Successive approximation of solutions to doubly perturbed stochastic differential equations with jumps

In this paper, we study the existence and uniqueness of solutions to doubly perturbed stochastic differential equations with jumps under the local Lipschitz conditions, and give the p-th exponential estimates of solutions. Finally, we give an example to illustrate our results.


Introduction
As the limit process from a weak polymers model, the following doubly perturbed Brownian motion x t = B t + α max 0≤s≤t x s + β min 0≤s≤t x s , ( was studied by P. Carmona [4] and J. R. Norris [11].Because of its important application, many people have devoted their investigation to this model and obtained a lot of results, for example, see [2,3,[5][6][7]12,14]. Motivated by above mentioned works, R. A. Doney and T. Zhang [8] studied the singly perturbed Skorohod equations they proved the existence and uniqueness of the solution to equation (1.2) where the coefficients b, σ satisfy the global Lipschitz conditions; Hu and Ren [9] and Luo [10] extended the global Lipschitz conditions of [8] to the case of non-Lipschitz conditions which are imposed by [13,17,18], they proved the existence and uniqueness of solutions to doubly perturbed neutral stochastic functional equations and doubly perturbed jump-diffusion processes, respectively.However, for many practical situations, the nonlinear terms do not obey the global Lipschitz and linear growth condition, even the non-Lipschitz condition.For example, consider the singly perturbed semi-linear stochastic differential equations dx(t) = ax(t)dt + σ(x(t))b(t, x(t))dB t + β max 0≤s≤t x(s), t ∈ [0, T]. (1.3) where a ∈ R, β ∈ (0, 1), and σ(x) satisfies the local Lipschitz condition: For any integer N > 0, there exists a positive constant k N such that for all x, y ∈ R with |x|, |y| ≤ N, it follows that |σ(x) − σ(y)| ≤ k N |x − y|. (1.4) Let us take where r ∈ [0, 1  2 ) and δ ∈ (0, 1) is sufficiently small, Assume b(t, x) satisfies the non-Lipschitz condition |b(t, x) − b(t, y)| ≤ ρ(|x − y|). (1.5) From the analysis of Section 5, the coefficients of equation (1.3) do not satisfy the global Lipschitz condition [8] or non-Lipschitz condition [9,10].In other words, the main results of [8][9][10] do not apply to equation (1.3).Therefore, it is very important to establish the existence and uniqueness theory of perturbed stochastic differential equations under some weaker conditions.The purpose of this paper is to study the existence and uniqueness of solutions to equation (2.1) with the local non-Lipschitz coefficients.Meantime, we will give the pth exponential estimates and the pth moment continuity of solutions.This paper is organized as follows.In Section 2, we first give some preliminaries and assumptions on equation (2.1).In Section 3, we state and prove our main results.While in Section 4, we show that the pth moment of solution will grow at most exponentially.As an application of the pth exponential estimates, we give the continuity of the pth moment of solutions.Finally, we give an example to illustrate the theory in Section 5.

Preliminaries
Let (Ω, F , {F t } t≥0 , P) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions (i.e. it is increasing and right continuous while F 0 contains all P-null sets).Let {w(t)} t≥0 be a one-dimensional Brownian motion defined on the probability space (Ω, F , P).Let { p = p(t), t ≥ 0} be a stationary F t -adapted and R-valued Poisson point process.Then, for A ∈ B(R − {0}), here B(R − {0}) denotes the Borel σ-field on R − {0} and 0 ∈ the closure of A, we define the Poisson counting measure N associated with p by where # denotes the cardinality of set {•}.It is known that there exists a σ-finite measure π such that This measure π is called the Lévy measure.Moreover, by Doob-Meyer's decomposition theorem, there exists a unique {F t }-adapted martingale Ñ((0, t] × A) and a unique {F t }-adapted natural increasing process N((0, t] × A) such that Here Ñ((0, t] × A) is called the compensated Lévy jumps and N(( , consider the following doubly perturbed stochastic differential equations (SDEs) with Lévy jumps where α, β ∈ (0, 1), the initial value In this paper, we assume that Lévy jumps N is independent of Brownian motion w and the random variable x 0 is independent of w, N and satisfies E|x 0 | p < ∞.

Assumption 2.2. There exist two positive constants
Assumption Remark 2.5.Clearly, Assumptions 2.1 and 2.2 imply the linear growth condition.Since k(•) is concave and k(0) = 0, we can find a pair of positive constants a and b such that k(u) ≤ a + bu, for u ≥ 0. Therefore, for any x ∈ R and t ∈ [0, T], Similarly, we can obtain In the sequel, to prove our main results we recall the following two lemmas.

