is said to be elliptic, and for 1 < λ, it is said to be hyperbolic. The parabolic case λ = 1, not generic, is omitted [BeK 91]. When n ≥ 3, the Bishop's result is not valid in general. 6.3. Elliptic points.

Behaviour of local CR orbits.

Under the notations of [DTZ 09], follow the construction of the complex tangent space E(z, ϕ(z)) to the CR orbit at z; compare with E 0 (z, Q(z)). We know the integral manifold, the orbit of E 0 (z, Q(z)); deduce an evaluation of the integral manifold of E(z, ϕ(z)). 6.6.4. Lemma. Under the above hypotheses, if k = 1, the local orbit Σ corresponding to Σ 0 has two connected components in the neighborhood of 0.

Proof. Use Remark 6.6.2 and the adaptation of the technique of [DTZ 09].

⊔ ⊓ 6.7. CR-orbits near a subvariety containing a special 1-hyperbolic point.

6.7.2. Proposition. Assume that S ⊂ C n (n ≥ 3), is a locally closed (2n-2)-submanifold, nowhere minimal at all its CR points, which has a unique spcial 1-hyperbolic flat complex point p, and such that: (i) the orbit Σ whose closure Σ ′ contains p is compact;

(ii) Σ has two connected components σ 1 , σ 2 , whose closures are homeomorphic to spheres of dimension 2n -3.

Then, there exists a neighborhood V of Σ ′ such that V \ Σ ′ is foliated by compact real (2n -3)dimensional CR orbits whose equation, in a neighborhood of p is (3), and, the w(= x n )-axis being assumed to be vertical, each orbit being diffeomorphic to the sphere S 2n-3 above Σ ′ , the union of two spheres S 2n-3 under Σ ′ , and there exists a smooth function ν, having the CR orbits as the level surfaces.

6.8. Geometry of the complex points of S.

6.8.1. Let G be the manifold of the oriented real linear (2n -2)-subspaces of C n . The submanifold S of C n has a given orientation which defines an orientation of the tangent space to S at any point p ∈ S. By mapping each point of S into its oriented tangent space, we get a smooth Gauss map

t : S → G 6.8..2. Dimension of G. dim G = 2(2n -2).
6.8..3. Proposition. For n ≥ 2, in general, S has isolated complex points.

Proof. Let π ∈ G be a complex hyperplane of C n whose orientation is induced by its complex structure; the set of such π is H = G C n-1,n = CP n-1 * ⊂ G, as real submanifold. If p is a complex point of S, then t(p) ∈ H or -t(p) ∈ H. The set of complex points of S is the inverse image by t of the intersections t(S) ∩ H and -t(S) ∩ H in G. Since dim t(S) = 2n -2, dim H = 2(n -1), dim G = 2(2n -2), the intersection is 0-dimensional, in general. 6.8.4. Homology of G. (cf [P 08]). G has the structure of a complex quadric; let S 1 , S 2 be generators of H 2n-2 (G, Z Z); we assume that S 1 and S 2 are fundamental cycles of complex projective subspaces of complex dimension (n -1) of G. Then, denoting also S, the fundamental cycle of the submanifold S and t * the homomorphism defined by t, we have:

t * (S) ∼ u 1 S 1 + u 2 S 2
where ∼ means homologous to. 6.8.5. Lemma (proved for n = 2 in [CS 51]). With the notations of 6.8.1 ′ , we have:

u 1 = u 2 ; u 1 + u 2 = χ(S), Euler-Poincaré characteristic of S.
The proof for n = 2 works for any n ≥ 3.

6.8.6. Local intersection numbers of H and t(S) when all complex points are flat. Proposition (known for n = 2 [Bi 65], here for n ≥ 3). Let S be a smooth, oriented, compact, 2codimensional, real submanifold of C n whose all complex points are flat and special. Then, on S, ♯ (special elliptic points) + ♯ (special k-hyperbolic points, with k even) -♯ (special k-hyperbolic points, with k odd) = χ(S). If S is a sphere, this number is 2.

7. Levi-flat hypersurfaces with prescribed boundary: particular cases.

7.1. To solve the boundary problem by Levi-fllat hypersurfaces, S has to satisfy necessary and sufficient local conditions. A way to prove that these conditions can occur is to construct an example for which the solution is obvious. 7.2. Sphere with elliptic points. 7.2.1. Example. In C 3 , Let S be defined by the equations:

(S) z 1 z 1 + z 2 z 2 + z 3 z 3 = 1 z 3 = z 3
We have CR-dim S = 1 except at the points z 1 = z 2 = 0; z 3 = ±1 where CR-dim S = 2. S is the unit sphere in C 2 × IR; it bounds the unit ball M in C 2 × IR, which is foliated by the complex balls C 2 × {x 3 } ∩ M . The leaves are relatively compact of real dimension 4 and are bounded by compact leaves (3-spheres) of a foliation of M . 7.2.2. Theorem [DTZ 05]. Let S ⊂ C n , n ≥ 3, be a compact connected smooth real 2-codimensional submanifold satisfying the conditions (i) S is nonminimal at every CR point;

(ii)every complex point of S is flat and elliptic and there exists at least one such point;

(iii) S does not contain complex manifold of dimension (n -2).

Then S is a topological sphere, and there exists a Levi-flat (2n-1)-subvariety M ⊂ C×C n with boundary S (in the sense of currents) such that the natural projection π : C × C n → C n restricts to a bijection which is a CR diffeomorphism between S and S outside the complex points of S. 7.3. Sphere with one special 1-hyperbolic point (sphere with two horns). 7.3.1. Example. In C 3 , let (z j ), j = 1, 2, 3, be the complex coordinates and z j = x j + iy j . In R 6 ∼ = C 3 , consider the 4-dimensional subvariety (with negligible singularities) S defined by: y

3 = 0 0 ≤ x 3 ≤ 1; x 3 (x 2 1 + y 2 1 + x 2 2 + y 2 2 + x 2 3 -1) + (1 -x 3 )(x 4 1 + y 4 1 + x 4 2 + y 4 2 + 4x 2 1 -2y 2 1 + x 2 2 + y 2 2 ) = 0 -1 ≤ x 3 ≤ 0; x 3 = x 4 1 + y 4 1 + x 4 2 + y 4 2 + 4x 2 1 -2y 2 1 + x 2 2 + y 2 2
The singular set of S is the 3-dimensional section x 3 = 0 along which the tangent space is not everywhere (uniquely) defined.

S being in the real hyperplane {y 3 = 0}, the complex tangent spaces to S are {x 3 = x 0 } for convenient x 0 .

The set S will be smoothed along the complement of 0 (origin of C 3 ) in its section by the hyperplane {x 3 = 0} by a small deformation leaving h unchanged. In the following S will denote this smooth submanifold.

From elementary analytic geometry, complex points of S are defined by their coordinates: e 3 : x j = 0, y j = 0, (j = 1, 2), x 3 = 1. h: x j = 0, y j = 0, (j = 1, 2), x 3 = 0; e 1 , e 2 : x 1 = 0, y 1 = ±1, x 2 = 0, y 2 = 0, x 3 = -1.

Lemma. The complex points are flat and special. The points e 1 , e 2 , e 3 are special elliptic; the point h is special 1-hyperbolic.

Remark that the numbers of special elliptic and special hyperbolic points satisfy the conclusion of Proposition 6.8.6. 7.3.1'. Shape of Σ ′ = S ∩ {x 3 = 0} in the neighborhood of the origin 0 of C 3 .

Lemma. Under the above hypotheses and notations, (i) Σ = Σ ′ \ 0 has two connected components σ 1 , σ 2 .

(ii) The closures of the three connected components of S \ Σ ′ are submanifolds with boundaries and corners. Proof. (i) The only singular point of Σ ′ is 0. We work in the ball B(0, A) of C 2 (x 1 , y 1 , x 2 , y 2 ) for small A and in the 3-space π λ = {y 2 = λx 2 }, λ ∈ IR. For λ fixed, π λ ∼ = IR 3 (x 1 , y 1 , x 2 ), and Σ ′ ∩ π λ is the cone of equation 4x 2 1 -2y 2 1 + (1 + λ 2 )x 2 2 + O(|z| 3 ) = 0 with vertex 0 and basis in the plane x 2 = x 0 2 the hyperboloid H λ of equation 4x 2 1 -2y 2 1 + (1 + λ 2 )x 02 2 + O(|z| 3 ) = 0; the curves H λ have no common point outside 0. So, when λ varies, the surfaces Σ ′ ∩ π λ are disjoint outside 0. The set Σ ′ is clearly connected; Σ ′ ∩ {y 1 = 0} = {0}, the origin of C 3 ; from above: σ 1 = Σ ∩ {y 1 > 0}; σ 2 = Σ ∩ {y 1 < 0}. the first case, we say that p is 1-up, (resp. 2-up), in the second that p is down. Such a T will be called an elementary model.

For instance, T is 1-up and has one special elliptic point, we solve the boundary problem as in S 1 in the proof of Theorem 7.3.3. 7.5.3. The gluing (to be precised) happens between two compatible elementary models along boundaries, for instance down and 1-up. 7.6. Other possible generalizations.

The mixed Plateau problem can be set up in projective space CP n and in subspaces of CP n on which the complex Plateau problem can be solved, using Statement 5.2.6, its gemeralisation to any n ≥ 3 and a better geometric condition on the given boundary.

  dz j and the exterior form (standard Kähler) ω = i 2 n j=1 dz j ∧ dz j = n j=1 dx j ∧ dy j . 1

  4.2.1. The conditions of regularity of M have been weakened, first in C n , and for p = 1, to a condition, a little stronger than the rectifiability, by H. Alexander [Al 88 ] who, moreover, has given an essential counterexample [ Al 87], then by Lawrence [Lce 95] and finally, and for any p, in C n and CP n , by T.C. Dinh [Di 98]: M is a rectifiable current whose tangent cone is a vector subspace almost everywhere. Moreover, Dinh has obtained the reduction of the boundary problem in CP n to the case p = 1, with weaker conditions than above and by an elementary analytic procedure [Di 98]. All the previous results are obtained as Corollaries. New progress by Harvey and Lawson [HL 04].

  ≤|| ζ ||; equality uniquely for ζ ∈ P pp [W 36].2.1.2. Corollary. Let V be a smooth real oriented 2p-dimensional submanifold of a Hermitian manifold X = (X, ω) of complex dimension n. Then V ω p /p! ≤ vol 2p (V ) with equality iff V is complex.

2.2. Currents with measure coefficients. [H 77] 2.2.1. Comass of an r-form; mass of a current with measure coefficients.

Proposition ([DTZ 05], [DTZ 09]

). Assume that S ⊂ C n , (n ≥ 3) is nowhere minimal at all its CR points and has an elliptic flat complex point p. Then there exists a neighborhood V of p such that V \ {p} is foliated by compact real (2n -3)-dimensional CR orbits diffeomorphic to the sphere S 2n-3 and there exists a smooth function ν, having the CR orbits as the level surfaces. Sketch of Proof (see [DTZ 09]).

