A higher order approximation method for jump-diffusion SDEs with discontinuous drift coefficient

We present the first higher-order approximation scheme for solutions of jump-diffusion stochastic differential equations with discontinuous drift. For this transformation-based jump-adapted quasi-Milstein scheme we prove $L^p$-convergence order 3/4. To obtain this result, we prove that under slightly stronger assumptions (but still weaker than anything known before) a related jump-adapted quasi-Milstein scheme has convergence order 3/4 - in a special case even order 1. Order 3/4 is conjectured to be optimal.


Introduction
We consider time-homogeneous jump-diffusion stochastic differential equations (SDEs) of the form dX t = µ(X t ) dt + σ(X t ) dW t + ρ(X t− ) dN t , t ∈ [0, T ], where ξ ∈ R, µ, σ, ρ : R → R are measurable functions, T ∈ (0, ∞), W = (W t ) t≥0 is a standard Brownian motion, and N = (N t ) t≥0 is a homogeneous Poisson process with intensity λ ∈ (0, ∞) on the probability space (Ω, F, P).We define the filtration F = (F t ) t≥0 as the augmented natural filtration of N and W , i.e. for all t ≥ 0 we set F t = σ({(W s , N s ) : s ≤ t} ∪ N ), where N = {A ∈ F : P(A) = 0}.With this, (Ω, F, F, P) is a filtered probability space that satisfies the usual conditions.Existence and uniqueness of the solution of the above SDE under our assumptions is well settled by [49,50].This paper aims to provide a higher order scheme for SDE (1) for the case where the drift is allowed to be discontinuous.This setting is, for example, relevant for modeling energy prices or control actions on energy markets ( [1,53,54]) as well as for modeling physical phenomena, cf.[55,44].
We would like to highlight the contributions that were most influential for the current paper: In [44] jump-adapted schemes are studied.Transformation-based schemes have first been introduced in [25,26] for the case without jumps (ρ ≡ 0), where transformation-based Euler-Maruyama schemes as well as a proof technique have been introduced, which inspired many of the subsequent contributions.A transformation-based quasi-Milstein scheme for the case without jumps (ρ ≡ 0) has been introduced in [34]; it has convergence order 3/4 in L p , which is optimal for the class of all algorithms with deterministic grid points, cf.[32].Approximation of discontinuous-drift-SDEs with jumps has been studied in [49].
In the current paper we construct the first higher-order scheme for jump-diffusion SDEs with discontinuous drift -a transformation-based jump-adapted quasi-Milstein scheme.Our main contribution is to prove convergence order 3/4 in L p , p ∈ (1, ∞) for this scheme.This is conjectured to be optimal.As a side result we prove that under slightly stronger assumptions (but still weaker than anything known in the literature) a related jump-adapted quasi-Milstein scheme has convergence order 3/4; in a special case even order 1.
For the proof we proceed in two steps.First we introduce the jump-adapted quasi-Milstein scheme and prove convergence order 3/4 (respectively 1) under the stronger assumptions.Then we make use of a transformation and the previous result to prove our target result in Theorem 4.4.

Setting
In the following we denote by Ñ = ( Ñt ) t≥0 the compensated Poisson process, i.e.Ñt = N t − λt for all t ∈ [0, ∞).It is well-known that Ñ is a square integrable F-martingale.Further we denote by L f the Lipschitz constant of a generic Lipschitz continuous function f , and by Id the identity mapping.
(iii) The jump coefficient ρ : R → R is Lipschitz continuous.
The following lemma is a direct consequence of Assumption 2.1.
Lemma 2.2 ([49, Lemma 2]).Let Assumption 2.1 hold.Then µ, σ, and ρ satisfy a linear growth condition, that is there exist constants c µ , c σ , c ρ ∈ (0, ∞) such that The existence and uniqueness of the strong solution of the SDE (1) is guaranteed by the following result.3 Convergence of the jump-adapted quasi-Milstein scheme We are going to introduce the jump-adapted quasi-Milstein scheme and prove under stronger assumptions than Assumption 2.1 convergence order 3/4, respectively order 1.This result in combination with a transformation technique will be used later to prove our main result.In this section we consider the time-homogeneous jump-diffusion SDE where ξ ∈ R, µ, σ, ρ : R → R are measurable functions, T ∈ (0, ∞), W = (W t ) t≥0 , N = (N t ) t≥0 , and (Ω, F, F, P) are defined as in Section 2.
