Existence and uniqueness of solutions of SDEs with discontinuous drift and finite activity jumps

In this letter we prove existence and uniqueness of strong solutions to multi-dimensional SDEs with discontinuous drift and finite activity jumps.


Introduction
On a filtered probability space (Ω, F, F = (F t ) t∈[0,T ] , P) that satisfies the usual conditions we consider a time-homogeneous stochastic differential equation (SDE) with finite activity jumps, where µ : R d → R d , σ : R d → R d×d , and ρ : R d × R → R d are measurable functions, T ∈ (0, ∞), and W = (W t ) t∈[0,T ] is a d-dimensional standard Brownian motion. Furthermore, ν is a Poisson random measure with finite jump intensity, associated with a scalar compound Poisson process. We assume that both W and ν are adapted to F.
In case the coefficient µ is Lipschitz, it is known that SDE (1) admits a unique strong solution, cf. [14,p. 79,Theorem 117] or [9,Theorem 1.9.3]. However, there are applications where jump SDEs with discontinuous drift appear. For example control actions on energy markets often lead to discontinuities in the drift of the controlled process, cf., e.g., [12,13]. So to be able to study control problems on energy markets in the most common models for energy prices, i.e. jump SDE models, existence and uniqueness needs to be settled.
In this paper we provide sufficient conditions for the existence and uniqueness of strong solutions to (1), where we allow µ to be discontinuous.
In the jump-free case, existence and uniqueness results for strong solutions to SDEs with discontinuous drift have been provided in the classical papers [18,15,16,17] and in the more recent papers [6,12,3,4]. In the case of presence of jumps in the driving process, to the best of our knowledge the only result so far is [11,Theorem 3.1] for the scalar case with Poisson jumps, but the focus of the latter paper is on numerics.

Assumptions
Assumption 2.1. The jump measure ν is a Poisson random measure with finite jump intensity, associated with a compound Poisson process L = (L t ) t∈[0,T ] , that is L t = Nt k=1 ξ k , where N = (N t ) t∈[0,T ] is a Poisson process and (ξ k ) k∈N is a family of iid random variables independent of N with associated distribution φ that has finite second moments and P(ξ k = 0) = 0 for all k ∈ N.
Since the process L has only a finite number of jumps in any interval (0, t], we get for all ρ(X s− , y)ν(dy, ds)ds.
In order to define assumptions on the drift coefficient, we recall the following definitions from differential geometry.
The intrinsic metric ρ on A is given by the piecewise Lipschitz constant of f .
We will assume that the drift µ is piecewise Lipschitz with exceptional set Θ, where Θ is a fixed, sufficiently regular hypersurface, see Assumption 2.2 below.
If Θ ∈ C 4 , locally there exists a unit normal vector, that is a continuously differentiable C 3 function n : D ⊆ Θ → R d (cf. [2, Theorem 4.1]) such that for every ζ ∈ U , n(ζ) = 1, and n(ζ) is orthogonal to the tangent space of Θ in ζ. Recall the following definition.
Therefore, we can define a mapping p : Θ ε → Θ assigning to each x the point p(x) on Θ, which is closest to x.

2.
A set Θ is said to be of positive reach, if there exists ε ∈ (0, ∞) such that Θ ε has the unique closest point property. The reach r Θ of Θ is the supremum over all such ε if such an ε exists, and 0 otherwise.
We make the following assumptions on the functions µ, σ, ρ: and every unit normal vector n of Θ has bounded second and third derivative.
(loc. bound) The coefficients µ and σ satisfy (techn. ass.) The function is well defined, bounded, and belongs to C 3 b (Θ; R d ).

Preliminary results
As a preliminary for the proof of our main result on existence and uniqueness of solutions of SDE (1), we need to adopt the transformation introduced in [4] and recall its properties. Furthermore, we require an Itô-type formula for this transform and its inverse.