In the case of a quadric S 0 (w = Q(z)), the CR orbits are defined by w 0 = Q(z), where w 0 is constant. Using (1), we approximate the tangent space to S by the tangent space to S 0 at a point with the same coordinate z; the same is done for the tangent spaces to the CR orbits on S and S 0 ; then we construct the global CR orbit on S through any given point close enough to p. 6.4. Special flat complex points. We say that the flat complex point p ∈ S is special if in convenient holomorphic coordinates,

(2)

Let z j = x j + iy j , x j , y j real, j = 1, . . . , n -1, then:

). A flat point p ∈ S is said to be special elliptic if 0 ≤ λ j < 1 for any j. A flat point p ∈ S is said to be special k-hyperbolic if 1 < λ j for j ∈ J ⊂ {1, . . . , n -1} and 0 ≤ λ j < 1 for j ∈ {1, . . . , n -1} \ J = ∅, where k denotes the number of elements of J.

Special elliptic (resp. k-hyperbolic) points are elliptic (resp. hyperbolic). 6.5. Special hyperbolic points. 6.5.1. We will not consider special parabolic points (one λ j = 1 at least) which don't appear generically.

S being given by (1), let S 0 be the quadric of equation w = Q(z). Suppose that S 0 is flat at 0 and that 0 is a special k-hyperbolic point. Then, in a neighborhood of 0, and with the above local coordinates, it is CR and nowhere minimal outside 0, and the CR orbits of S 0 are the (2n -3)-dimensional submanifolds given by w = const. = 0.

The section w = 0 of S 0 is a real quadratic cone Σ ′ 0 in R 2n whose vertex is 0 and, outside 0, it is a CR orbit Σ 0 in the neighborhood of 0. 6.6. Foliation by CR-orbits in the neighborhood of a special 1-hyperbolic point.

We mimic the begining of the proof of 2.4.2. in ([DTZ 05], [DTZ 09]). 6.6.1. Local 2-codimensional submanifolds.

In C 3 , consider the 4-dimensional submanifold S locally defined by the equation

and the 4-dimensional submanifold S 0 of equation

having a special 1-hyperbolic point at 0, (λ 1 > 1, 0 ≤ λ 2 < 1), and the cone Σ ′ 0 whose equation is: Q = 0. On S 0 , a CR orbit is the 3-dimensional submanifold K w0 whose equation is w 0 = Q(z). If w 0 > 0, K w0 does not cut the line L = {x 1 = x 2 = y 2 = 0}; if w 0 < 0, K w0 cuts L at two points.

y 2 2 = 0 whose only zero , in the neighborhood of 0, is {0}: the connected components are obtained for y 1 > 0 and y 1 < 0 respectively. ⊔ ⊓ (ii) The three connected components of S \ Σ ′ are the components which contain, respectively e 1 , e 2 , e 3 and whose boundaries are σ 1 , σ 2 , σ 1 ∪ σ 2 ; these boundaries have corners as shown in the first part of the proof.

⊔ ⊓

The connected component of C 2 × IR \ S containing the point (0, 0, 0, 0, 1/2) is the Levi-flat solution, the complex leaves being the sections by the hyperplanes x 3 = x 0 3 , -1 < x 0 3 < 1. The sections by the hyperplanes x 3 = x 0 3 are diffeomorphic to a 3-sphere for 0 < x 0 3 < 1 and to the union of two disjoint 3-spheres for -1 < x 0 3 < 0, as can be shown intersecting S by lines through the origin in the hyperplane x 3 = x 0 3 ; Σ ′ is homeomorphic to the union of two 3-spheres with a common point. 7.3.2. Proposition (cf [D 08], Proposition 2.6.1). Let S ⊂ C n be a compact connected real 2-codimensional manifold such that the following holds:

(i) S is a topological sphere; S is nonminimal at every CR point;

(ii) every complex point of S is flat; there exist three special elliptic points e j , j = 1, 2, 3 and one special 1-hyperbolic point h;

(iii) S does not contain complex manifolds of dimension (n -2);

(iv) the singular CR orbit Σ ′ through h on S is compact and Σ ′ \ {h} has two connected components σ 1 and σ 2 whose closures are homeomorphic to spheres of dimension 2n -3;

(v) the closures S 1 , S 2 , S 3 of the three connected components S ′ 1 , S ′ 2 , S ′ 3 of S \ Σ ′ are submanifolds with (singular) boundary.

Then each S j \ {e j ∪ Σ ′ }, j = 1, 2, 3 carries a foliation F j of class C ∞ with 1-codimensional CR orbits as compact leaves.

Proof. From conditions (i) and (ii), S satisfying the hypotheses of Proposition 6.3.2, near any elliptic flat point e j , and of Proposition 6.7.2 near Σ ′ , all CR orbits are diffeomorphic to the sphere S 2n-3 . The assumption (iii) guarantees that all CR orbits in S must be of real dimension 2n -3. Hence, by removing small connected open saturated neighborhoods of all special elliptic points, and of Σ ′ , we obtain, from S \ Σ ′ , three compact manifolds S j ", j = 1, 2, 3, with boundary and with the foliation F j of codimension 1 given by its CR orbits whose first cohomology group with values in R is 0, near e j . It is easy to show that this foliation is transversely oriented. 7.3.2'. Recall the Thurston's Stability Theorem ( [ CaC], Theorem 6.2.1). Let (M, F ) be a compact, connected, transversely-orientable, foliated manifold with boundary or corners, of codimension 1, of class C 1 . If there is a compact leaf L with H 1 (L, R) = 0, then every leaf is homeomorphic to L and M is homeomorphic to L × [0, 1], foliated as a product, Then, from the above theorem, S j " is homeomorphic to S 2n-3 × [0, 1] with CR orbits being of the form S 2n-3 × {x} for x ∈ [0, 1]. Then the full manifold S j is homeomorphic to a half-sphere supported by S 2n-2 and F j extends to S j ; S 3 having its boundary pinched at the point h. 7.3.3. Theorem. Let S ⊂ C n , n ≥ 3, be a compact connected smooth real 2-codimensional submanifold satisfying the conditions (i) to (v) of Proposition 7.3.2. Then there exists a Levi-flat (2n -1)-subvariety M ⊂ C × C n with boundary S (in the sense of currents) such that the natural projection π : C × C n → C n restricts to a bijection which is a CR diffeomorphism between S and S outside the complex points of S.

Proof. By Proposition 6.3.2 , for every e j , a continuous function ν ′ j , C ∞ outside e j , can be constructed in a neighborhood U j of e j , j = 1, 2, 3, and by Proposition 6.7.2, we have an analogous result in a neighborhood of Σ ′ . Furthermore, from section 7.3.2', a smooth function ν" j whose level sets are the leaves of F j can be obtained globally on S ′ j \ {e j ∪ Σ ′ }. With the functions ν ′ j and ν" j , and analogous functions near Σ ′ , then using a partition of unity, we obtain a global smooth function ν j : S j → R without critical points away from the complex points e j and from Σ ′ .

Let σ 1 , resp. σ 2 the two connected, relatively compact components of Σ \ {h}, according to condition (iv); σ 1 , resp. σ 2 are the boundary of S 1 , resp. S 2 , and σ 1 ∪ σ 2 the boundary of S 3 . We can assume that the three functions ν j are finite valued and get the same values on σ 1 and σ 2 . Hence a function ν : S → R.

The submanifold S being, locally, a boundary of a Levi-flat hypersurface, is orientable. We now set S = N = gr ν = {(ν(z), z) : z ∈ S}. Let S s = {e 1 , e 2 , e 3 , σ 1 ∪ σ 2 }. λ : S → S z → ν((z), z) is bicontinuous; λ| S\Ss is a diffeomorphism; moreover λ is a CR map. Choose an orientation on S. Then N is an (oriented) CR subvariety with the negligible set of singularities τ = λ(S s ).

At every point of S \ S s , d x1 ν = 0, then condition (H) (section 5.1.1) is satisfied at every point of N \ τ . Then all the assumptions of Theorem 5.1.2 being satisfied by N = S, in a particular case, we conclude that N is the boundary of a Levi-flat (2n -2)-variety (with negligible singularities) M in R × C n .

Taking π : C × C n → C n to be the standard projection, we obtain the conclusion. 7.4. Case of a torus. 7.4.1. Euler-Poincaré characteristic of a torus is χ(T k ) = 0. 7.4.2. Example. In C 3 , let (z j ), j = 1, 2, 3, be the complex coordinates and z j = x j + iy j . In R 6 ∼ = C 3 , consider the 4-dimensional subvariety (with negligible singularities) S defined by:

glue it with the symmetric with respect to the real hyperplane x 3 = -1 2 , and and smooth along {x 3 = 0}, {x 3 = ± 1 2 }. The complex points are flat and special. 7.4.3. Theorem. Let S ⊂ C n , n ≥ 3, be a compact connected smooth real 2-codimensional submanifold satisfying the following conditions:

(i) S is a topological torus; S is nonminimal at every CR point;

(ii) every complex point of S is flat; there exist two special elliptic points e 1 , e 2 and two special 1hyperbolic points h 1 , h 2 ;

(iii) S does not contain complex manifolds of dimension (n -2); (iv) the singular CR orbits Σ ′ 1 , Σ ′ 2 through h 1 and h 2 on S are compact and, for j = 1, 2, Σ ′ j \ {h j } have two connected components σ 1 j and σ 2 j ; (v) the closures S 1 , S 2 , S 3 , S 4 of the four connected components

Then there exists a Levi-flat (2n -1)-subvariety M ⊂ C × C n with boundary S (in the sense of currents) such that the natural projection π : C × C n → C n restricts to a bijection which is a CR diffeomorphism between S and S outside the complex points. 7.5. Generalizations. 7.5.1. Elementary models and their gluing. The examples and the proofs of the theorems when S is homeomorphic to a sphere (sections 7.3) or a torus (section 7.4) suggest the following definitions. 7.5.2. Definitions.

Let T ′ be a smooth, locally closed (i.e. closed in an open set), connected submanifold of C n , n ≥ 3. We assume that T ′ has the following properties:

(i) T ′ is relatively compact, non necessarily compact, and of codimension 2.

(ii) T ′ is nonminimal at every CR point;

(iii) T ′ has exactly 2 complex points which are flat and either special elliptic or special 1-hyperbolic. (iv) If p ∈ T ′ is 1-hyperbolic, the singular orbit Σ ′ through p is compact, Σ ′ \ p has two connected components σ 1 , σ 2 , whose closures are homeomorphic to spheres of dimension 2n -3.