Assumption 3.1.We assume on the coefficients of SDE ( 2) that In the following lemma we state some direct consequences of Assumption 3.1.
(i) Then there exists a constant c f ∈ (0, ∞) such that for all x ∈ R, (iii) There exists a constant b g ∈ (0, ∞) such that for all x, y ∈ R for which g is differentiable on [x, y] it holds that Now we define our jump-adapted time discretisation.For this, let δ = T M for all M ∈ N and define the equidistant time discretisation (s m ) m∈{0,...,M } by s m = δm for m ∈ {0, . . ., M }.We add to these deterministic points the family (ν i ) i∈N of all jump times of the Poisson process.
Then we order all points so that they are increasing, yielding the time discretisation (τ n ) n∈N with the maximum step size less or equal than δ.Formally, we define (τ n ) n∈N as follows: Given a time grid (τ n ) n∈N we define the time-continuous jump-adapted quasi-Milstein scheme (Z and for all n ∈ N 0 by and or equivalently Between the points of the time discretisation, i.e. for t ∈ τ n , τ n+1 , n ∈ N 0 , we set Note that the following integral notation is equivalent to ( 5), ( 6), (7): Similarly, we can express the solution on SDE (2) by To simplify the notation we denote for given M ∈ N and time point t ∈ [0, T ], the largest value of the sequence (s m ) m=0,...,M which is smaller or equal t, by t,

Preparatory lemmas
In this section we provide some basic properties of the time discretisation, moment estimates, and Markov properties.We start providing some basic properties of the time discretisation (τ n ) n∈N .We consider two filtrations, which we both need later on.In addition to F, we define the filtration F = ( F t ) t≥0 for all t ∈ [0, ∞) by Recall that N is the set of all nullsets of F. Because W and N are independent W is also a Brownian motion with respect to the filtration ( F t ) t≥0 .Further it is obvious that Z (M ) as well as Z are F -adapted càdlàg processes, since for all t ∈ [0, ∞), F t ⊂ F t .
In the following lemma we summarise some obvious properties.
Lemma 3.2.Let M ∈ N and let the sequence (τ n ) n∈N be defined by (3) and (4).Then for all n ∈ N, (i) τ n is an F-stopping time and also an F-stopping time; (ii) Z (M ) τn is F τn -measurable and also F τn -measurable; (iii) (W τn+s − W τn ) s≥0 is a Brownian motion with respect to the filtration (F τn+s ) s≥0 , (N τn+s − N τn ) s≥0 is a Poisson process with respect to the filtration (F τn+s ) s≥0 , both processes are independent of F τn , and it holds that (W τn+s −W τn ) s≥0 is independent of (N τn+s −N τn ) s≥0 ; (iv) (W τn+s − W τn ) s≥0 is a Brownian motion with respect to the filtration ( F τn+s ) s≥0 , which is independent of F τn .
Next we provide moment bounds for the jump-adapted quasi-Milstein scheme.
Then there exists a constant ĉ ∈ (0, ∞) such that for all t ∈ [a, b], Proof.In the case that Z = Id the claim is proven directly using Jensen's inequality.In the case that Z = W , we know that ( t a Y (s) dW (s)) t∈[a,b] is a martingale.Hence we apply the Burkholder-Davis-Gundy inequality and afterwards Jensen's inequality to obtain that there exists some c ∈ (0, ∞) such that For the case that Z = N the claim is proven by applying [30, Lemma 2.1].
