Decomposition of stochastic flows generated by Stratonovich SDEs with jumps

Consider a manifold $M$ endowed locally with a pair of complementary distributions $\Delta^H \oplus \Delta^V=TM$ and let $\text{Diff}(\Delta^H, M)$ and $\text{Diff}(\Delta^V, M)$ be the corresponding Lie subgroups generated by vector fields in the corresponding distributions. We decompose a stochastic flow with jumps, up to a stopping time, as $\varphi_t = \xi_t \circ \psi_t$, where $\xi_t \in \text{Diff}(\Delta^H, M)$ and $\psi_t \in \text{Diff}(\Delta^V, M)$. Our main result provides Stratonovich stochastic differential equations with jumps for each of these two components in the corresponding infinite dimensional Lie groups. We present an extension of the It\^o-Ventzel-Kunita formula for stochastic flows with jumps generated by classical Marcus equation (as in Kurtz, Pardoux and Protter, Annales de L'I.H.P. section B, 1995, among others). The results here correspond to an extension of Catuogno, da Silva and Ruffino, Stoch. Dyn. 2013, where this decomposition was studied for the continuous case.


Introduction
Let M be a compact differentiable manifold, with Diff(M ) the corresponding infinite dimensional Lie group of diffeomorphisms of M generated by the Lie algebra of smooth vector fields. Consider a stochastic flow ϕ t of local diffeomorphisms in M , generated by a Stratonovich SDE driven by a semimartingale Z t with jumps. The question we address in this paper is the possibility of decomposing ϕ t in components that belong to specific subgroups of Diff(M), which provide dynamical or geometrical information of the system. In the continuous case, decompositions of this kind for stochastic flows has been studied in the literature, with many distinct frameworks and with different aimed subgroups. Among others see Bismut [2], Kunita [8], [9], Ming Liao [11] and some of our previous work [3], [6], [13]. In the last few papers mentioned, for example, geometrical conditions on a Riemannian manifold have been stated to guarantee the existence of the decomposition where the first component lies in the subgroups of isometries or affine transformations.
The geometrical framework here is the following: consider that the manifold M is equipped with a pair of complementary distributions ∆ H and ∆ V (horizontal and vertical, respectively) in the sense of sections in Grassmanian bundles of M . In this context, two Lie subgroups of Diff(M ) appear naturally: Diff(∆ H , M ) and Diff(∆ V , M ), the set of diffeomorphisms generated by vector fields in the corresponding distributions. We want to decompose the original flow, up to a stopping time, as ϕ t = ξ t • ψ t , where ξ t ∈ Diff(∆ H , M ) and ψ t ∈ Diff(∆ V , M ). Our main result provides Stratonovich stochastic differential equations with jumps for each of these two components in the corresponding infinite dimensional Lie groups (Theorem 4.5). One of the key points on obtaining the equations for each component ξ t and ψ t is an extension of the classical Itô-Ventzel-Kunita formula for discontinuous càdlàg processes, which are also generated by SDEs with jumps (Theorem 3.2); here, we generalize the definition of the integration with respect to a change of variables in Kurtz, Pardoux and Protter [10].
A motivation for this decomposition appears in many dynamical systems. For instance, in a foliated space (say, assuming that the horizontal distribution is integrable) the decomposition measures how far the trajectories are from being foliation-preserving: namely, the closer the second component ψ t is to the identity, closer the system is to be (horizontal) foliation-preserving. Vice-versa, changing the order of the vertical and horizontal distributions and the order of the components in the decomposition, the same interpretation holds if the vertical component is integrable. For example: a principal bundle with a connection is a natural state space with this feature, where the vertical distribution is determined by the fibres and the horizontal distribution is given by the geometry. The dynamics in averaging principles for transversal perturbation in foliated systems also illustrates the relevance of this decomposition, see e.g. the Liouville torus in Hamiltonian systems in Li [12], or in a general compact foliation in Gargate and Ruffino [7].
In this paper, we consider the Stratonovich SDEs driven by semimartingales with jumps Z t , in the sense of the so called Marcus equation (see e.g. [10]). This means, among other aspects (recalled in Section 2), that the jumps of the solution occur in the direction of the deterministic flow generated by the corresponding vector fields. This property guarantees that if the manifold M is embedded in an Euclidean space and the vector fields are in T M then, after the jumps, the solution flow remains in M . Also, we assume that the number of jumps in Z t is finite in a bounded interval of time a.s.. Hence, it includes as a possible integrator the so called Lévy-jump diffusion (see Applebaum [1]), but not Lévy process in general. This idea of finite jumps in bounded intervals has parallel in the theory of chain control sets, see Colonius and Kliemann [5], Patrão and San Martin [14], and references therein.
The paper is organized as follows: In Section 2, we describe some basic properties and definitions of Stratonovich SDE driven by a semimartingale with jumps, in the sense of Marcus equation. In Section 3, our extension of the Itô-Ventzel-Kunita formula is presented, for two stochastic flows generated by Marcus equations with respect to the same semimartingale with jumps. Finally, in section 4, we deal with the decomposition of the stochastic flow ϕ t = ξ t • ψ t , obtaining the corresponding SDEs for each component. The results here correspond to an extension of the article Catuogno, da Silva and Ruffino [4], where the same decomposition is studied for continuous flows.