(v) If p ∈ T ′ is 1-hyperbolic, in the neighborhood of p, with convenient coordinates, the equation of T ′ , up to third order terms is

(z j z j + λ j Re z 2 j ); λ 1 > 1; 0 ≤ λ j < 1 for j = 1 or in real coordinates x j , y j with z j = x j + iy j ,

x n = (λ 1 + 1)x 2 1 -(λ 1 -1)y 2 1 + n-1 j=2

(1 + λ j )x 2 j + (1λ j )y 2 j + O(|z| 3 ) Other configurations are easily imagined. up-and down-1-hyperbolic points. Let T be the (2n -2)-submanifold with (singular) boundary contained into T ′ such that either σ 1 (resp. σ 2 ) is the boundary of T near p, or Σ ′ is the boundary of T near p. In


Amel Bentata

Rama Cont

Mimicking the marginal distributions of a semimartingale

Keywords: MSC Classification Numbers: 60J75, 60H10 Contents, preserving property, Lévy processes

whether they are published or not. The documents may come

Mimicking the marginal distributions of a semimartingale

Amel Bentata, Rama Cont

Introduction

Stochastic processes with path-dependent / non-Markovian dynamics used in various fields such as physics and mathematical finance present challenges for computation, simulation and estimation. In some applications where one is interested in the marginal distributions of such processes, such as option pricing or Monte Carlo simulation of densities, the complexity of the model can be greatly reduced by considering a low-dimensional Markovian model with the same marginal distributions. Given a process ξ, a Markov process X is said to mimick ξ on the time interval [0, T ], T > 0, if ξ and X have the same marginal distributions:

∀t ∈ [0, T ],

ξ t d = X t . (1)
X is called a Markovian projection of ξ. The construction of Markovian projections was first suggested by Brémaud [START_REF] Brémaud | Point processes and queues[END_REF] in the context of queues. Construction of mimicking processes of 'Markovian' type has been explored for Ito processes [START_REF] Gyöngy | Mimicking the one-dimensional marginal distributions of processes having an Itô differential[END_REF] and marked point processes [START_REF] Cont | Recovering portfolio default intensities implied by CDO tranches[END_REF]. A notable application is the derivation of forward equations for option pricing [START_REF] Bentata | Forward equations for option prices in semimartingales[END_REF][START_REF] Dupire | Pricing with a smile[END_REF]. We proposer in this paper a systematic construction of such Markovian projections for (possibly discontinuous) semimartingales. Given a semimartingale ξ, we give conditions under which there exists a Markov process X whose marginal distributions are identical to those of ξ, and give an explicit construction of the Markov process X as the solution of a martingale problem for an integrodifferential operator [START_REF] Bass | Stochastic differential equations with jumps[END_REF][START_REF] Komatsu | Markov processes associated with certain integro-differential operators[END_REF][START_REF] Stroock | Diffusion processes associated with Lévy generators[END_REF][START_REF]Markov Processes from Ito's Perspective[END_REF].

In the martingale case, the Markovian projection problem is related to the problem of constructing martingales with a given flow of marginals, which dates back to Kellerer [START_REF] Kellerer | Markov-Komposition und eine Anwendung auf Martingale[END_REF] and has been recently explored by Yor and coauthors [START_REF] Baker | A brownian sheet martingale with the same marginals as the arithmetic average of geometric brownian motion[END_REF][START_REF] Hirsch | Unifying constructions of martingales associated with processes increasing in the convex order, via Lévy and Sato sheets[END_REF][START_REF] Madan | Making Markov martingales meet marginals[END_REF] using a variety of techniques. The construction proposed in this paper is different from the does not rely on the martingale property of ξ. We shall see nevertheless that our construction preserves the (local) martingale property. Also, whereas the approaches described in [START_REF] Baker | A brownian sheet martingale with the same marginals as the arithmetic average of geometric brownian motion[END_REF][START_REF] Hirsch | Unifying constructions of martingales associated with processes increasing in the convex order, via Lévy and Sato sheets[END_REF][START_REF] Madan | Making Markov martingales meet marginals[END_REF] use as a starting point the marginal distributions of ξ, our construction describes the mimicking Markov process X in terms of the local characteristics [START_REF] Jacod | Limit theorems for stochastic processes[END_REF] of the semimartingale ξ. Our construction thus applies more readily to solutions of stochastic differential equations where the local characteristics are known but not the marginal distributions.

Section 2 presents a Markovian projection result for a R d -valued semimartingale given by its local characteristics. We use these results in section 2.4 to derive a partial integro-differential equation for the one-dimensional distributions of a discontinuous semimartingale, thus extending the Kolmogorov forward equation to a non-Markovian setting. Section 3 shows how this result may be applied to processes whose jumps are represented as the integral of a predictable jump amplitude with respect to a Poisson random measure, a representation often used in stochastic differential equations with jumps. In Section 4 we show that our construction applied to a large class of semimartingales, including smooth functions of a Markov process (Section 4.1), and time-changed Lévy processes (Section 4.2).

A mimicking theorem for discontinuous semimartingales

Consider, on a filtered probability space (Ω, F , (F t ) t≥0 , P), an Ito semimartingale, on the time interval [0, T ], T > 0, given by the decomposition

ξ t = ξ 0 + t 0 β s ds + t 0 δ s dW s + t 0 y ≤1
y M (ds dy)

+ t 0 y >1
y M (ds dy),

(2) where ξ 0 is in R d , W is a standard R n -valued Wiener process, M is an integervalued random measure on [0, T ] × R d with compensator measure µ and M = M -µ is the compensated measure [16, Ch.II,Sec.1], β (resp. δ) is an adapted process with values in R d (resp. M d×n (R)).

Our goal is to construct a Markov process, on some filtered probability space (Ω 0 , B, (B t ) t≥0 , Q) such that X and ξ have the same marginal distributions on [0, T ], i.e. the law of X t under Q coincides with the law of ξ t under P. We will construct X as the solution to a martingale problem [START_REF] Ethier | Markov Processes: Characterization And Convergence[END_REF][START_REF] Stroock | Diffusion processes associated with Lévy generators[END_REF][START_REF] Stroock | Multidimensional diffusion processes[END_REF][START_REF] Mikulevičius | On the martingale problem associated with non-degenerate lévy operators[END_REF] on the canonical space Ω 0 = D([0, T ], R d ).

Martingale problems for integro-differential operators

Let Ω 0 = D([0, T ], R d ) be the Skorokhod space of right-continuous functions with left limits. Denote by X t (ω) = ω(t) the canonical process on Ω 0 , B 0 t its filtration and B t ≡ B 0 t+ . Our goal is to construct a probability measure Q on Ω 0 such that X is a Markov process under Q and ξ and X have the same one-dimensional distributions:

∀t ∈ [0, T ], ξ t d = X t .
In order to do this, we shall characterize Q as the solution of a martingale problem for an appropriately chosen integro-differential operator L.

Let C 0 b (R d ) denote the set of bounded and continuous functions on R d and

C ∞ 0 (R d ) the set of infinitely differentiable functions with compact support on R d . Consider a time-dependent integro-differential operator L = (L t ) t∈[0,T ] defined, for f ∈ C ∞ 0 (R d ), by L t f (x) = b(t, x).∇f (x) + d i,j=1 a ij (t, x) 2
∂ 2 f ∂x i ∂x j (x) + R d [f (x + y) -f (x) -1 { y ≤1} y.∇f (x)]n(t, dy, x), (3)
where a :

[0, T ] × R d → M d×d (R), b : [0, T ] × R d → R d are measurable functions and, for each (n(t, . , x), (t, x) ∈ [0, T ] × R d ) is a measurable family of positive measures on R d -{0}. For x 0 in R d , we recall that a probability measure Q x0 on (Ω 0 , B T ) is a solution to the martingale problem for (L, C ∞ 0 (R d )) on [0, T ] if Q (X 0 = x 0 ) = 1 and for any f ∈ C ∞ 0 (R d ), the process f (X t ) -f (x 0 ) - t 0 L s f (X s ) ds is a (Q x0 , (B t ) t≥0 )-martingale on [0, T ].
Existence, uniqueness and regularity of solutions to martingale problems for integro-differential operators have been studied under various conditions on the coefficients [START_REF] Stroock | Multidimensional diffusion processes[END_REF][START_REF] Jacod | Calcul stochastique et problèmes de martingales[END_REF][START_REF] Ethier | Markov Processes: Characterization And Convergence[END_REF][START_REF] Komatsu | Markov processes associated with certain integro-differential operators[END_REF][START_REF] Mikulevičius | On the martingale problem associated with non-degenerate lévy operators[END_REF].

We make the following assumptions on the coefficients:

Assumption 1 (Boundedness of coefficients). There exists

K 1 > 0 ∀(t, z) ∈ [0, T ] × R d b(t, z) + a(t, z) + 1 ∧ y 2 n(t, dy, z) ≤ K 1 and lim R→∞ T 0 sup z∈R d n (t, { y ≥ R}, z) dt = 0.
where . denotes the Euclidean norm.

Assumption 2 (Continuity). For all t ∈ [0, T ] and of positive measures such that

B ∈ B(R d -{0}), b(t, .), a(t, .
∀(t, z) ∈ [0, T ] × R d n(t, dy, z) = n β (t, dy, z) + C y d+β dy, 1 ∧ y β n β (t, dy, z) ≤ K 2 , lim ǫ→0 sup z∈R d y ≤ǫ y β n β (t, dy, z) = 0.
Mikulevicius and Pragarauskas [START_REF] Mikulevičius | On the martingale problem associated with non-degenerate lévy operators[END_REF] show that if L satisfies Assumptions 1-3 (in which corresponds to a "non-degenerate Lévy operator" in the terminology of [START_REF] Mikulevičius | On the martingale problem associated with non-degenerate lévy operators[END_REF]) the martingale problem for (L, C ∞ 0 (R d ) ) has a unique solution for every initial condition x 0 ∈ R d : Proposition 1. [21, Theorem 5] Under Assumptions 1, 2 and 3 the martingale problem for

((L t ) t∈[0,T ] , C ∞ 0 (R d )) on [0, T ] is well-posed : for any x 0 ∈ R d ,there exists a unique probability measure Q x0 on (D([0, T ], R d ), F T ) such that Q x0 (X 0 = x 0 ) = 1 and for any f ∈ C ∞ 0 (R d ) f (X t ) -f (x 0 ) - t 0 L s f (X s ) ds is a Q x0 -martingale. Under Q x0 , (X t
) is a Markov process and the evolution operator

(Q t ) t∈[0,T ] defined by ∀f ∈ C 0 b (R d ) Q t f (x 0 ) = E Qx 0 [f (X t )] (4)
is strongly continuous on [0, T [.

A uniqueness result for the Kolmogorov forward equation

An important property of continuous-time Markov processes is their link with partial (integro-)differential equation (PIDE) which allows to use analytical tools for studying their probabilistic properties. In particular the transition density of a Markov process solves the forward Kolmogorov equation (or Fokker-Planck equation) [START_REF]Markov Processes from Ito's Perspective[END_REF]. The following result shows that under Assumptions 1, 2 and 3 the forward equation corresponding to L has a unique solution:

Theorem 1 (Kolmogorov Forward equation). Under Assumptions 1, 2 and 3, each x 0 in R d , there exists a unique family (p t (x 0 , dy), t ∈ [0, T ]) of bounded measures on R d such that p 0 (x 0 , .) = ǫ x0 is the point mass at x 0 and

∀g ∈ C ∞ 0 (R d , R), g(y) dp dt (x 0 , dy) = p t (x 0 , dy)L t g(y). (5)
p t (x 0 , .) is the marginal distribution at time t of the unique solution associated to the martingale problem for (L, C ∞ 0 (R d ) ) starting from x 0 on [0, T ].

