Well-posedness and order preservation for neutral type stochastic di ﬀ erential equations of inﬁnite delay with jumps

: In this work, we are concerned with the order preservation problem for multidimensional neutral type stochastic di ﬀ erential equations of inﬁnite delay with jumps under non-Lipschitz conditions. By using a truncated Euler-Maruyama scheme and adopting an approximation argument, we have developed the well-posedness of solutions for a class of stochastic functional di ﬀ erential equations which allow the length of memory to be inﬁnite, and the coe ﬃ cients to be non-Lipschitz and even unbounded. Moreover, we have extended some existing conclusions on order preservation for stochastic systems to a more general case. A pair of examples have been constructed to demonstrate that the order preservation need not hold whenever the di ﬀ usion term contains a delay term, although the jump-di ﬀ usion coe ﬃ cient could contain a delay term.


Introduction
In [1], Asker studied well-posedness for a class of neutral type stochastic differential equations driven by Brownian motions with infinite delay; Bao et al. [2] also investigated the exponential ergodicity, weak convergence, and asymptotic Log-Harnack inequality for several kinds of models with infinite memory.So far, there is no order preservation available for stochastic differential equations with infinite memory.Moreover, the order preservation theorems play an essential role in the theory of stochastic systems and their applications because, in many fields of analysis, they constitute an effective way to control a complicated stochastic system by using a simpler one.These types of theorems are used in a wide range of practical problems in fields such as finance, economics, biology, and mathematics; see also [3][4][5][6][7][8].Consequently, we focus on establishing order preservation for neutral-type stochastic differential equations of infinite memory with jumps and obtaining the well-posedness for these stochastic systems under non-Lipschitz conditions.
The pioneering work on order preservation for stochastic differential equations is detailed in [9], and was later generalized in [10].Since their works, the order preservation for two stochastic differential equations driven by continuous noise processes has been investigated extensively.With regard to the order preservation under various settings, we can refer to, for example, [11] for one-dimensional stochastic differential equations, [12] for one-dimensional stochastic hybrid delay systems, and [13] for multidimensional stochastic functional differential equations.
Meanwhile, the order preservation for two stochastic differential equations subject to the discontinuous case has also garnered much attention.For example, applying criteria of a "viability condition", the authors of [14] showed a comparison theorem of stochastic differential equations with jumps under Lipschitz and linear growth conditions; using a Tanaka-type formula, [15] further established a comparison theorem for one-dimensional stochastic differential delay equations with jumps, where the coefficients satisfy local Lipschitz and linear growth conditions; adopting an approximation argument, the work in [16] extends the results on one-dimensional equations to multidimensional stochastic functional differential equations with jumps, where the coefficients satisfy a non-Lipschitz condition.
It is worth pointing out that [13,15,17,18] focus on order preservation for stochastic functional differential equations with Lipschitz coefficients, which rules out the case of non-Lipschitz conditions.On the other hand, few studies have focused on stochastic functional differential equations with non-Lipschitz coefficients, and, in the existing literature, most have focused on stochastic functional differential equations of finite delay.Yet, the corresponding issue for stochastic functional differential equations with infinite memory is rarely addressed in the literature.Moreover, the multidimensional order preservation theorem affords a further widening of the field of application, especially for those processes whose dynamics are influenced by each other.Based on the above motivations, in this work, we aimed to develop an approximation method to investigate order preservation for multidimensional neutral type stochastic functional differential equations, which allow the coefficients to be non-Lipschitz and depend on the whole history of the system.Compared to the existing results on order preservation, the innovations of our work can be described as follows: (i) We introduce the truncated Euler-Maruyama scheme method into the analysis of the wellposedness problem of neutral-type stochastic differential equations of infinite delay with jumps, and we establish the existence of the solutions; (ii) Our model is more applicable and practical, as we deal with neutral-type stochastic differential equations under non-Lipschitz conditions.
The rest of the paper is arranged as follows.In Section 2, we introduce some notations and present the framework of our paper; Section 3 is devoted to the existence and uniqueness of solutions for a class of neutral stochastic functional differential equations of infinite delay for pure jumps; Section 4 focuses on the order preservation for this system.

