Stochastic invariance of closed sets for jump-diffusions with non-Lipschitz coefficients

We provide necessary and sufficient first order geometric conditions for the stochastic invariance of a closed subset of R^d with respect to a jump-diffusion under weak regularity assumptions on the coefficients. Our main result extends the recent characterization proved in Abi Jaber, Bouchard and Illand (2016) to jump-diffusions. We also derive an equivalent formulation in the semimartingale framework.


Introduction
We consider a weak solution to the following stochastic differential equation with jumps dX t = b(X t )dt + σ(X t )dW t + ρ(X t− , z) (µ(dt, dz) − F (dz)dt) , X 0 = x, (1.1) that is: a filtered probability space (Ω, F, F = (F) t≥0 , P) satisfying the usual conditions and supporting a d-dimensional Brownian motion W , a Poisson random measure µ on R + × R d with compensator dt ⊗ F (dz), and a F-adapted process X with càdlàg sample paths such that (1.1) holds P-almost surely.
Throughout this paper, we assume that b : R d → R d , σ : R d → M d and ρ : R d × R d → R d are measurable, where M d denotes the space of d × d matrices. In addition, we assume that b, σ and ρ(., z) H(ρ(., z))ρ(., z)F (dz) are continuous for any We also assume that there exist q, L > 0 such that, for all x ∈ R d , ρ(x, z) q ln ρ(x, z) F (dz) ≤ L(1 + x q ), Let D denote a closed subset of R d . Our aim is to characterize the stochastic invariance (a.k.a viability) of D under weak regularity assumptions, i.e. find necessary and sufficient conditions on the coefficients such that, for all x ∈ D, there exists a D-valued weak solution to (1.1) starting at x.
Invariance and viability problems have been intensively studied in the literature, first in a deterministic setup [3] and later in a random environment. For the diffusion case, see [1,9,4] and the references therein. In the presence of jumps, we refer to [22,24,14]. Note that a first order characterization for a smooth volatility matrix σ is given in [14], where the Stratonovich drift appears (see [9] for the diffusion case). For a second order characterization, we refer to [24,Propositions 2.13 and 2.15].
Combining the techniques used in [1,24], we derive for the first time in Theorem 2.2 below, a first order geometric characterization of the stochastic invariance with respect to (1.1) when the volatility matrix σ can fail to be differentiable. We also provide an equivalent formulation of the stochastic invariance with respect to semimartingales in Theorem 3.2. This extends [1] to the jump-diffusion case. From a practical perspective, this is the first known first order characterization that could be directly applied to construct affine [11,18] and polynomial processes [8] on any arbitrary closed sets, since for these processes the volatility matrix can fail to be differentiable (on the boundary of the domain).
In fact, in the sequel, we only make the following assumption on the covariance matrix C := σσ on D can be extended to a C 1,1 in which C 1,1 loc means C 1 with a locally Lipschitz derivative and S d denotes the set of d × d symmetric matrices. Note that we do not impose the extension of C to be positive semi-definite outside D, so that σ might only match with its square-root on D. Also, it should be clear that the extension needs only to be local around D.
From now on we use the same notation C for σσ on D and for its extension defined in Assumption (H 2 ). All identities involving random variables have to be considered in the a.s. sense, the probability space and the probability measure being given by the context. Elements of R d are viewed as column vectors. We use the standard notation I d to denote the d × d identity matrix and denote by M d the collection of d × d matrices. We say that A ∈ S d (resp. S d + ) if it is a symmetric (resp. and positive semi-definite) element of M d . Elements of R d are viewed as column vectors. Given x = (x 1 , . . . , x d ) ∈ R d , diag [x] denotes the diagonal matrix whose i-th diagonal component is x i . If A is a symmetric positive semi-definite matrix, then A 1 2 stands for its symmetric square-root.
The rest of the paper is organized as follows. Our main results are stated and proved in Sections 2-3. In the Appendix, we adapt to our setting some technical results, mainly from [1].

Stochastic invariance for SDEs
In order to ease the comparison with [1], we first provide in Theorem 2.2 below a characterization of the invariance for stochastic differential equations with jumps. An equivalent formulation in terms of semimartingales is also provided in the next section (see Theorem 3.2 below). We insist on the fact that the two formulations are equivalent by the representation theorem of semimartingales with characteristics as in (3.1) below in terms of a Brownian motion and a Poisson random measure (see [ We start by making precise the definition of stochastic invariance 1 for stochastic differential equations with jumps.