Proof. Under Assumptions 1, 2 and 3 Proposition 1 implies that the martingale problem for L on the domain C ∞ 0 (R d ) is well-posed. For any x 0 in R d , denote (X, Q x0 ) the unique solution of the martingale problem for L. Define

∀t ∈ [0, T ] ∀f ∈ C 0 b (R d ) Q t f (x 0 ) = E Qx 0 [f (X t )] . ( 6
)
Q t is the evolution operator on [0, T ] of X on C 0 b (R d ). (Q t ) t∈[0,T ]
is then strongly continuous on [0, T [. If q t (x 0 , dy) denotes the law of (X t ) under Q x0 , the martingale property shows that q t (x 0 , dy) satisfies the equation [START_REF] Brunick | Matching statistics of an Ito process by a process of diffusion type[END_REF]. Integration of (5) with respect to time yields q t (x 0 , dy)g(y) = g(x 0 ) + t 0 q s (x 0 , dy)L s g(y) ds.

(

) 7
We have thus constructed one solution q t of (5) with initial condition q 0 (dy) = ǫ x0 . This solution of ( 5) is in particular positive with mass 1.

To show uniqueness of solutions of (5), we will rewrite (5) as the forward Kolmogorov equation associated with a homogeneous operator on space-time domain and use uniqueness results for the corresponding homogeneous equation. Let f ∈ C ∞ 0 (R d ) and γ ∈ C 1 ([0, T ]) and consider the (homogeneous) dependent operator A mapping functions of the form (t, x) ∈ [0, T ] × R d → f (x)γ(t), which will be denoted

C 1 ([0, T ]) ⊗ C ∞ 0 (R d ), into : A(f γ)(t, x) = γ(t)L t f (x) + f (x)γ ′ (t). ( 8
)
For any

x 0 in R d , if (X, Q x0
) is a solution of the martingale problem L, then the law of η t = (t, X t ) is a solution of the martingale problem for A: for any

f ∈ C ∞ 0 (R d ) and γ ∈ C([0, T ]), q t (x 0 , dy)f (y)γ(t) = f (x 0 )γ(0) + t 0 q s (x 0 , dy)A(f γ)(s,
A on C 1 ([0, T ]) ⊗ C ∞ 0 (R d ). Define for all t in [0, T ], for all g in C 1 ([0, T ]) ⊗ C ∞ 0 (R d ): Q t g(x 0 ) = R d q t (x 0 , dy)g(t, y) (10)
One observes that Q t .(x 0 ) corresponds to the extension of Q t defined on the domain

C ∞ 0 (R d ) to the domain C 1 ([0, T ]) ⊗ C ∞ 0 (R d ).
Consider now a family p t (dy) of positive measures solution of [START_REF] Cont | Recovering portfolio default intensities implied by CDO tranches[END_REF] with p 0 (dy) = ǫ x0 (dy). After integration by parts

R d p t (dy)f (y)γ(t) = f (x 0 )γ(0) + t 0 R d p s (dy)A(f γ)(s, y) ds (11)
holds true. Define for all t in [0, T ], for all g in C 1 ([0, T ]) ⊗ C ∞ 0 (R d ):

P t g = R d
p t (dy)g(t, y).

Given [START_REF] Cont | Recovering portfolio default intensities implied by CDO tranches[END_REF] and [START_REF] Ethier | Markov Processes: Characterization And Convergence[END_REF], for all ǫ > 0:

Q t (f γ)(x 0 ) -Q ǫ (f γ)(x 0 ) = t ǫ R d q u (x 0 , dy)A(f γ)(u, y) du = t ǫ Q u (A(f γ))(x 0 ) du, P t (f γ) -P ǫ (f γ) = t ǫ R d p u (dy)A(f γ)(u, y) du = t ǫ P u (A(f γ)) du. (12)
For any λ > 0, we have

λ ∞ 0 e -λt Q t (f γ)(x 0 ) dt = f (x 0 )γ(0) + λ ∞ 0 e -λt t 0 Q s (A(f γ))(x 0 ) ds dt = f (x 0 )γ(0) + λ ∞ 0 e -λt ∞ s e -λt dt Q s (A(f γ))(x 0 ) ds = f (x 0 )γ(0) + ∞ 0 e -λs Q s (A(f γ))(x 0 ) ds Consequently, ∞ 0 e -λt Q t (λ -A)(f γ)(0, x 0 ) dt = f (x 0 )γ(0) = ∞ 0 e -λt P t (λ -A)(f γ) dt. ( 13
) Q t defines a strongly continuous semigroup on C 0 b ([0, T ] × R d ).
The Hille-Yosida theorem [11, Proposition 2.1 and Theorem 2.6] then implies that for all λ > 0,

R(λ -A) = C 0 b ([0, T ] × R d ), where R(λ -A) denotes the image of C 1 ([0, T ]) ⊗ C ∞ 0 (R d ) by the mapping g → (λ -A)g. Hence, since (13) holds then for all h in C 0 b ([0, T ] × R d ) ∞ 0 e -λt Q t h (0, x 0 ) dt = ∞ 0 e -λt P t h dt (14)
so the Laplace transform of t → Q t h (0, x 0 ) is uniquely determined. We will now show that t → Q t h (0, x 0 ) is right-continuous. Furthermore t → b(t, .), t → a(t, .), and t → n(t, . , .) are bounded in t on [0, T ] (Assumption 1). It implies that for any fixed [START_REF] Gyöngy | Mimicking the one-dimensional marginal distributions of processes having an Itô differential[END_REF] shows that Q t .(x 0 ) and P t . are weakly right-continuous in

f ∈ C ∞ 0 (R d ) and any fixed γ ∈ C 1 ([0, T ]), t → Q t A(f γ)(x 0 ) and t → P t A(f γ) are bounded on [0, T ]. [11, Theorem 2.6] implies also that the set {A(f γ), f ∈ C ∞ 0 (R d ), γ ∈ C 1 ([0, T ])} is dense in C 0 b ([0, T ]×R d ) and
t on C 1 ([0, T ]) ⊗ C ∞ 0 (R d ), i.e, for T ≥ t ′ ≥ t: lim t ′ →t P t ′ (f γ) = P t (f γ) lim t ′ →t Q t ′ (f γ)(x 0 ) = Q t (f γ)(x 0 ). Since C 0 b ([0, T ] × R d ) is separating [11, Proposition 4.4, Chapter 3], Q t .( x
0 ) and P t . are weakly right-continuous and ( 14) holds for any λ > 0, we have h(y)q t (x 0 , dy) = h(y)p t (dy) for all h ∈ C 0 b (R d ). This ends the proof.

Remark 2.1. Assumptions 1, 2 and 3 are sufficient but not necessary for the well-posedness of the martingale problem. For example, the boundedness Assumption 1 may be relaxed to local boundedness, using localization techniques developed in [START_REF] Stroock | Diffusion processes associated with Lévy generators[END_REF][START_REF] Stroock | Multidimensional diffusion processes[END_REF]. Such extensions are not trivial and, in the unbounded case, additional conditions are needed to ensure that X does not explode (see [START_REF] Stroock | Multidimensional diffusion processes[END_REF]Chapter 10]).

Markovian projection of a semimartingale

We will make the following assumptions, which are almost-sure analogs of Assumptions 1, 2 and 3, on the local characteristics of the semimartingale ξ: Assumption 4. β, δ are bounded on [0, T ]:

∃K 1 > 0, ∀t ∈ [0, T ] β t ≤ K 1 , δ t ≤ K 1 a.s.
Assumption 5. µ has a density m(ω, t, dy) with respect to the Lebesgue measure on [0, T ] which satisfies

∃K 2 > 0, ∀t ∈ [0, T ] R d 1 ∧ y 2 m(., t, dy) ≤ K 2 < ∞ a.s.
and lim

R→∞ T 0 m (., t, { y ≥ R}) dt = 0 a.s. Assumption 6. Either (i) ∃ǫ > 0, ∀t ∈ [0, T [ t δ t δ t ≥ ǫ I d a.s.
or (ii) δ ≡ 0 and there exists β ∈]0, 2[, c, K 3 > 0, and a family m β (t, dy) of positive measures such that

∀t ∈ [0, T [ m(t, dy) = m β (t, dy) + c y d+β dy a.s., 1 ∧ y β m β (t, dy) ≤ K 3 , lim ǫ→0 y ≤ǫ y β m β (t, dy) = 0 a.s.
Note that Assumption 5 is only slightly stronger than stating that m is a Lévy kernel since in that case we already have 1 ∧ y 2 m(., t, dy) < ∞. Assumption 6 extends the "ellipticity" assumption to the case of pure-jump semimartingales and holds for a large class of semimartingales driven by stable or tempered stable processes.

Theorem 2 (Markovian projection). Define, for (t, z) ∈ [0, T ]×R d , B ∈ B(R d - {0}), b(t, z) = E [β t |ξ t -= z] , a(t, z) = E t δ t δ t |ξ t -= z , n(t, B, z) = E [m(., t, B)|ξ t -= z] . (15)
If (β, δ, n) satisfies Assumptions 4, 5, 6 and (b, a, n) satisfies Assumption 2 then there exists a Markov process ((X t ) t∈[0,T ] , Q ξ0 ), with infinitesimal generator L defined by (3), whose marginal distributions mimick those of ξ:

∀t ∈ [0, T ] X t d = ξ t .
X is the weak solution of the stochastic differential equation

X t = ξ 0 + t 0 b(u, X u ) du + t 0 Σ(u, X u ) dB u + t 0 y ≤1 y Ñ (du dy) + t 0 y >1 y N (du dy), (16)
where

(B t ) is an n-dimensional Brownian motion, N is an integer-valued ran- dom measure on [0, T ] × R d with compensator n(t, dy, X t-) dt, Ñ = N -n the associated compensated random measure and Σ ∈ C 0 ([0, T ]×R d , M d×n (R)) such that t Σ(t, z)Σ(t, z) = a(t, z).
We will call (X, Q ξ0 ) the Markovian projection of ξ.