Preliminaries
For a fixed number r > 0, set Then, (D r , • r ) is a Banach space.Under the uniform norm • r , the space D r is complete but not separable.Let (W(t)) t≥0 be an m-dimensional Brownian motion and N(dt, du) a Poisson counting process with characteristic measure λ on a measurable subset Y defined on the probability space (Ω, F , P) with the filtration (F t ) t≥0 satisfying the usual condition (i.e., F 0 contains all P-null sets and F t = F t+ := s>t F s ).We assume that W(t) and N(dt, du) are independent.
where U is a class of control functions and is increasing and concave .
(A3) For any T > 0, there exists a constant C(T ) such that P-a.s.
(A4) G(ξ) ≤ G(η) for ξ ≤ η and there exists a constant α ∈ (0, 1  2 ) such that Under (A1)-(A3), (2.1) admits a unique strong solution (X(t)) t≥0 ; see Theorem 3.1 below for more details.For the existence and uniqueness of strong solutions to stochastic functional differential equations with infinite delay, we refer the reader to [1,19,20] and the references therein.In particular, using the Picard approximation, Ren and Chen [19] studied the existence and uniqueness for a class of neutral-type stochastic differential equations of infinite delay with Poisson jumps in an abstract space under non-Lipschitz.We remark that we provide an alternative method to establishing the wellposedness of neutral type stochastic differential equations of infinite delay with jumps.The Lipschitz coefficient α in (A1) is set to less than one-half rather than 1 40 , as detailed in [19].So, in some sense, our result is more general.Assumption (A4) is just imposed for the sake of the monotonicity principle of the solution process; see Theorem 4.1 below for more details.
Meanwhile, to establish the order preservation for multidimensional neutral-type stochastic differential equations of infinite delay, in view of [21], we introduce the partial orders on R d and C r as follows: for In this section, we finally recall the definition of D-order preservation (see, e.g., [

Existence and uniqueness of solutions
In the case that G = 0 and N = 0, the existence and uniqueness of solutions to (2.1) with weak one-sided local Lipschitz conditions has been studied in [2].On the other hand, under the same conditions the authors of [1] has extended the result to neutral-type stochastic differential equations of infinite delay.Compared with these, we point out that the following result is included in [1,2].In contrast to the assumptions put forward in [1,2], the assumptions (A1)-(A3) are more general.Moreover, in [16], where order preservation of a stochastic functional differential equation with non-Lipschitz coefficients is given, a tried-and-true method shows that we can approximate the non-Lipschitz stochastic functional differential equations by using those with Lipschitz coefficients to prove the existence of solutions.It is worth pointing out that the Bismut formula for stochastic functional differential equations of finite delay plays a crucial role in the analysis of the existence of those with non-Lipschitz coefficients.Alternatively, for the neutral-type stochastic differential equations of infinite delay, this method is no longer valid.To prove the well-posedness of solutions, we adopt a truncated Euler-Maruyama approximation argument (see, e.g., [1,2]), where the essential ingredient is to construct the associated segment process and introduce an approximate function in a good way.
For any k ≥ 1, let Then, one has Theorem 3.1.Let (A1)-(A3) hold with b = 0, σ = 0, and γ = 0.Then, for any t ≥ 0 and ξ ∈ D r , (2.1) has a unique solution such that Proof.In what follows, we write X t in lieu of X ξ t for brevity.(a) First, we shall show that Then, by (A1), one infers that Combining the Itô formula with the assumption (A1), one has, for any 0 By taking the Young inequality into consideration, one gets The Burkholder-Davis-Gundy inequality, together with the assumptions (A2) and (A3), implies that It follows from the assumptions (A2) and (A3) that The Young inequality implies that Therefore, from the above inequalities, we obtain Applying the Gronwall inequality leads to which, together with (3.2), implies that Then, by the Bihari inequality, we have P-a.s. where (b) Second, we aim to derive the uniqueness of the solution.Let X(t) and Y(t) be two solutions to (2.1) with the same initial value X 0 .Set , and, in the last step we apply the assumption (A1).Then, carrying out the same technique to deduce (3.3), one has Due to the fact that the function su(s) is increasing, and by the Gronwall inequality, we have Furthermore, using Jensen's inequality, we get (c) Finally, we shall divide two cases to show the existence of the solution to (2.1).We shall adopt the truncated Euler-Maruyama scheme approach (see, e.g., [1,2]), where the essential ingredient is to construct an approximation of the segment process in a good way.Case 1.In this part, we shall show existence of the solution for bounded b, σ and By the definition of ψ k , it is easy to see that A straightforward calculation leads to the following for x ∈ R d and i = 1, 2, • • • , d: where δ i j = 1 if i = j, or 0 otherwise.Then, it follows from (3.1) that, for Set N 0 := {n ∈ N : n ≥ r log 2 } and s := sup{k ∈ Z; k ≤ s}, i.e., the integer part of s > 0. For any n ∈ N 0 , consider a stochastic differential equation: where Xn In view of a similar technique as in the proof of the uniqueness in (b), (3.5) has a unique solution by piecewise solving piece-wisely using the time step length 1  n .And, beyond that, we can find an n ∈ N 0 satisfying that e r/n ≤ 2; then, , and H n t = X n t − Xn t .Using (3.5) and the assumption (A1), the Young inequality leads to the following for any ε > 0: where , and Moreover, since b and σ are bounded on bounded subsets of [0, ∞) × D r , then and |σ(t, It follows from the definition of τ n R and (3.6) that, for t ≤ τ n R , In addition, it is easy to see from (3.5) and (A1) that where, in the last step, we have used the fact that In view of (3.8) and (3.9), besides the Burkholder-Davis-Gundy inequality, we get Combining these inequalities with (3.10), one has In what follows, we shall prove that X n (•) is a Cauchy sequence.Fix T > 0 and set Then, D 2 r is a complete metric space with .
By the Itô formula, one has the following for any m, n ∈ N 0 : (3.12) By applying the elementary inequality, the assumption (A2) and (3.4) yield that 2 and where, in the penultimate inequality, we have used the fact that, for any 0 by aid of the definition of Xn t and (A1).Here, because u ∈ U .By the Burkholder-Davis-Gundy inequality, (A2), and (3.4), one gets By virtue of (3.4) and a Taylor expansion, one infers that, for t ∈ [0, T ], This and (A2) imply that Substituting (3.13)-(3.16)into (3.12),we infer that, for any k ≥ 1 and m, n ∈ N 0 , Let k ↑ ∞; using Jensen's inequality and noting (3.7), we get the following for any n, m ∈ N 0 : where because of (3.11).Thus, to prove that X n • converges in probability to a solution of (2.1), it is sufficient to show that lim Indeed, this and (3.17) yield that, for any ε > 0, This implies that X n (t) is a Cauchy sequence in D 2 r with the norm ρ and has a unique limit X(t) on D 2 r due to the completeness of (D 2 r , ρ).Then, by using a standard argument, we can show that (X(t)) t∈[0,T ] is the unique solution to (2.1).So, to achieve the desired assertion, it is sufficient to show that (3.18) holds true.By a simple calculation, and using the assumption (A1), we have where, in the second step, we have used the Chebyshev inequality.Therefore, (3.18) holds.Case 2. Next, we present the existence of the solution for unbounded b, σ and β.
Then, b n , σ n , and has a unique solution X n (t), t ≥ 0. For any m ≥ n ≥ 1 and ξ ∈ D r with ξ r ≤ n, one has Then it follows that X n (t) = X m (t) for t ≤ τn , where τn := n ∧ inf{t ≥ 0 : X n t r ≥ n}.Let n ↑ ∞; then, τn ↑ ∞.So, for t < τn , X(t) := X n (t) is a solution of (2.1).