We have not yet proven finiteness of moments of the approximation scheme, but this is not required for the following lemma; the lemma is proven in Appendix A.
and the constants c Wq as in Lemma 3.3 it holds that In the next lemma we provide moment estimates for the jump-adapted quasi-Milstein scheme, which depend on the fineness of the discretisation scheme.We use these afterwards to prove moment estimates with constants independent of M .The proof can be found in Appendix A. Lemma 3.6.Let Assumption 3.1 hold and let p ∈ N, p ≥ 2. Then for all M ∈ N, δ = T M , and all ξ ∈ R there exists a constant c M ∈ (0, ∞) such that Next we provide several moment estimates for the jump-adapted quasi-Milstein scheme.Recall that ξ denotes the initial value of Z (M ) .Also this lemma is proven in Appendix A. Lemma 3.7.Let Assumption 3.1 hold and let p ∈ N, p ≥ 2. Then there exist constants Let Z (M ), ξ be the solution to SDE (2) with initial value ξ.
Using these equalities the Markov property follows by direct computations. 1

Occupation time estimates
In this section we provide occupation time estimates for our approximation scheme.For this we compose ideas of [34] and [49].The novelty here lies in the additional complexity caused by the adapted scheme.Lemma 3.9.Let Assumption 3.1 hold.Let ζ ∈ R with σ(ζ) = 0. Then there exists a constant c 5 ∈ (0, ∞) such that for all ξ ∈ R, all M ∈ N, δ = T M , and all ε ∈ (0, ∞) we have Proof.Recall that we denote by Z (M ), ξ t the solution of SDE (2) with initial value ξ ∈ R. It holds by equation (8) that By [55,Lemma 158] we obtain for all a ∈ R that where sgn(x) = 1 (0,∞) (x) − 1 (−∞,0] (x) for all x ∈ R and L a t Z (M ), ξ is the local time of Z (M ), ξ in a.By (15) we obtain Now we estimate the expectation of each summand on the right hand side of ( 16) separately.
For the first one we use Lemma 3.7 (11) to obtain that there exists c 1 ∈ (0, ∞) such that Using (14) we get for the second summand of ( 16) that We estimate the expectation of each summand of (18) separately.Lemma 3.7 (11) ensures Next we estimate the second summand of (18) using the Cauchy-Schwarz inequality, and the Itô isometry, Lemma 3.5 (10), and Lemma 3.7 ( 11): For the third summand of (18) we use Cauchy-Schwarz inequality, Lemma 3.4, and Lemma 3.7 (11).This shows that there exists ĉ ∈ (0, ∞) such that Combining (18), (19), (20), and (21) we obtain that there exists a constant c 1 ∈ (0, ∞) such that Next we consider the expectation of the third summand of (16).We use the Cauchy-Schwarz inequality, Lemma 3.4, and Lemma 3.7 (11) to obtain that (23) Plugging ( 17), (22), and ( 23) into ( 16) we obtain that there exists a constant Note that the continuous martingale part of ( 14) is For this we get using [55, Theorem 88], • Applying [55, Lemma 159] for the Borel measurable function and using ( 25) and ( 24) yields For estimating the first factor in (27) we will use This, Lemma 3.5, and Lemma 3.7 (11) and ( 12) yield For estimating the second factor of ( 27) we will use This, Lemma 3.5, and Lemma 3.7 (11) show Plugging ( 28) and ( 29) into ( 27) we obtain that there exists a constant c 3 ∈ (0, ∞) such that Hence ( 31), (26), and ( 30) ensure for all ε ∈ (0, ε 0 ), Then there exists a constant Since Y and are independent we have for the first summand of ( 32), By the Cauchy-Schwarz inequality, Observe that Lemma 3.5 equation (10) gives Plugging ( 35) and ( 36) into (34) we obtain For estimating the second summand of (32) note that Since t ∈ (s m , s m+1 ∧ min{ν i > s m }], s m = t.Moreover, the sum in the above calculation has at most one summand, since the indicator functions are disjoint.Hence (38) can be rewritten as follows: Next we define It holds that W1 , W2 , and W3 are standard normally distributed and independent of each other.
Further they are all independent of F s , since s ≤ t − δ, and W1 and W2 are independent of F t−(t−t) .