Proof. First, we observe that n is a Lévy kernel : for any (t, z)

∈ [0, T ] × R d R d 1 ∧ y 2 n(t, dy, z) = E R d 1 ∧ y 2 m(t, dy)|ξ t -= z < ∞ a.s.,
using Fubini's theorem and Assumption 5. Consider now the case of a pure jump semimartingale verifying (ii) and define, for

B ∈ B(R d -{0}), ∀z ∈ R d n β (t, B, z) = E B m(t, dy, ω) - c dy y d+β |ξ t -= z .
As argued above, n β is a Lévy kernel on R d . Assumptions 4 and 5 imply that (b, a, n) satisfies Assumption 1. Furthermore, under assumptions either (i) or (ii) for (δ, m), Assumption 3 holds for (b, a, n). Together with Assumption 2 yields that L is a non-degenerate operator and Proposition 1 implies that the martingale problem for (L t ) t∈[0,T ] on the domain

C ∞ 0 (R d ) is well-posed. Denote ((X t ) t∈[0,T ] , Q ξ0 ) its unique solution starting from ξ 0 and q t (ξ 0 , dy) the marginal distribution of X t . Let f in C ∞ 0 (R d ). Itô's formula yields f (ξ t ) = f (ξ 0 ) + d i=1 t 0 d i=1 ∂f ∂x i (ξ s -) dξ i s + 1 2 t 0 tr ∇ 2 f (ξ s -) t δ s δ s ds + s≤t f (ξ s -+ ∆ξ s ) -f (ξ s -) - d i=1 ∂f ∂x i (ξ s -)∆ξ i s = f (ξ 0 ) + t 0 ∇f (ξ s -).β s ds + t 0 ∇f (ξ s -).δ s dW s + 1 2 t 0 tr ∇ 2 f (ξ s -) t δ s δ s ds + t 0 y ≤1 ∇f (ξ s -).y M (ds dy) + t 0 R d f (ξ s -+ y) -f (ξ s -) -1 { y ≤1} y.∇f (ξ s -) M (ds dy).
We note that

• since ∇f is bounded t 0 y ≤1 ∇f (ξ s -).y M (ds dy) is a square-integrable martingale. • t 0 y >1 ∇f (ξ s -).y M (ds dy) < ∞ a.s. since ∇f is bounded.
• since ∇f (ξ s -) and δ s are uniformly bounded on [0, T ], t 0 ∇f (ξ s -).δ s dW s is a martingale.

Hence, taking expectations, we obtain:

E P [f (ξ t )] = E P [f (ξ 0 )] + E P t 0 ∇f (ξ s -).β s ds + E P 1 2 t 0 tr ∇ 2 f (ξ s -) t δ s δ s ds + E P t 0 R d f (ξ s -+ y) -f (ξ s -) -1 { y ≤1} y.∇f (ξ s -) M (ds dy) = E P [f (ξ 0 )] + E P t 0 ∇f (ξ s -).β s ds + E P 1 2 t 0 tr ∇ 2 f (ξ s -) t δ s δ s ds + E P t 0 R d f (ξ s -+ y) -f (ξ s -) -1 { y ≤1} y.∇f (ξ s -) m(s, dy) ds .
Observing that:

E P t 0 ∇f (ξ s -).β s ds ≤ ∇f E P t 0 β s ds < ∞, E P 1 2 t 0 tr ∇ 2 f (ξ s -) t δ s δ s ≤ ∇ 2 f E P t 0 δ s 2 ds < ∞, E P t 0 R d f (ξ s -+ y) -f (ξ s -) -1 { y ≤1} y.∇f (ξ s -) m(s, dy) ds ≤ ∇ 2 f 2 E P t 0 y ≤1 y 2 m(s, dy) ds + 2 f E P t 0 y >1
m(s, dy) ds < +∞, we may apply Fubini's theorem to obtain

E P [f (ξ t )] = E P [f (ξ 0 )] + t 0 E P [∇f (ξ s -).β s ] ds + 1 2 t 0 E P tr ∇ 2 f (ξ s -) t δ s δ s ds + t 0 E P R d f (ξ s -+ y) -f (ξ s -) -1 { y ≤1} y.∇f (ξ s -) m(s, dy) ds.
Conditioning on ξ t-and using the iterated expectation property,

E P [f (ξ t )] = E P [f (ξ 0 )] + t 0 E P ∇f (ξ s -).E P [β s |ξ s-] ds + 1 2 t 0 E P tr ∇ 2 f (ξ s -) E P t δ s δ s |ξ s-] ds + t 0 E P E P R d f (ξ s -+ y) -f (ξ s -) -1 { y ≤1} y.∇f (ξ s -) m(s, dy)|ξ s-ds = E P [f (ξ 0 )] + t 0 E P [∇f (ξ s -).b(s, ξ s-)] ds + 1 2 t 0 E P tr ∇ 2 f (ξ s -) a(s, ξ s-) ds + t 0 R d E P f (ξ s -+ y) -f (ξ s -) -1 { y ≤1} y.∇f (ξ s -) n(s, dy, ξ s-) ds.
Hence

E P [f (ξ t )] = E P [f (ξ 0 )] + E P t 0 L s f (ξ s-) ds . (17)
Let p t (dy) denote the law of (ξ t ) under P, (17) writes:

R d p t (dy)f (y) = R d p 0 (dy)f (y) + t 0 R d p s (dy)L s f (y) ds. (18)
Hence p t (dy) satisfies the Kolmogorov forward equation ( 5) for the operator L with the initial condition p 0 (dy) = µ 0 (dy) where µ 0 denotes the law of ξ 0 . Applying Theorem 1, the flows q t (ξ 0 , dy) of X t and p t (dy) of ξ t are the same on [0, T ]. This ends the proof.

Remark 2.2 (Mimicking conditional distributions). The construction in Theorem 2 may also be carried out using [START_REF] Jacod | Calcul stochastique et problèmes de martingales[END_REF]. If (b 0 , a 0 , n 0 ) satisfies Assumption (3), then following the same procedure we can construct a Markov process (X, Q 0 ξ0 ) whose infinitesimal generator has coefficients (b 0 , a 0 , n 0 ) such that

b 0 (t, z) = E [β t |ξ t -= z, F 0 ] , a 0 (t, z) = E t δ t δ t |ξ t -= z, F 0 , n 0 (t, B, z) = E [m(., t, B)|ξ t -= z, F 0 ] . instead of (b, a, n) in
∀f ∈ C 0 b (R d ), ∀t ∈ [0, T ] E P [f (ξ t )|F 0 ] = E Q 0 ξ 0 [f (X t )] ,
i.e. the marginal distribution of X t matches the conditional distribution of ξ t given F 0 .

Remark 2.3. For Ito processes (i.e. continuous semimartingales of the form (2) with µ = 0), Gyöngy [12, Theorem 4.6] gives a "mimicking theorem" under the non-degeneracy condition t δ t .δ t ≥ ǫI d which corresponds to our Assumption 6, but without requiring the continuity condition (Assumption 2) on (b, a, n).

Brunick & Shreve [START_REF] Brunick | Matching statistics of an Ito process by a process of diffusion type[END_REF] extend this result by relaxing the ellipticity condition of [START_REF] Gyöngy | Mimicking the one-dimensional marginal distributions of processes having an Itô differential[END_REF]. In both cases, the mimicking process X is constructed as a weak solution to the SDE (16) (without the jump term), but this weak solution does not in general have the Markov property: indeed, it need not even be unique under the assumptions used in [START_REF] Gyöngy | Mimicking the one-dimensional marginal distributions of processes having an Itô differential[END_REF][START_REF] Brunick | Matching statistics of an Ito process by a process of diffusion type[END_REF]. In particular, in the setting used in [START_REF] Gyöngy | Mimicking the one-dimensional marginal distributions of processes having an Itô differential[END_REF][START_REF] Brunick | Matching statistics of an Ito process by a process of diffusion type[END_REF], the law of X is not uniquely determined by its 'infinitesimal generator' L. This makes it difficult to 'compute' quantities involving X, either through simulation or by solving a partial differential equation. By contrast, under the additional continuity condition 2 on the projected coefficients, X is a Markov process whose law is uniquely determined by its infinitesimal generator L and whose marginals are the unique solution of the Kolmogorov forward equation [START_REF] Brunick | Matching statistics of an Ito process by a process of diffusion type[END_REF]. This makes it possible to compute the marginals of X by simulating the SDE [START_REF] Jacod | Limit theorems for stochastic processes[END_REF] or by solving a forward PIDE.

It remains to be seen whether the additional Assumption 2 is verified in most examples of interest. We will show in Section 4 that this is indeed the case.

Remark 2.4 (Markovian projection of a Markov process). The term Markovian projection is justified by the following remark: if the semimartingale ξ is already a Markov process and satisfies the assumption of Theorem 2, then the uniqueness in law of the solution to the martingale problem for L implies that the Markovian projection (X, Q ξ0 ) of ξ has the same law as (ξ, P ξ0 ). So the map which associates (the law Q ξ0 of ) X to ξ may indeed be viewed as a projection; in particular it is involutive.

This property contrasts with other constructions of mimicking processes [START_REF] Baker | A brownian sheet martingale with the same marginals as the arithmetic average of geometric brownian motion[END_REF][START_REF] Cont | Recovering portfolio default intensities implied by CDO tranches[END_REF][START_REF] Gyöngy | Mimicking the one-dimensional marginal distributions of processes having an Itô differential[END_REF][START_REF] Hamza | A family of non-gaussian martingales with gaussian marginals[END_REF][START_REF] Madan | Making Markov martingales meet marginals[END_REF]] which fail to be involutive. A striking example is the construction, by Hamza & Klebaner [START_REF] Hamza | A family of non-gaussian martingales with gaussian marginals[END_REF], of discontinuous martingales whose marginals match those of a Gaussian Markov process.

Forward equations for semimartingales

Theorem 1 and Theorem 2 allow us to obtain a forward PIDE which extends the Kolmogorov forward equation to semimartingales which verify the Assumptions of Theorem 2: Theorem 3. Let ξ be a semimartingale given by (2) satisfying the assumptions of Theorem 2. Denote p t (dx) the law of ξ t on R d . t → p t is the unique solution, in the sense of distributions, of the forward equation

∀t ∈ [0, T ] ∂p t ∂t = L ⋆ t . p t , (19)
with initial condition p 0 = µ 0 , where µ 0 denotes the law of ξ 0 , where L ⋆ is the adjoint of L, defined by

∀g ∈ C ∞ 0 (R d , R), L ⋆ t g(x) = -∇ [b(t, x)g(x)] + ∇ 2 a(t, x) 2 g(x) (20)
+

R d g(x -z)n(t, z, x -z) -g(x)n(t, z, x) -1 z ≤1 z.∇ [g(x)n(t, dz, x)] ,
where the coefficients b, a, n are defined as in [START_REF] Jacod | Calcul stochastique et problèmes de martingales[END_REF].

Proof. The existence and uniqueness is a direct consequence of Theorem 1 and Theorem 2. To finish the proof, let compute L ⋆ t . Viewing p t as an element of the dual of

C ∞ 0 (R d ), ( 5
) rewrites : for f ∈ C ∞ 0 (R d , R) ∀f ∈ C ∞ 0 (R d , R), f (y) dp dt (dy) = p t (dy)L t f (y).
We have

∀f ∈ C ∞ 0 (R d ), ∀t ≤ t ′ < T < p t ′ -p t t ′ -t , f > t ′ →t → < p t , L t f >=< L * t p t , f >,
where < ., . > is the duality product.