Definition 2.1 (Stochastic invariance).
A closed subset D ⊂ R d is said to be stochastically invariant with respect to the jump-diffusion (1.1) if, for all x ∈ D, there exists a weak solution X to (1.1) starting at X 0 = x such that X t ∈ D for all t ≥ 0, almost surely.
The following theorem provides a first order geometric characterization of the stochastic invariance using the (first order) normal cone N D (x) at x consisting of all outward pointing vectors, for all x ∈ D and u ∈ N D (x), in which DC j (x) denotes the Jacobian of the j-th column of C(x) and (CC + ) j (x) is the j-th column of (CC + )(x) with C(x) + defined as the Moore-Penrose pseudoinverse 2 of C(x). Figure 1: Interplay between the geometry/curvature of D and the coefficients (b, C, ρ).
Before moving to the proof, we start by giving the geometric interpretation of conditions (2.1a)-(2.1d), also shown in Figure 1. Condition (2.1c) states that at the 1 The concept is also often known as viability. We use the term invariance here in order to stay coherent with the affine/polynomial literature. 2 The Moore-Penrose pseudoinverse of a m×n matrix A is the unique n×m matrix A + satisfying: AA + A = A, A + AA + = A + , AA + and A + A are Hermitian. boundary of the domain, the column of the covariance matrix should be tangential to the boundary, while (2.1a) requires from D to capture all the jumps of the process. Moreover, at the boundary, the jumps can have infinite variation only if they are tangent to the boundary, by (2.1b). Finally, it follows from (2.1d) that the compensated drift should be inward pointing. We notice that the compensated drift extends the Stratonovich drift (see [9,14]) when the volatility matrix can fail to be differentiable. In fact, if the volatility matrix is smooth, [ , for all x ∈ D and u ∈ Ker σ(x) .
Conversely, the example of the square root process C(x) = x and σ(x) = √ x on D := R + shows that σ may fail to be differentiable at 0 while C satisfies (H 2 ).
The proof of Theorem 2.2 adapts the argument of [1] combined with techniques taken from [24] to handle the jump component. For the necessity, we use the same conditioning/projection argument together with the small time behavior of double stochastic integrals as in [1]. For this we need to inspect the regularity of σ, this is the object of Lemma 2.3 below. For the sufficiency, we show that conditions (2.1a)-(2.1d) imply the positive maximum principle for the infinitesimal generator and we conclude by applying [13,Theorem 4.5.4], which is possible by Lemma 2.4 below. The latter lemma highlights the role of the growth condition (H 0 ). In fact, (H 1 ) would only yield that Lφ is bounded. This is not enough to apply [13,Theorem 4.5.4].
We first recall the following crucial lemma. This is an immediate consequence of the implicit function theorem giving the regularity of the distinct eigenvalues of C and their corresponding eigenvectors under (H 2 ). We refer to [1, Lemma 3.1] for the proof.
. Let x ∈ D be such that the spectral decomposition of C(x) is given by Then there exist an open (bounded) neighborhood N (x) of x and two measurable M d -valued functions on R d , We will also need the continuity of the infinitesimal generator of (1.1) acting on smooth functions φ where Dφ (resp. D 2 φ) is the gradient (resp. Hessian) of φ. In the sequel, we denote by C(D) the space of continuous functions on D. We add the superscript p on C to denote functions with p-continuous derivatives for all p ≤ ∞, and the subscript c (resp. 0) stands for functions with compact support (resp. vanishing at infinity). This is the object of the following lemma (a similar formulation in the semimartingale set-up can be found in [23, Lemma A.1]).
Observe that I 2 (x, y) → 0 when y → x, since D 2 φ is uniformly continuous (recall that φ has compact support). In addition, it follows from (H C ) that I 1 (x, y) → Φ(x) when y → x, which ends the proof.
We can now move to the proof of Theorem 2.2.
Proof of Theorem (2.2). Part a. We first prove that our conditions are necessary. Let X denote a weak solution starting at X 0 = x such that X t ∈ D for all t ≥ 0. If x / ∈ ∂D, then N D (x) = {0} and there is nothing to prove. We therefore assume from now on that x ∈ ∂D. Let 0 < η < 1. Throughout the proof, we fix ψ η a bounded continuous function is the open ball with center x and radius η.
Step 2. By the proof of [1, Proposition 3.5], it suffices to consider the case where the positive eigenvalues of the covariance matrix C at the fixed point x ∈ D are all distinct as in Lemma 2.3. We can also restrict the study to σ = C for the jump part. Fix u ∈ N D (x) and let φ be a smooth function (with compact support in N (x)) such that max , for all t ≥ 0. Let w η := (η − 1)ψ η . By reapplying Step 1, with the test function φ (resp. w η ) instead of φ (resp. ψ η ), we obtain Let (FB s ) s≥0 be the completed filtration generated byB. SinceB,B ⊥ are independent andB has independent increments, conditioning by FB t yields, by Lemma A.3 in the appendix, We now apply Lemma A.1 of the Appendix to (Dφσ)(X) and reapply the same conditioning argument to find a bounded adapted process η such that Step 3. We now check that we can apply Lemma A.2 below. First note that all the above processes are bounded. This follows from Lemmas 2.3 and 2.4, (H 1 ) and the fact that φ has compact support. In addition, given T > 0, the independence of the increments ofB implies that θ s = E FB T Lφ(X s ) for all s ≤ T . From Lemma 2.4 and since X has almost surely no jumps at 0, it follows that θ is a.s. continuous at 0. Moreover, since Dφσ is C 1,1 , D(Dφσ)σ is Lipschitz which, combined with (A.5), implies (A.2).