Stratonovich SDEs with jumps
For reader's convenience, we recall the definition of a Stratonovich SDE driven by a semimartingale with jumps in R d , in the sense of Marcus equation (for more details, see e.g. Kurtz, Pardoux e Protter [10]): with initial condition x(0) = x 0 . The solution, in its integral form is: where x t is an adapted stochastic process taking values in R d , the integrator {Z i s : s ≥ 0} is a semimartingale with jumps and X i are vector fields in R d for all i ∈ {0, 1, . . . , m}. For sake of simplicity we use the matricial notation X = (X 1 , . . . , X m ) and Z = (Z 1 , . . . , Z m ). The solution is interpreted as a stochastic process that satisfies the equation: in the following sense: the first term on the right hand side of (3) is a standard Itô integral of the predictable process X(x t − ) with respect to the semimartingale Z t . The second term is a Stieltjes integral with respect to the continuous part of the quadratic variation of Z t . In the third term: φ(X∆Z s , x s − ) indicates the solution at time one of the ODE generated by the vector field X∆Z s and initial condition x s − . Thus, the jumps of this equation occurs in deterministic directions. Regularity conditions on vector fields X implies that there exists a unique stochastic flow of diffeomorphisms ϕ t . Conditions on the derivatives of the vector fields guarantee that there exists a stochastic flow of local diffeomorphisms. Moreover, for an embedded submanifold M in an Euclidean space, if the vector fields of the equation (1) are in T M then, for each initial condition on M , the solution stays in M a.s., see [10].
Let Y be smooth vector fields in R d , and consider G t a d-dimensional process where φ(Y ∆Z s , G s− , u) has the same interpretation as before: it is the solution of the ODE generated by the vector field Y ∆Z s , and initial condition G s− , at time t = u. More specifically, given t 0 > 0, the jump of the integral on the left hand side of equation (4) is given by: The dynamical interpretation of the expression above is that, after opening a unit interval with a 'fictitious curve' which connects X(G t 0− ) and X(G t 0 ), the jump of the integral is given by the mean of X along this curve multiplied by the jump of the semimartingale Z t , see [10].
We consider that the number of jumps in the integrator process Z t is finite in bounded intervals a.s.. For example, this hypothesis includes the so called Levy- Brownian motion, N t is a Poisson process and the random variables (J k ) are i.i.d. (see e.g. Applebaum [1]).