For z ∈ R d , define the translation operator τ z by τ z f (x) = f (x + z). Then

p t (dx) L t f (x) = p t (dx) b(t, x)∇f (x) + 1 2 tr ∇ 2 f (x) a(t, x) + |z|>1 (τ z f (x) -f (x))n(t, dz, x) + |z|≤1 (τ z f (x) -f (x) -z.∇f (x)) n(t, dz, x) = -f (x) ∂ ∂x [b(t, x)p t (dx)] + f (x) ∂ 2 ∂x 2 [ a(t, x) 2 p t (dx)] + |z|>1 f (x)(τ -z (p t (dx)n(t, dz, x)) -p t (dx)n(t, dz, x)) + |z|≤1 f (x)(τ -z (p t (dx)n(t, dz, x)) -p t (dx)n(t, dz, x)) -z ∂ ∂x (p t (dx)n(t, dz, x)) ,
allowing to identify L ⋆ .

Martingale-preserving property

An important property of the construction of ξ in Theorem 2 is that it preserves the (local) martingale property: if ξ is a local martingale, so is X:

Proposition 2 (Martingale preserving property).

1. If ξ is a local martingale which satisfies the assumptions of Theorem 2, then its Markovian projection (X t ) t∈[0,T ] is a local martingale on (Ω 0 , B t , Q ξ0 ).

If furthermore

∀t ∈ [0, T ] E P y 2 µ(dt dy) < ∞, then (X t ) t∈[0,T ] is a square-integrable martingale.
Proof. 1) If ξ is a local martingale then the uniqueness of its semimartingale decomposition entails that

β t + y ≥1 y m(t, dy) = 0 dt × P -a.e. hence Q ξ0 ∀t ∈ [0, T ], t 0 ds b(s, X s-) + y ≥1 y n(s, dy, X s-) = 0 = 1.
The assumptions on m, δ then entail that X, as a sum of an Ito integral and a compensated Poisson integral, is a local martingale.

2) If E P y 2 µ(dt dy) < ∞ then

E Q ξ 0 y 2 n(t, dy, X t-) < ∞,
and the compensated Poisson integral in X is a square-integrable martingale.

3 Mimicking a semimartingale driven by a Poisson random measure

The representation (2) is not the most commonly used in applications, where a process is constructed as the solution to a stochastic differential equation driven by a Brownian motion and a Poisson random measure

∀t ∈ [0, T ] ζ t = ζ 0 + t 0 β s ds + t 0 δ s dW s + t 0 ψ s (y) Ñ (ds dy), (21)
where ξ 0 ∈ R d , W is a standard R n -valued Wiener process, β and δ are nonanticipative càdlàg processes, N is a Poisson random measure on [0, T ] × R d with intensity ν(dy) dt where

R d 1 ∧ y 2 ν(dy) < ∞, Ñ = N -ν(dy)dt, (22)
and the random jump amplitude ψ :

[0, T ] × Ω × R d → R d is P ⊗ B(R d )-
measurable, where P is the predictable σ-algebra on [0, T ] × Ω. In this section, we shall assume that ∀t ∈ [0, T ], ψ t (ω, 0) = 0 and

E t 0 R d 1 ∧ ψ s (., y) 2 ν(dy) ds < ∞.
The difference between this representation and (2) is the presence of a random jump amplitude ψ t (ω, .) in [START_REF] Mikulevičius | On the martingale problem associated with non-degenerate lévy operators[END_REF]. The relation between these two representations for semimartingales has been discussed in great generality in [START_REF] Karoui | Representation de processus ponctuels multivariés à l'aide d'un processus de poisson[END_REF][START_REF] Kabanov | On the representation of integral-valued random measures and local martingales by means of random measures with deterministic compensators[END_REF]. Here we give a less general result which suffices for our purpose. The following result expresses ζ in the form (2) suitable for applying Theorem 2.

Lemma 1 (Absorbing the jump amplitude in the compensator).

ζ t = ζ 0 + t 0 β s ds + t 0 δ s dW s + t 0 ψ s (z) Ñ (ds dz)
can be also represented as

ζ t = ζ 0 + t 0 β s ds + t 0 δ s dW s + t 0 y M (ds dy), ( 23
)
where M is an integer-valued random measure on [0, T ] × R d with compensator µ(ω, dt, dy) given by

∀A ∈ B(R d -{0}), µ(ω, dt, A) = ν(ψ -1 t (ω, A)) dt,
where ψ -1 t (ω, A) = {z ∈ R d , ψ t (ω, z) ∈ A} denotes the inverse image of A under the partial map ψ t .

Proof. The result can be deduced from [10, Théorème 12] but we sketch here the proof for completeness. A Poisson random measure N on [0, T ] × R d can be represented as a counting measure for some random sequence

(T n , U n ) with values in [0, T ] × R d N = n≥1 1 {Tn,Un} . (24)
Let M be the integer-valued random measure defined by:

M = n≥1 1 {Tn,ψT n (.,Un)} . (25)
µ, the predictable compensator of M is characterized by the following property Using formulae [START_REF]Markov Processes from Ito's Perspective[END_REF] and ( 25):

E t 0 A χ(s, y) M (ds dy) = E   n≥1 χ(T n , ψ Tn (., U n ))   = E t 0 ψ -1 s (.,A) χ(s, ψ s (., z)) N (ds dz) = E t 0 ψ -1 s (.,A)
χ(s, ψ s (., z)) ν(dz) ds Formula (26) and the above equalities lead to:

E t 0 A χ(s, y) µ(ds dy) = E t 0 ψ -1 s (.,A)
χ(s, ψ s (., z)) ν(dz) ds .

Since ψ is a predictable random function, the uniqueness of the predictable compensator µ (take φ ≡ Id in [16, Thm 1.8.]) entails

µ(ω, dt, A) = ν(ψ -1 t (ω, A)) dt. ( 27
)
Formula ( 27) defines a random measure µ which is a Lévy kernel

t 0 1 ∧ y 2 µ(dy ds) = t 0 1 ∧ ψ s (., y) 2 ν(dy) ds < ∞.
In the case where ψ t (ω, .) : R d → R d is invertible and differentiable, we can characterize the density of the compensator µ as follows:

Lemma 2 (Differentiable case). If the Lévy measure ν(dz) has a density ν(z) and if ψ t (ω, .) : [START_REF] Mikulevičius | On the martingale problem associated with non-degenerate lévy operators[END_REF], has the representation

R d → R d is a C 1 (R d , R d )-diffeomorphism, then ζ, given in
ζ t = ζ 0 + t 0 β s ds + t 0 δ s dW s + t 0 y M (ds dy),
where M is an integer-valued random measure with compensator

m(ω; t, y) dt dy = 1 ψt(ω,R d ) (y) |det∇ y ψ t | -1 (ω, ψ -1 t (ω, y)) ν(ψ -1 t (ω, y)) dt dy,
where ∇ y ψ t denotes the Jacobian matrix of ψ t (ω, .).

Proof. We recall from the proof of Lemma 1:

E t 0 A χ(s, y) µ(ds dy) = E t 0 ψ -1 s (.,A)
χ(s, ψ s (., z)) ν(z) ds dz .

Proceeding to the change of variable ψ s (., z) = y:

E t 0 ψ -1 s (.,A) χ(s, ψ s (., z)) ν(z) ds dz = E t 0 A 1 {ψs(R d )} (y) χ(s, y) |det∇ψ s | -1 (., ψ -1 s (., y)) ν(ψ -1 s (., y))ds dy .
The density appearing in the right hand side is predictable since ψ is a predictable random function. The uniqueness of the predictable compensator µ yields the result.

Let us combine Lemma 2 and Theorem 2. To proceed, we make a further assumption.

Assumption 7. The Lévy measure ν admits a density ν(y) with respect to the Lebesgue measure on R d and:

∀t ∈ [0, T ] ∃K 2 > 0 t 0 y >1
m(t, y) = 1 {y∈ψt(R d )} |det∇ψ t | -1 (ψ -1 t (y)) ν(ψ -1 t (y)), ( 28
) and b(t, z) = E [β t |ζ t -= z] , a(t, z) = E t δ t δ t |ζ t -= z , j(t, y, z) = E [m(t, y) |ζ t -= z] . (29)
If δ, m satisfy Assumptions 5-6 and Assumption 2 holds for (b, a, j), then the stochastic differential equation

X t = ζ 0 + t 0 b(u, X u ) du + t 0 Σ(u, X u ) dB u + t 0 y J(du dy), (30)
where (B t ) is an n-dimensional Brownian motion, J is an integer valued random measure on [0, T ] × R d with compensator j(t, dy, X t-) dt, J = J -j and

Σ : [0, T ] × R d → M d×n (R) is a continuous function such that t Σ(t, z)Σ(t, z) = a(t, z), admits a unique weak solution ((X t ) t∈[0,T ] , Q ζ0 ) whose marginal distri- butions mimick those of ζ: ∀t ∈ [0, T ] X t d = ζ t .
Under Q ζ0 , X is a Markov process with infinitesimal generator L given by (3).

Proof. We first use Lemma 2 to obtain the representation (23) of ζ:

ζ t = ζ 0 + t 0 β s ds + t 0 δ s dW s + t 0 y M (ds dy)
Then, we observe that ψ s (., y) 2 ν(y) dy ds.

Hence:

ζ t = ζ 0 + t 0 β s ds - t 0 ψs(y) >1 ψ s (.,

Examples

We now give some examples of stochastic models used in applications, where Markovian projections can be characterized in a more explicit manner than in the general results above. These examples also serve to illustrate that the continuity assumption (Assumption 2) on the projected coefficients (b, a, n) in ( 15) can be verified in many useful settings.

Semimartingales driven by a Markov process

In many examples in stochastic modeling, a quantity Z is expressed as a smooth function f : R d → R of a d-dimensional Markov process Z: ξ t = f (Z t ). We will show that in this situation our assumptions will hold as soon as Z has an infinitesimal generator whose coefficients satisfy Assumptions 1, 2 and 3. Consider a time-dependent integro-differential operator

L = (L t ) t∈[0,T ] defined, for f ∈ C ∞ 0 (R d ), by L t f (z) = b Z (t, z).∇f (z) + d i,j=1 Σ ij (t, x) 2
∂ 2 f ∂x i ∂x j (x) + R d [f (z + ψ(t, z, y) -f (z) -ψ(t, y, z).∇f (z)]ν(y)dy, (31) where
Σ : [0, T ] × R d → M d×d (R), b Z : [0, T ] × R d → R d and ψ : [0, T ] × R d × R d
are measurable functions and ν is a Lévy density. We assume that ψ(., ., 0

) = 0 ψ(t, z, .) is a C 1 (R d , R d ) -diffeomorphism ∃K 2 > 0 ∀t ∈ [0, T ] ∀z ∈ R d t 0 { y ≥1}
1 ∧ ψ(s, z, y) 2 ν(y) dy ds < K 2 .