In the following we set c = max{c µ , c σ }, κ = c 1 + ζ 1 + L σ , and For the first summand of (40) we have (41) Next we observe Plugging ( 42) into ( 41) and using M ≥ M 0 we obtain For the second summand of (40), define the set This, the fact that for all a, b ∈ R, Analogously to the derivation of ( 44) we get and Further it holds that For i ∈ {1, 2, 3} we estimate Combining ( 44) and (48) ensures By ( 45) and ( 48) we obtain that Plugging ( 50) into (46) we get Using ( 47), (49), and (51) we obtain We further estimate (53) Combining ( 52) and ( 53) we obtain Hence we have proven that Next we estimate the second summand of (40).We use (54), the fact that Y is measurable, [13, p. 33], and the fact that W1 and W2 are independent of F t−(t−t) to obtain Since W1 and W2 are independent, standard normally distributed random variables, Plugging ( 56) into (55) we obtain Furthermore, Combining ( 32), ( 37), ( 40), (43), and (57) we obtain that there exists a constant c ∈ (0, ∞) such that for all M ≥ M 0 , For all M < M 0 similar calculations as in (33) and Lemma 3.5 yield This and (58) prove the claim.
For ω ∈ {ω ∈ Ω : Since Y is F s -measurable, it is F s m+1 -measurable.Hence, using (61) and Hölder's inequality for the conditional expectation we get For the first conditional expectation in (62), recalling that t ≥ s m + δ, we get (63) For the second conditional expectation in (62), Lemma 3.8 and Lemma 3.7 equation ( 13) assure Then there exists a constant c 2 ∈ (0, ∞) such that Plugging ( 63) and ( 64) into (62) we obtain Hence, we obtain Lemma 3.12.Let Assumption 3.1 hold.Let ζ ∈ R such that σ(ζ) = 0 and Then for all p, q ∈ N there exists a constant c 9 ∈ (0, ∞) such that for all M ∈ N, δ = T M it holds that The proof of this lemma works analogously to the proof of [34, equation (68)].The only changes are the notation due to the jump-adapted time grid and that we use Lemma 3.11 instead of [34,Lemma 8].
Then for all p, q ∈ N there exists a constant c 10 ∈ (0, ∞) such that for all M ∈ N, δ = T M it holds that Proof.We start calculating that there exist constants Using this and Minkowski's inequality we obtain that there exists a constant Next we consider each expectation of (65) separately.We start with the first one and apply Jensen's inequality to obtain For the first expectation in (66) we calculate using Lemma 3.7 equation ( 11), For the second expectation in (66), Lemma 3.5 and Lemma 3.7 give Plugging (67) and ( 68) into (66) we obtain that there exists a constant c 4 ∈ (0, ∞) such that For the second expectation in (65), Lemma 3.12 gives For the third expectation in (65) we use Jensen's inequality, the Cauchy-Schwarz inequality, Lemma 3.5, and Lemma 3.7 to obtain that there exists a constant c 5 ∈ (0, ∞) such that (71) Combining ( 65), ( 69), (70), and (71) we obtain that there exists a constant c 10 ∈ (0, ∞) such that

Convergence result
In this section we provide the convergence rate of the jump-adapted quasi-Milstein scheme.For the proof we make use of ideas from [34].
Theorem 3.14.Let Assumption 3.1 hold.Then for all p ∈ [1, ∞) there exists a constant c 11 ∈ (0, ∞) such that for all M ∈ N and δ = T M it holds that If we additionally assume that m σ = 0, then for all p ∈ [1, ∞) there exists a constant ĉ11 ∈ (0, ∞) such that for all M ∈ N and δ = T M it holds that Remark 3.15.We formulate our theorems for general p ∈ [1, ∞), while the lemmas we use are formulated, for simplicity, for p ∈ N. Extending the statements of the lemmas to p ∈ [1, ∞) is however straightforward and therefore omitted.