An extension of Itô-Ventzel-Kunita formula
In the continuous case, we have the well known Itô-Ventzel-Kunita formula: Let ϕ t be a stochastic flow of (local) diffeomorphisms, which is solution of the following Stratonovich SDE where N i are continuous semimartingales, and X i are smooth vector fields in R d , be a continuous semimartingale. Then: For a proof, see e.g. Kunita [9, Thm 8.3]). A direct corollary of this result is a Leibniz formula for the composition of stochastic flows of diffeomorphisms in the continuous case, that is, where the differential form is considered in the Stratonovich sense.
In the proof of the main result of the next section, one needs an extension of this Leibniz formula for flows generated by Stratonovich SDE driven by semimartingales with jumps. Precisely, let ψ t and ξ t be flows of diffeomorphisms generated by Marcus equations with respect to the same semimartingale with jumps Z t : where X, Y are smooth vectors fields. We assume that Z t has the property of finite jumps, as described in the previous section. We define the following integral which generalizes the classical Marcus integral (3), (i.e. [10, Eq. 2.2]): The first term on the right hand side of (6) is the standard Itô integral of the predictable process ψ s * (Y (ξ s − )). The second term corresponds to the finite variation, such that the continuous part of the expression satisfies the classical Itô-Ventzel-Kunita formula (see, e.g. Kunita [8,Thm. 8.3]). In the summation, we use the notation φ(X, x 0 , u) for the solution of ordinary differential equation with respect to the vector field X, and initial condition x 0 at time u (if omitted, it means that u = 1). Note that if ψ s is the identity for all s ∈ [0, t], one recovers equation (3).
With the same notation as before, our extension of Itô-Ventzel-Kunita formula is given by Proof. Consider t 0 = sup {s ∈ [0, a] such that the formula holds in [0, s]}. We want to prove that t 0 = a. The proof follows by contradiction considering two cases: when the driven process Z t is continuous at t 0 , or when it jumps at t 0 . In the first case, if t 0 < a, there exists a.s. an ε > 0 such that the process Z t is continuous in [t 0 , t 0 + ε). So, by Itô-Ventzel formula for the continuous case (see e.g. Theorem 8.3 in [9]), the relation is valid up to time t 0 + ε, hence we have a contradiction.
On the other hand, if the process Z t jumps at t 0 , and t 0 < a, we prove that the relation is also valid for t 0 plus some positive ε. In fact: So, the formula also holds for t = t 0 . By assumption, there exists a. s. an ε > 0 such that Z t is continuous in (t 0 , t 0 + ε). Applying again the classical Itô-Ventzel-Kunita formula, we have a contradiction. Hence t 0 = a.
Next corollary contains the Leibniz formula which we are going to use to obtain the SDEs for the components of our decomposition in each corresponding Lie group. Its proof follows straightforward from Theorem 3.2. Corollary 3.3. Let ψ t and ξ t be flows generated by Marcus equations with respect to the same semimartingale Z t , with the property of finite jumps. Therefore:

Decomposition of stochastic flows with jumps
In this section, following the framework of [10], we assume that the manifold M is embedded in an Euclidean space. Hence, smooth vector fields in M can be extended to smooth vector fields in a tubular neighbourhood of M . Let ϕ t be a stochastic flow of (local) diffeomorphisms in M generated by a Stratonovich SDE in the sense of Marcus equation, where the integrator has the property of finite jumps, as described in the previous section.
Assume that the manifold M has (locally) a pair of differentiable distributions, that we call horizontal and vertical. We denote the first one by ∆ H : U ⊆ M → Gr k (M ) and the second by ∆ V : U ⊆ M → Gr n−k (M ). Also, we assume that the horizontal and vertical distributions are complementary, that is, for each x ∈ U ,

Existence of the decomposition
We are interested in decomposing the stochastic flow as ϕ t = ξ t • ψ t , where the components ξ t ∈ Diff(∆ H , M ) and ψ t ∈ Diff(∆ V , M ). Initially, consider the following definitions:  A straightforward extension of this result holds for a stochastic flow ϕ t generated by a Stratonovich SDE with respect to a semimartingale Z t with jumps: Moreover, if t 0 is a point of discontinuity of Z t , t 0 ≤ τ and the decomposition holds for ϕ t 0 , then t 0 < τ , a.s..
Proof. By assumption, the first discontinuity of the integrator Z t happens at a stopping time t 1 > 0 a.s.. Hence, Theorem 4.3 guarantees the existence of the decomposition at least up to t 1 . For the last statement, take δ > 0 such that the integrator Z t is continuous in (t 0 , t 0 + δ) and apply again Theorem 4.3.
In the last statement of the Propostition above, the hypothesis on the decomposition of the diffeomorphism ϕ t 0 cannot be removed: In fact, it may happen that the decomposition holds up to the time t − 0 , and the process jumps to a non decomposable diffeomorphism ϕ t 0 . As a basic example to illustrate this fact, consider the canonical horizontal and vertical foliations in R 2 . Take the pure rotation system dx t = Ax t ⋄ dZ t , where Z t is a semimartingale such that Z t / ∈ {π/2 + kπ, k ∈ Z} for t < t 0 . The corresponding flow and its decompostion for t < t 0 is given by If at time t 0 the process jumps to Z t 0 ∈ {π/2 + kπ, k ∈ Z}, then ϕ t 0 is not decomposable. On the other hand, if the stochastic flow jumps to a decomposable diffeomorphism ϕ t 0 , then the decomposition holds at least an additional δ > 0 longer than t 0 . In Morgado and Ruffino [12], we have used this fact to construct a stochastic flow ϕ t generated by a Stratonovich SDE with controlled jumps that approximates arbitrarily a stochastic flow ϕ t generated by a continuous Stratonovich SDE, with the property that ϕ t can be decomposable for all t > 0.