18

(28) can then be expressed as

m Z (t, y, z) = 1 {y∈ψt(R d )} |det∇ψ| -1 (t, z, ψ -1 (t, z, y)) ν(ψ -1 (t, z, y)).
Consider the stochastic differential equation

∀t ∈ [0, T ] Z t = Z 0 + t 0 b Z (u, Z u-) du + t 0 Σ(u, Z u-) dW u + t 0 ψ(u, Z u-, y) Ñ (du dy), (32)
where (W t ) is an n-dimensional Brownian motion, N is a Poisson random measure on [0, T ] × R d with compensator ν(y) dy dt, Ñ the associated compensated random measure. Throughout this section we shall assume that (b Z , Σ, m Z ) satisfy Assumptions 1, 2 and 3. Proposition 1 then implies that for any Z 0 ∈ R d , the above SDE admits a weak solution ((Z t ) t∈[0,T ] , Q Z0 ), unique in law. Under Q Z0 , Z is a Markov process with infinitesimal generator L. Assume furthermore that Z t has a density q t with respect to the Lebesgue measure on R d . Consider now the process

∀t ∈ [0, T ] ξ t = f (Z t ) f : R d → R (33)
The aim of this section is to build in an explicit manner the Markovian Projection of ξ t for a sufficiently large class of functions f . Before stating the main result, we start with an useful Lemma, which will be of importance when one wants to characterize the yielding Markovian projection of ξ t .

Lemma 3. Let Z be a R d -valued random variable with density q(z) and

f ∈ C 1 (R d , R) such that ∀z ∈ R d , ∂f ∂z d (z) = 0. ( 34
)
Define the function

F : R d → R such that f (z 1 , • • • , z d-1 , F (z)) = z d . Then for any measurable function g : R d → R such that E [|g(Z)|] < ∞ and any w ∈ f (R d ), E [g(Z)|f (Z) = w] = R d-1 dz 1 ...dz d-1 g(z 1 , • • • , z d-1 , w) q(z1,••• ,z d-1 ,F (z1,••• ,z d-1 ,w)) ∂f ∂z d (z1,••• ,z d-1 ,F (z1,••• ,z d-1 ,w))
.

Proof. Consider the d-dimensional random variable κ(Z), where κ is defined by

κ(z) = (z 1 , • • • , z d-1 , f (z)) ,
and let us compute the law of κ(Z).

(∇ z κ) =      1 0 0 0 . . . . . . . . . . . . 0 • • • 1 0 ∂f ∂z1 • • • ∂f ∂z d-1 ∂f ∂z d      . One observes that |det(∇ z κ)|(z) = ∂f ∂z d (z) > 0. Hence κ is a C 1 (R d , R d )- diffeomorphism with inverse κ -1 . κ(κ -1 (z)) = (κ -1 1 (z), • • • , κ -1 d-1 (z), f (κ -1 1 (z), • • • , κ -1 d (z)) = z. For 1 ≤ i ≤ d-1, κ -1 i (z) = z i and f (z 1 , • • • , z d-1 , κ -1 d (z)) = z d that is κ -1 d (z) = F (z). Hence κ -1 (z 1 , • • • , z d ) = (z 1 , • • • , z d-1 , F (z)).
Define q κ (z) dz the inverse image of the measure q(z) dz under the partial map κ by

q κ (z) = 1 {κ(R d )} (z) |det(∇ z κ -1 )|(z) q(κ -1 (z)) = 1 {κ(R d )} (z) ∂f ∂z d -1 (z 1 , • • • , z d-1 , F (z)) q(z 1 , • • • , z d-1 , F (z)).
q κ (z) is the density of κ(Z). So, for any w ∈ f (R d ),

E [g(Z)|f (Z) = w] = E [g(Z)|κ(Z) = (z 1 , • • • , z d-1 , w)] = R d-1 dz 1 ...dz d-1 g(z 1 , • • • , z d-1 , w) q(z 1 , • • • , z d-1 , F (z 1 , • • • , z d-1 , w)) ∂f ∂z d (z 1 , • • • , z d-1 , F (z 1 , • • • , z d-1 , w))
.,

We can now formulate the main result of this section:

Theorem 5. Let f ∈ C 2 (R d , R) with bounded derivatives such that ∀z ∈ R d ∂f ∂z d (z) = 0. (35) Define, for w ∈ f (R d ), t ∈ [0, T ], b(t, w) = R d-1 ∇f (.).b Z (t, .) + 1 2 tr ∇ 2 f (.) t Σ(t, .)Σ(t, .) + R d (f (. + ψ(t, ., y)) -f (.) -ψ(t, ., y).∇f (.)) ν(y) dy, (z 1 , • • • , z d-1 , w) × q t (z 1 , • • • , z d-1 , F (z 1 , • • • , z d-1 , w)) ∂f ∂z d (z 1 , • • • , z d-1 , F (z 1 , • • • , z d-1 , w)) ., σ(t, w) = R d-1 ∇f (.)Σ(t, .) 2 (z 1 , • • • , z d-1 , w) × q t (z 1 , • • • , z d-1 , F (z 1 , • • • , z d-1 , w)) ∂f ∂z d (z 1 , • • • , z d-1 , F (z 1 , • • • , z d-1 , w)) . 1/2 . ( 36
)
and the measure j(t, du, w) defined on R -{0} by

j(t, [u, ∞[, w) = R d-1 R d 1 {f (.+ψ(t,.,y))-f (.)≥u} (z 1 , • • • , z d-1 , w) ν(y) dy × q t (z 1 , • • • , z d-1 , F (z 1 , • • • , z d-1 , w)) ∂f ∂z d (z 1 , • • • , z d-1 , F (z 1 , • • • , z d-1 , w)) . ( 37
)
for u > 0 and analogously for u < 0. Then the stochastic differential equation

X t = ξ 0 + t 0 b(s, X s ) ds + t 0 σ(s, X s ) dB s + t 0 y ≤1 y J(ds dy) + t 0 y >1 y J(ds dy), (38)
where (B t ) is a Brownian motion, J is an integer-valued random measure on [0, T ] × R with compensator j(t, du, X t-) dt, J = J -j, admits a weak solution ((X t ) t∈[0,T ] , Q ξ0 ), unique in law, whose marginal distributions mimick those of ξ:

∀t ∈ [0, T ] X t d = ξ t .
Under Q ξ0 , X is a Markov process with infinitesimal generator L given by

∀f ∈ C ∞ 0 (R),L t f (w) = b(t, w)f ′ (w) + σ 2 (t, w) 2 f ′′ (w) + R d [f (w + u) -f (w) -uf ′ (w)]j(t, du, w).
Proof. Applying Itô's formula to f (Z t ) yields

ξ t = ξ 0 + t 0 ∇f (Z s -).b Z (s, Z s-) ds + t 0 ∇f (Z s -).Σ(s, Z s-)dW s + 1 2 t 0 tr ∇ 2 f (Z s-) t Σ(s, Z s-)Σ(s, Z s-) ds + t 0 ∇f (Z s -).ψ(s, Z s-, y) Ñ (ds dy) + t 0 R d (f (Z s -+ ψ(s, Z s-, y)) -f (Z s -) -ψ(s, Z s-, y).∇f (Z s -)) N (ds dy) = ξ 0 + t 0 ∇f (Z s -).b Z (s, Z s-) + 1 2 tr ∇ 2 f (Z s-) t Σ(s, Z s-)Σ(s, Z s-) + R d (f (Z s -+ ψ(s, Z s-, y)) -f (Z s -) -ψ(s, Z s-, y).∇f (Z s -)) ν(y) dy ds + t 0 ∇f (Z s -).Σ(s, Z s-)dW s + t 0 R d (f (Z s -+ ψ(s, Z s-, y)) -f (Z s -)) Ñ (
κ z (y) : R d → R d y → (y 1 , • • • , y d-1 , f (z + y) -f (z)). Since for any z ∈ R d , |det∇ y κ z | (y) = ∂ ∂y d f (z + y) > 0, one can define κ -1 z (y) = (y 1 , • • • , y d-1 , F z (y)) F z (y) : R d → R f (z+(y 1 , • • • , y d-1 , F z (y)))-f (z) = y d .
Considering φ the inverse of ψ that is φ(t, ψ(t, z, y), z) = y, define Φ(t, z, y) = φ(t, z, κ -1 z (y)).

Φ corresponds to the inverse of Ψ and Φ is differentiable on R d with image R d . Define the random measure m(t, z, y) dt dy m(t, z, y) = |det∇ y Φ(t, z, y)| ν(Φ(t, z, y))

= det∇ y φ(t, z, κ -1 z (y))

∂f ∂y d (z + κ -1 z (y)) -1
ν(φ(t, z, κ -1 z (y))).

Using Assumption 7,

t 0 y >1
1 ∧ Ψ(s, z, y) 2 ν(y) dy ds

= t 0 y >1 1 ∧ ψ 1 (s, z, y) 2 + • • • + ψ d-1 (s, z, y) 2 + (f (z + ψ(s, z, y)) -f (z)) 2 ν(y) dy ds ≤ t 0 y >1 1 ∧ ψ 1 (s, z, y) 2 + • • • + ψ d-1 (s, z, y) 2 + ∇f 2 ψ(s, z, y) 2 ν(y) dy ds ≤ t 0 y >1 1 ∧ (2 ∨ ∇f 2 ) ψ(s, z, y) 2 ν(y)
ξ t = ξ 0 + t 0 β s ds + t 0 δ s dB s + t 0 u K(ds du), where          β t = ∇f (Z t -).b Z (t, Z t-) + 1 2 tr ∇ 2 f (Z t-) t Σ(t, Z t-)Σ(t, Z t-) + R d (f (Z t -+ ψ(t, Z t-, y)) -f (Z t -) -ψ(t, Z t-, y).∇f (Z t -)) ν(y) dy, δ t = ∇f (Z t-)Σ(t, Z t-) , and K is an integer-valued random measure on [0, T ] × R with compensator k(t, Z t-, u) du dt defined for all z ∈ R d via k(t, z, u) = R d-1 m(t, (y 1 , • • • , y d-1 , u), z) dy 1 • • • dy d-1 = R d-1 |det∇ y Φ(t, z, (y 1 , • • • , y d-1 , u))| ν(Φ(t, z, (y 1 , • • • , y d-1 , u))) dy 1 • • • dy d-1 , (39)
and K its compensated random measure. In particular for any u > 0,

k(t, z, [u, ∞[) = R d 1 {f (z+ψ(t,z,y))-f (z)≥u} ν(y) dy. (40)
Let us show that if (b Z , Σ, ν) satisfy Assumptions 1, 2 and 3 then the triplet (δ t , β t , k(t, Z t-, u)) satisfies the assumptions of Theorem 2. First, note that β t and δ t satisfy Assumption 4 since b Z (t, z) and Σ(t, z) satisfy Assumption 1 and ∇f and ∇ 2 f are bounded. As argued before, one sees that if Σ is non-degenerate then δ t is. In the case δ t ≡ 0, for t ∈ [0, T [, R > 0, z ∈ B(0, R) and g ∈ C 0 0 (R) ≥ 0, consider C and