The transform
In the proof of our main result, we will apply a transform G : R d → R d that has the property that the process formally defined by Z = G(X) satisfies an SDE with Lipschitz coefficients and therefore has a solution by classical results, see [9, Theorem 1.9.3].
The function G is chosen so that it impacts the coefficients of the SDE (1) only locally around the points of discontinuity of the drift. This behaviour is ensured by incorporating a bump function into G. Adopted from [4], G is given by with r Θ > ε 0 > 0, see Assumption 2.2, α as in Assumption 2.2, and with a constant c ∈ (0, ∞) and φ : R → R, We recall the following properties of G. Thereby, we denote G ′ := ∇G and by G ′′ the Hessian of G.
(ii) G ′ is Lipschitz, G ′′ exists on R d \ Θ and is piecewise Lipschitz with exceptional set Θ; (iii) G ′ and G ′′ are bounded; (iv) for c sufficiently small, see [5,Lemma 1], G is globally invertible; (v) G and G −1 are Lipschitz continuous.

A change of variable formula
We require a multidimensional Itô-type formula, which is in a very specific sense slightly more general than [8, Theorem 3.1], and which is, up to our best knowledge, novel. Note that if the Itôtype formula holds for the scalar components of G = (G 1 , . . . , G d ) and G −1 = (G −1 1 , . . . , G −1 d ), then it also holds for G, G −1 .
There exists an open set U ⊆ Θ ε 0 with y 0 ∈ U such that there exists ψ U ∈ C 2 (R d−1 , R) with the property that all y = (y 1 , . . . , y d−1 , y d ) ∈ Θ∩U can be represented as y = (y 1 , . . . , y d−1 , ψ U (y 1 , . . . , y d−1 )), or such that we can rotate the state space in a smooth way so that the representation holds.
Hence without loss of generality we assume that such a rotation is not necessary. This means that the set of discontinuities of f ′ is locally given by some {y d = ψ U (y 1 , . . . , y d−1 )}.
Since ψ U ∈ C 2 (R d−1 , R), it holds that ψ U (Y 1 , . . . , Y d−1 ) is a semimartingale. Due to the properties of f from Lemma 3.1, on these U the Itô-type formula [8,Theorem 3.1] holds. Since f ′ is a continuous function, the first order terms in [8, equation (3.6)] reduce to the form of the first order terms in (3). The same holds for all mixed derivatives except ∂ 2 f Furthermore, since f ′′ is bounded, we may apply the local time formula from [10, Chapter IV, Corollary 1, p. 219], which says that formally the integral of )} with respect to the quadratic variation of Y d from [8, equation (3.6)] can be expressed as space integral of the same function times a local time. Since for all s ∈ (0, t], P(Y s ∈ Θ) = 0, we have

Existence and uniqueness result
In this section we are going to prove our main result. Proof. The idea of the proof is to construct a process Z for which we can verify that G −1 (Z) solves (1). For this we will first heuristically apply the Itô-type formula from Proposition 3.2 to G componentwise. Observe that Combining this with Proposition 3.2 yields (G(X s− + ρ(X s− , y)) − G(X s− )) ν(dy, ds).

So we have
Now we define the process Z = G(X), which for all t ∈ [0, T ] is given by where for all y ∈ R, z ∈ R d In [4] it is shown thatμ andσ are Lipschitz. Due to the global Lipschitz continuity of G and G −1 , the linear growth of G, G −1 , and ρ, and the finiteness of the second moment of ξ 1 we get that there exists cρ ∈ (0, ∞) such that for all z, z 1 , z 2 ∈ R, R ρ(z 1 , y) −ρ(z 2 , y) 2 φ(dy) ≤ cρ z 1 − z 2 2 , R ρ(z, y) 2 φ(dy) ≤ cρ(1 + z 2 ).
This closes the proof.

Examples
We mention the two most important classes of jump processes that are covered in our setup.
Example 5.1 (Poisson process). Let ξ k = 1 for all k ∈ N and for all y ∈ R let ρ(x, y) = ρ(x), i.e. ρ depends only on the first variable. Therefore, Theorem 4.1 holds for the case that the jump process governing (1) is a Poisson process, that is for all t ∈ [0, T ], Example 5.2 (Compound Poisson process). Now let ρ(x, y) = y · ρ(x). Therefore, Theorem 4.1 also holds for the case that the jump process governing (1) is a compound Poisson process, that is for all t ∈ [0, T ],