Stratonovich SDEs with jumps for the components
Using the extension of Itô-Ventzel-Kunita formula for Stratonovich SDE with jumps, one can obtain the correspondent SDEs (in the sense of Marcus equation) for the components ξ t and ψ t in the decomposition above. Initially, we have that ϕ t satisfies the following right invariant equation in the Lie group terminology: where R ϕt * is the derivative of the right translation of ϕ t in the identity of Diff(M). In fact, for all i ∈ {0, 1, . . . , m}, it holds that R ϕt * X i = X i (ϕ t ), since if γ is a curve in Diff(M) such that γ(0) = Id and γ ′ (0) = X i , we have: By assumption in Proposition 4.4, we consider that ∆ H and ∆ V preserves transversality along Diff(∆ H , M ). So, for each ξ ∈ Diff(∆ H , M ), the distributions ∆ H and Ad(ξ)∆ V are complementary. For each x ∈ U , we have a unique decomposition of X i (x) in the corresponding subspaces, given by: . Now, consider the autonomous Stratonovich SDE in the subgroup Diff(∆ H , M ), given by: Note that, by construction, we have that v i (ξ t , x) ∈ Ad(ξ t )∆ V (x). In this sense, for all x ∈ U , and using support theorem again, one conclude that ψ t ∈ Diff(∆ V , M ). So, with the same notation as before, we have proved the following result: Theorem 4.5. The components ξ t and ψ t in the decomposition above can be described, respectively, as the solutions of the following Stratonovich SDE's with jumps:

Compatibility of the vector fields and the pair of distributions
In some geometrical structures where we have a sort of compatibility of the vector fields with the pair of complementary distributions, it is possible to guarantee that the decomposition holds for all positive time t ≥ 0.
Given an initial condition x 0 ∈ M , consider a local coordinate system α 0 : With respect to these coordinate systems one writes the original flow as ϕ t = (ϕ 1 t (x, y), ϕ 2 t (x, y)).
Proposition 4.6. Suppose that the decomposition ϕ t = ξ t • ψ t holds up to the stopping time τ in a neighbourhood of the initial condition x 0 . Writing in coordinates, τ = sup{t > 0; det ∂ϕ 2 t (x, y) ∂y = 0}.
Proof. Straightforward by the inverse function theorem: The local decomposition exists if and only if the (n − p) × (n − p) matrix ∂ϕ 2 (x,y) ∂y is invertible. That is, ϕ = (ξ 1 (x, y), Id 2 ) • (Id 1 , ϕ 2 (x, y)), where Id 1 and Id 2 are the identities in R p and in R n−p , respectively.
The proposition above says that the hypothesis for the decomposition can be weakened to conditions only on the original flow ϕ t instead of using the transversality preserving property (Definition 4.2). Corollary 4.7. Suppose that the vertical distribution is integrable (a foliation). If the the original flow ϕ t sends each vertical leaf into a vertical leaf, then the decomposition ϕ t = ξ t • Ψ t holds for all time t ≥ 0.
Proof. Straightforward from the proposition above since, for all t ≥ 0, Example 1. Consider in R n \ {0} the distribution ∆ H given by the spherical foliation and the distribution ∆ V given by the radial foliation. Then, for this pair of distribution, any linear system ϕ t has the decomposition ϕ t = ξ t • ψ t for all t ≥ 0 since each radial line is sent to a radial line.
Example 2. Most of relevant dynamics in fibre bundles send fibre into another fibre. The fibres generate the vertical distributions, the horizontal distributions are given by connections. Hence the decomposition for these flows with respect to these pairs of distributions exists for all t ≥ 0. A standard example in this context is the linearized flow ϕ t * : T x 0 M → T ϕt(x 0 ) in the linear frame bundle π : BM → M . By the Corollary above, the decomposition ϕ t * = ξ t • ψ t holds for all t ≥ 0 since ϕ t * is an isomorphism between the fibres π −1 (x 0 ) and π −1 (ϕ t (x 0 )).