K T > 0 chosen via Assumption 3 k(t, z, u) = R d-1 |det∇ y Φ(t, z, (y 1 , • • • , y d-1 , u))| ν(Φ(t, z, (y 1 , • • • , y d-1 , u))) dy 1 • • • dy d-1 = R d-1 det∇ y φ(t, z, κ -1 z (y 1 , • • • , y d-1 , u)) ∂f ∂y d (z + κ -1 z (y 1 , • • • , y d-1 , u)) -1 ν(φ(t, z, κ -1 z (y 1 , • • • , y d , u))) dy 1 • • • dy d-1 ≥ R d-1 ∂f ∂y d (z + κ -1 z (y 1 , • • • , y d-1 , u)) -1 C κ -1 z (y 1 , • • • , y d-1 , u) d+β dy 1 • • • dy d-1 = R d-1 C (y 1 , • • • , y d-1 , u) d+β dy 1 • • • dy d-1 = 1 |u| d+β R d-1 C (y 1 /u, • • • , y d-1 /u, 1) d+β dy 1 • • • dy d-1 = C ′ 1 |u| 1+β , with C ′ = R d-1 C (w 1 , • • • , w d-1 , 1) -1 dw 1 • • • dw d-1 . Similarly 1 ∧ |u| β k(t, z, u) - C ′ |u| 1+β du = 1 ∧ |u| β R d-1 ∂f ∂y d (z + κ -1 z (y 1 , • • • , y d-1 , u)) -1 det∇ y φ(t, z, κ -1 z (y 1 , • • • , y d-1 , u)) ν(φ(t, z, κ -1 z (y 1 , • • • , y d-1 , u))) - C κ -1 z (y 1 , • • • , y d-1 , u) d+β dy 1 • • • dy d-1 du = R d 1 ∧ |f (z + (y 1 , • • • , y d-1 , u)) -f (z)| β |det∇ y φ(t, z, (y 1 , • • • , y d-1 , u))| ν(φ(t, z, (y 1 , • • • , y d-1 , u))) - C (y 1 , • • • , y d-1 , u) d+β dy 1 • • • dy d-1 du ≤ R d 1 ∧ ∇f (y 1 , • • • , y d-1 , u) β |det∇ y φ(t, z, (y 1 , • • • , y d-1 , u))| ν(φ(t, z, (y 1 , • • • , y d-1 , u))) - C (y 1 , • • • , y d-1 , u) d+β dy 1 • • • dy d-
(t, w) = E [β t |f (Z t-) = w] , σ(t, w) = E δ 2 t |f (Z t-) = w 1/2 , j(t, u, w) = E [k(t, Z t-, u)|f (Z t-) = w] .
Applying Lemma 3, one can compute explicitly the conditional expectations above. For example, Proceeding similarly, one can show that that Assumption 2 holds for σ and j so Theorem 2 may be applied to yield the result.

b(t, w) = R d-1 ∇f (.).b Z (t, .) + 1 2 tr ∇ 2 f (.) t Σ(t, .)Σ(t, .) + R d (f (. + ψ(t, ., y)) -f (

Time changed Lévy processes

Models based on time-changed Lévy processes have been the focus of much recent work especially in mathematical finance [START_REF] Carr | Stochastic volatility for Lévy processes[END_REF]. Let L t be a Lévy process, (b, σ 2 , ν) be its characteristic triplet, N its jump measure. Define The impact of the random time change on the marginals can be captured by making the characteristics state dependent ( bα(t, X t-), σ 2 α(t, X t-), α(t, X t-)ν ) by introducing the same adjustment factor α(t, X t-) to the drift, diffusion coefficient and Lévy measure. In particular if α(t, x) is affine in x we get an affine process [START_REF] Duffie | Transform Analysis and Asset Pricing for Affine Jump-diffusions[END_REF] where the affine dependence of the characteristics with respect to the state are restricted to be colinear, which is rather restrictive. This remark shows that time-changed Lévy processes, which in principle allow for a wide variety of choices for θ and L, may not be as flexible as apparently simpler affine models when it comes to reproducing marginal distributions.

X t =

  ) and , n(t, B, .) are continuous on R d , uniformly in t ∈ [0, T ]. Assumption 3 (Non-degeneracy). Either ∀R > 0 ∀t ∈ [0, T ] inf z ≤R inf x∈R d t x.a(t, z).x > 0 or a ≡ 0 and there existsβ ∈]0, 2[, C > 0, and a family n β (t, dy, z)

[ 16 ,

 16 Thm 1.8.]: for any positive P ⊗ B(R d )-measurable map χ : [0, T ]× Ω× R d → R + and any A ∈ B(R d -{0}), y) N (ds dy) = E t 0 B χ(s, y) ν(dy) ds .

  ds dy), where the terms above are well-defined thanks to Assumption 7. Lemma 2 leads to:

t 0 1 ∧ 0 1 0 1

 100 u 2 k(s, Z s-, u) du ds = t ∧ |f (Z s -+ ψ(s, Z s-, y)) -f (Z s -)| 2 ν(y) dy ds ≤ t ∧ ∇f 2 ψ(s, Z s-, u) 2 ν(y)dy ds, is bounded by Assumption 7. Hence k satisfies Assumption 5.

1 0 k

 10 du is also bounded. Similar arguments would show that lim ǫ→0 |u|≤ǫ |u| β k(t, Z t-, u) -C |u| 1+β du = 0 a.s. and lim R→∞ T (t, Z t-, {|u| ≥ R}) dt = 0 a.s., since this essentially hinges on the fact that f has bounded derivatives. Define as in Theorem 2 b

  .) -ψ(t, ., y).∇f (.)) ν(y) dy, (z1 , • • • , z d-1 , w) × q t (z 1 , • • • , z d-1 , F (z 1 , • • • , z d-1 , w)) | ∂f ∂z d (z 1 , • • • , z d-1 , F (z 1 , • • • , z d-1 , w))| . with F : R d → R defined by f (z 1 , • • • , z d-1 , F (z)) = z d .Since f is C 2 with bounded derivatives, (b Z , Σ, ν) satisfy Assumption 2 and (t, z) → q t (z) is continuous in (t, z) on [0, T ] × R d (see Theorem 1) then b(t, .) is continuous on R uniformly in t ∈ [0, T ]. Hence Assumption 2 holds for b.

  y) 2 ν(y) dy ds + This representation has the form (2) and Assumptions 4 and 7 guarantee that the local characteristics of ζ satisfy the assumptions of Theorem 2. Applying Theorem 2 yields the result.

					t
					δ s dW s
					0
	+	t	y M (ds dy) +	t	y M (ds dy).
	0	y ≤1	0	y >1

  ds dy).

	is a continuous local martingale with [B] t = t. Lévy's theorem implies that B
	is a Brownian motion.
	If Σ ≡ 0 and (ii) holds, then ξ t is a pure-jump semimartingale. Define K t
	K t =	t	Ψ(s, Z s-, y) Ñ (ds dy),
	0
	with Ψ(t, z, y) = ψ(t, z, κ z (y)) where
	If Σ satisfies assumption (i) then (B t ) t∈[0,T ] defined by
	dB t =	∇f (Z t -).Σ(t, Z t-)W t ∇f (Z t-)Σ(t, Z t-)	,

  dy ds is bounded. One may apply Lemma 2 and express K t as K t = Extracting the d-th component of K t , one obtains the semimartingale decomposition of ξ t on [0, T ]

	t 0 where M is a compensated integer-valued random measure on [0, T ] × R d with y M (ds dy)
	compensator m(t, Z t-, y) dy dt.

  L Θt Θ t =where (θ t ) is a locally bounded F t -adapted positive cadlag process, interpreted as the rate of time change.Theorem 6 (Markovian projection of time-changed Lévy processes). Let L be a scalar Lévy process with triplet (b, σ 2 , ν) and letξ t = L Θt with Θ t = t 0 θ s ds where θ t > 0 is a positive semimartingale. Define ∀t ∈ [0, T ] ∀z ∈ R α(t, z) = E[θ t |ξ t-= z],and suppose that α(t, .) is continuous onR d , uniformly in t ∈ [0, T ]. Assume that lim R→∞ ν ({ y ≥ R}) = 0and either (i) or (ii) holds for (σ, θ t ν), then• (ξ t ) has the same marginals as (X t ) on [0, T ], defined as the weak solution ofX t = ξ 0 +with the given initial condition p 0 (dy) = µ 0 (dy) where µ 0 denotes the law of ξ 0 .Proof. Consider the Lévy-Ito decomposition of L:L t = bt + σW t + (Θ t) is a continuous martingale starting from 0, with quadratic variation Θ t = t 0 θ s ds. The Dubins-Schwarz theorem (see[START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] Theorem 1.7]) implies that one can pick Z a Brownian motion independent of W , such that W (Θ t ) Hence X t is the weak solution of :X t = X 0 +Using the notations of Theorem 2, β t = b θ t , δ t = σ θ t , m(t, dy) = θ t ν(dy).

						d =	t	θ s dZ s .
						0
				t		t
				σ θ s dZ s +	bθ s ds
				0		0
	+	t		t zθ s Ñ (ds dz) +	t	t zθ s N (ds dz).
	0		|z|≤1		σ α(s, X s-)dB s + 0 |z|>1	bα(s, X s-)ds
					0	0
			+	t	z J(ds dz) +	t	zJ(ds dz),
				0		|z|≤1	0	|z|>1
						forward
	equation:					∂p t ∂t	= L ⋆ t . p t ,
	where, L * t is given by
	L ⋆ t g(x) = -b	∂ ∂x	[α(t, x)g(x)] +	σ 2 2	∂ 2 ∂x 2 [α 2 (t, x)g(x)]
	+
	R d
					t	z Ñ (dsdz) +	t	zN (dsdz).
					0	|z|≤1	0	|z|>1
	Then ξ rewrites					t	θ s ds,
	ξ t = ξ 0 + σW (Θ t ) + bΘ t	0
	+	Θt			z Ñ (ds dz) +	θt	zN (ds dz).
	0		|z|≤1	0	|z|>1

where B t is a real-valued brownian motion, J is an integer-valued random measure on [0, T ] × R with compensator α(t, X t-) ν(dy) dt.

• The marginal distribution p t of ξ t is the unique solution of the

ν(dz) g(x -z)α(t, x -z) -g(x)α(t, x) -1 z ≤1 z. ∂ ∂x [g(x)α(t, x)] ,

W Since Assumptions 4 and 5 are satisfied and :

b(t, .) = E [β t |ξ t -= .] = b α(t, .), σ(t, .) = E δ 2 t |ξ t -= . 1/2 = σ α(t, .), n(t, B, .) = E [m(t, B)|ξ t -= .] = α(t, .)ν(B),

are all continuous on R uniformly in t on [0, T ]. One may apply Theorem 2.