Proof.By Hölder's inequality it is enough to consider p ∈ N, p ≥ 2. First denote For all s ∈ [0, T ], (72) For the first summand we get (73) Observe that for all u ∈ (τ n , τ n+1 ] by Lemma 3.1, For the first summand of (73) this and Jensen's inequality ensure that there exists a constant Next we estimate each of the summands of (74) separately.For the first one we calculate For the second summand of (74) we obtain using Lemma 3.7 equation ( 12) that there exists For the third summand of (74) we apply Proposition 3.13 and obtain that there exists c 10 ∈ (0, ∞) such that For the fourth summand of (74) we use Lemma 3.7 equation ( 11) to obtain For the fifth summand of (74) we use Lemma 3.5 and Lemma 3.7 equation ( 11) to get Plugging ( 75), ( 76), ( 77), (78), and ( 79) into (74) we obtain that there exists c 2 ∈ (0, ∞) such that (80) Next we estimate the second summand of (73).For this we define for all M ∈ N and s ∈ [0, T ], It holds for all M ∈ N, m ∈ {0, . . ., M − 1}, and s ∈ [s m , s m+1 ] that The sequence (U M,sm ) m∈{0,...,M } is a discrete martingale.Observe that Next we estimate the expectation of each summand separately.For the first one we apply the discrete Burkholder-Davis-Gundy inequality and Jensen's inequality to obtain that there exists a constant c 4 ∈ (0, ∞) such that We define c 5 = c 4 c p σ L p µ and apply the Cauchy-Schwarz inequality, the Minkowski inequality, Jensen's inequality, and Lemma 3.7 (11) to obtain Next we apply Lemma 3.5 with p = 0 and q = 2p to obtain For the expectation of the second summand of (82), Jensen's inequality, the Cauchy-Schwarz inequality, Lemma 3.7 (11), and Lemma 3.5 ensure Combining (82), (83), and (84) we obtain that there exists a constant c 6 ∈ (0, ∞) such that For estimating the second summand of (72), observe that for u ∈ (τ n , τ n+1 ] by Lemma 3.1, Using this, the Burkholder-Davis-Gundy inequality, and Jensen's inequality we obtain for the second summand of (72) that there exist constants c 7 , c 8 ∈ (0, ∞) such that For the third summand of (86), Proposition 3.13 yields Plugging this, (75), ( 76), (78), and (79) into (86) that there exists a constant c 9 ∈ (0, ∞) such that For the third summand of (72), Lemma 3.4 ensures Combining ( 72), ( 73), ( 80), ( 81), ( 85), (87), and (88) we obtain that there exists a constant p is a Borel measurable mapping, because it is monotonically increasing.Hence we apply Gronwall's inequality and obtain that there exists a constant c 11 ∈ (0, ∞) such that This proves the first statement.For the second statement we assume S σ = ∅.This improves the estimate in (87).Hence, the claim follows from analogue calculations.

Convergence of the transformation-based jump-adapted quasi-Milstein scheme
To get from SDE (1) satisfying Assumption (2.1) to SDE (2) satisfying Assumption (3.1) we use a certain transformation.It is based on the transformation from [27]; we slightly modify it exactly as in [34].The function G is constructed such that the coefficients of SDE (1) are adjusted locally around the potential discontinuities of the drift.For the convenience of the reader we recall the definition and its properties.Let α 1 , . . ., α m be defined for all i ∈ {1, . . ., m} by using the conventions that min ∅ = ∞ and 1 0 = ∞, fix ν ∈ (0, ϕ) and define the bump function With this we define the transformation function G : R → R for all x ∈ R by The following lemma provides properties of the transformation G.
(i) The function G is differentiable on R.
(ii) The derivative G ′ is Lipschitz continuous.
We know that the function In the following we extend this mapping to G ′′ : R → R by defining Now we define the process Z : [0, T ] × Ω → R by Z = G(X).Itô's formula shows that Z satisfies Proof.As in the proof of [49,Therem 3.1] we can apply the Meyer-Itô formula of [45,p. 221,Theorem 7] to derive that Z is a solution of SDE (2).Because by Lemma 4.2 µ , σ, and ρ are Lipschitz continuous, we know by [45,p. 255,Theorem 6] that the solution of the SDE is unique.Now we have all tools at hand to prove that there exists an approximation scheme with strong rate of convergence 3/4.This scheme is the transformation-based jump-adapted quasi-Milstein scheme.