Part b.
We now prove that our conditions are sufficient. It follows from (2.1c) and the proof of [1, Proposition 4.1] that for any smooth function φ such that max D φ = φ(x) ≥ 0. Moreover, after noticing that In addition, it follows from (2.1a) that φ(x + ρ(x, z)) ≤ φ(x) for F -almost all z. Combining all the above with (2.1d) we finally get

Equivalent fomulation in the semimartingale framework
In this section, we provide an equivalent formulation of Theorem 2.2 in the semimartingale set-up which is more adapted to the construction of affine and polynomial jump-diffusions (see Remark 3.3 below). We stress once more that, by [12,5], (1.1) is a very general formulation, equivalent to the semimartingale formulation (3.2) below (see also [16,   The triplet ( b, c, K) is called the differential characteristics of X. In addition we assume that there exist q, L > 0 such that for all x ∈ R d . It follows that X is a locally square-integrable semimartingale (see [ where X c is a continuous local martingale with quadratic variation X c · = · 0 c(X s )ds and B := dz). Finally, we assume that the restriction of c to D can be extended to a C 1,1 and we denote by C this extended function.
We are now ready to state an equivalent formulation of Theorem 2.2 adapted to (3.2). We start by defining naturally the notion of stochastic invariance with respect to a semimartingale.

Definition 3.1 (Stochastic invariance).
A closed subset D ⊂ R d is said to be stochastically invariant with respect to the semimartingale (3.1) if, for all x ∈ D, there exists a filtered probability space (Ω, F, F := (F t ) t≥0 , P) supporting a semimartingale X with characteristics (3.1) starting at X 0 = x and such that X t ∈ D for all t ≥ 0, P-almost surely.
for all x ∈ D and u ∈ N D (x).

A Technical lemmas
For completeness, we provide in the sequel some technical lemmas with their proofs. They are either standard or minor modifications of already known results.
The generalized Itô's lemma derived in [1,Lemma 3.3] can easily be extended to account for jumps in the following way.
Lemma A.1. Assume that σ is continuous and that there exists a solution X to (1.1). Let f ∈ C 1,1 c (R d , R). Then, there exists an adapted bounded process η such that Proof. Since f ∈ C 1,1 has a compact support, we can find a sequence (f n ) n in C ∞ with compact support (uniformly) and a constant K > 0 such that for all n ≥ 1. This is obtained by considering a simple mollification of f . Set µ := µ − dtF (dz). Since f n is twice differentiable and bounded, Itô's formula [17,Theorem I.4.57] yields in which η n := 1 2 Tr[D 2 f n σσ ](X). Since σσ is continuous, (i) above implies that (η n ) n is uniformly bounded in L ∞ (dt×dP). By [10,Theorem 1.3], there exists (η n ) ∈ Conv(η k , k ≥ n) such thatη n → η dt ⊗ dP almost surely. Let N n ≥ 0 and (λ n k ) n≤k≤Nn ⊂ [0, 1] be such thatη n = Nn k=n λ n k η k and Nn k=n λ n k = 1. Setf n := Nn k=n λ n k f k . Then, in which η n := (Df n b)(X s )+η n s + f n (X s + ρ(X s , z)) −f n (X s ) − Df n (X s )ρ(X s , z) F (dz).
in L 2 (Ω, F, P) as n → ∞, and therefore a.s. after possibly considering a subsequence. It thus remains to send n → ∞ in (A.1) to obtain the required result.
t 0 γ s 2 ds < ∞, for all t ≥ 0, We also used the following elementary lemma which extends [25,Lemma 5.4] to account for jumps (see also [21, Corollaries 2 and 3 of Theorem 5.13]). Lemma A.3. Let B, B ⊥ denote two independent Brownian motions and µ a Poisson random measure on R + × R d with compensator dt ⊗ F on a filtered probability space (Ω, F, (F t ) t≥0 , P). Let (γ s ) s≥0 be an adapted square integrable process and ξ : R + ×R d → R d be a predictable process such that E t 0 ξ(s, z) 2 F (dz)ds < ∞, for all t ≥ 0. Define the sub-filtration F B t = σ{B s , s ≤ t} ⊂ F t and denote by µ = µ − dtF (dz). Then P − a.s., for all t ≥ 0, Moreover, it holds similarly for any integrable adapted process θ that For general ξ, the result follows from Itô's isometry and the fact that simple processes are dense in L 2 (dt ⊗ F ) (see [ for all s, t ≤ 1. (A.5) Proof. Set g t := E sup s≤t X s 2 . By convexity of y → y 2 , we have (a + b + c + d) 2 = 16( a+b+c+d 4 ) 2 ≤ 4(a 2 + b 2 + c 2 + d 2 ). Combined with Cauchy-Schwarz and Burkholder-Davis-Gundy inequalities, we get for all u ≤ t g u ≤ 4 x 2 + 4t