Order preservation for neutral-type stochastic differential equations of infinite delay with jumps
In this section, we first show the D-order preservation problems for a class of neutral-type stochastic differential equations of infinite delay with pure jump processes.
Next, we aim to study the order preservation for neutral-type stochastic differential equations with compensatory jump processes.Consider two multidimensional neutral-type stochastic differential equations of infinite delay with jumps for any t ∈ [0, T ]: and Then, (4.8) and (4.9) are respectively equivalent to  (iii) The jump diffusion term γ = (γ 1 , γ 2 , • • • , γ d ) satisfies that ξ i (0) − G i (ξ) + γ i (t, ξ, •) ≤ ξi (0) − G i ( ξ) + γ i (t, ξ, •) for any 1 ≤ i ≤ d, t ≥ 0 if ξ, ξ ∈ D r with ξ ≤ D ξ. Remark 4.1.In Theorems 4.1 and 4.2, it is easy to find that D-order preservation theorems hold in when the diffusion term does not contain a segment process.However, the jump-diffusion coefficient can contain a delay term.It is consistent with the result for one-dimensional stochastic differential delay equations in [15].
In the following part inspired by [15, Examples 2.2 and 2.3], we will also establish two examples to support the above two opinions respectively.This implies that P{ω ∈ Ω : X(t) > 0} > 0.
Therefore, it holds that D-order preservation need not hold if the diffusion coefficient includes a delay term.However, the following example shows that the jump diffusion can conclude a delay function.
For d, m ∈ N, i.e., the set of all positive integers, let (R d , •, • , | • |) be the d-dimensional Euclidean space with the inner product •, • inducing the norm | • | and R d ⊗ R m denote the collection of all d × m matrixes with real entries, which is endowed with the Hilbert-Schmidt norm • .D := D([−∞, 0]; R d ) denotes the family of all càdlàg functions dr = ∞, by the Bihari inequality, we have lim n,m→∞ du) are locally bounded.Therefore, according to (a), (b), and Case 1,