Theorem 4.4.Let Assumption 2.1 hold.Let p ∈ [1, ∞), G be the transformation defined in (89), let Z : [0, T ] × Ω → R be defined by Z = G(X), and let Z (M ) be the jump-adapted quasi-Milstein scheme for Z as introduced in (6) and (7).Then there exists a constant c ∈ (0, ∞) such that for all M ∈ N and δ = T M it holds that Proof.Using X = G −1 (Z), the Lipschitz continuity of G −1 (Lemma 4.1 (v)), the fact that the coefficients of Z satisfy Assumption 3.1 (Lemma 4.3), and Theorem 3.14 we obtain that there exists a constant c ∈ (0, ∞) such that V. Schwarz and M. Szölgyenyi are supported by the Austrian Science Fund (FWF): DOC 78.We would like to thank Thomas Müller-Gronbach, who at the Bȩdlewo workshop "Numerical analysis and applications of SDEs" pointed out to us that under Assumption 3.1 in the special case of m σ = 0 we could possibly prove convergence order 1, which was successful and improved Theorem 3.14.

A Additional proofs
Proof of Lemma 3.5.We recall (4) to obtain Hence, τ n+1 can be expressed as a measurable function of τ n and (N τn+s − N τn ) s≥0 .We denote this function by f and compute Next we use the conditional expectation and [13, p. 33] to calculate By Lemma 3.2 it holds that (W τn+s − W τn ) s≥0 and (N τn+s − N τn ) s≥0 are independent of F τn and that (W τn+s − W τn ) s≥0 is independent of (N τn+s − N τn ) s≥0 .Using this, Lemma 3.3, and [13, p. 33] we obtain which proves the first statement.
This proves the second statement.
Proof of Lemma 3.6.Observe that Using (7) we get that for all n, k ∈ N there exist constants c 1 , c 2 ∈ (0, ∞) such that where we used that N τn − N τ n−1 ∈ {0, 1}.Using (6) we obtain for all k, n ∈ N that Lemma 3.2 and Lemma 3.3 yield for all q ∈ N, Plugging (95) into (94) we get that there exist c 3 , c 4 ∈ (0, ∞) such that Combining this with (93) we get that there exist constants c Using (97) and the fact that ξ is deterministic, we recursively obtain Analogously using (98) recursively, then (96), and defining Recalling c 6 > 1 and plugging (99) into (92) we obtain Similar steps as in (92) together with (100), and the fact that c 8 > 1 ensure that there exists a constant c 10 ∈ (0, ∞) such that Here we set c 10 as the (p−1)-th moment of a Poisson distributed random variable with parameter c 8 λT .Choosing c M as the sum of the constants derived in (101) and (102) finishes the proof.
Proof of Lemma 3.7.By (8) we have for all s ∈ [0, T ], (103) In the following we estimate each summand of (103) separately.For the first one we apply Jensen's inequality, use the fact that u lies in at most one interval (τ n , τ n+1 ], and apply Lemma 3.6 to get (104) Next we consider the second summand of (103) and apply Lemma 3.4 to show that there exists ĉ ∈ (0, ∞) so that Estimating the first expectation of (105) using Lemma 3.6 we obtain that there exists a constant c M ∈ (0, ∞) such that (106) Next we estimate the second expectation of (105) using Lemma 3.2, Lemma 3.3, (9), and Lemma 3.6.We get that there exist c M , c Wp ∈ (0, ∞) such that Plugging ( 106) and ( 107) into (105) we obtain (108) Next we consider the last summand of (103).We recall that by ν i we denote the i-th jump time of N and use Lemma 3.6.This ensures that there exists c M ∈ (0, ∞) such that   This proves the first inequality.
Nest observe that there exists a constant c 2 ∈ (0, ∞) such that