Existence and uniqueness theorem
In this section, we study the existence and uniqueness of solutions to doubly perturbed SDEs with Lévy jumps and the local non-Lipschitz coefficients.
Let us consider the following equation with the initial data The proof of Proposition 3.1 is given in the Appendix.Now, we construct a successive approximation sequence using a Picard type iteration.Let x 0 (t) = x 0 , t ∈ [0, T], define the following Picard sequence: Obviously, according to proposition 3.1, the solution In what follows, C > 0 is a constant which can change its value from line to line.
Proof.For any s ≥ 0, it follows that from (3.2) Taking the maximal value on both sides of (3.4), by the Hölder inequality, the Doob's martingale inequality and Assumption 2.3, we have Therefore, we get By Assumptions 2.1 and 2.2, we have Then the Jensen inequality implies that 2 ), it follows that By Assumption 2.1, we have that γ is a non-decreasing continuous function, γ(0) = 0 and x , k + (x) and ρ + (x) are non-negative, non-increasing functions, we have that is a non-negative, non-increasing function, thus γ is a non-negative, non-decreasing concave function.Since γ(•) is concave and γ(0) = 0, we can find a pair of positive constants a and b such that is the solution to the following ordinary differential equation: By recurrence, it is easy to verify that for each n ≥ 0, Since r(t) is continuous and bounded on [0, T], we have Taking the maximal value on both sides of (3.7), by the Hölder inequality, the Doob's martingale inequality and Assumption 2.3, we have By Assumption 2.1 and Jensen's inequality, we get Similar to (3.5), we obtain By the inequality (3.3) and Fatou's lemma, it is easily seen that lim sup Owing to Lemma 2.6, we immediately get that lim sup Then {x n (t)} n≥1 is a Cauchy sequence.The proof is complete.Now, we state and prove our main results.Proof.According to (3.10), it follows that there exists Then the Borel-Cantelli lemma can be used to show that x n (t) converges to x(t) almost surely uniformly on [0, T] as n → ∞.Taking limits on both sides of (3.2) and letting n → ∞, we obtain that x(t) is a solution of equation (2.1).Now we devote to proving the uniqueness of equation (2.1).Suppose x(t) and y(t) are two solutions of equation (2.1) with initial value x 0 and y 0 , we have Then, in the same way as the proof of (3.8) one can show that where G(t) = t 1 ds γ(s) .In particular, if x 0 = y 0 , then Obviously, G is a strictly increasing function, then G has an inverse function which is strictly increasing, and G −1 (−∞) = 0. Finally, we obtain for any t ∈ [0, T] which implies the uniqueness.This completes the proof.
Proof.Let T 0 ∈ (0, T), for each N ≥ 1, we define the truncation function f N (t, x) as follows: and g N (t, x), h N (t, x, v) similarly.Then f N , g N and h N satisfy Assumption 2.1 due to that the following inequality about f N , g N and h N hold: where x, y ∈ R and t ∈ [0, T 0 ].Therefore, by Theorem 3.4, there exists a unique solution x N (t) and x N+1 (t), respectively, to the following equations Define the stopping times where we set inf{φ} = ∞ as usual.Similar to (3.7), we obtain Again the Hölder inequality, the Doob's martingale inequality imply that Noting that for any 0 we derive that Then it follows from Assumption 2.4 that 2 ).Obviously, by Assumption 2.4, we have that γ N (•) is a non-negative, non-decreasing concave function, γ N (0) = 0 and 0 + 1 γ N (x) dx = ∞.By using Lemma 2.6 again, it follows that Therefore, we obtain that For each ω ∈ Ω, there exists an N 0 (ω) > 0 such that 0 < T 0 ≤ τ N 0 .Now define x(t) by Letting N → ∞, then yields Since T 0 is arbitrary, then we have x(t) is the solution of equation (2.1) on [0, T].The proof is complete.

p-th moment exponential estimates
In this section, we will give the pth exponential estimates of solutions to equation (2.1).

Remark 4.2.
In particular, we see clearly that if let ρ(u) = Ku, L > 0, then Assumption 4.1 reduces to the linear growth condition.That is, for any x ∈ R and t ∈ [0, T], we have Theorem 4.3.Let Assumptions 2.1-2.3 and 4.1 hold, for any p ≥ 2 where C 4 and C 5 are two positive constants of the inequality (4.11).
Proof.For any t ≥ 0, it follows from (2.1) that Taking the maximal value on both sides of (4.2), by Holder's inequality, the Burkholder inequality and Assumption 2.3, we have Therefore, we get where C = (1−|α|−|β|) p .Using Hölder's inequality, we get By the basic inequality for any > 0, it follows that By Assumptions 2.1, 2.2 and letting = K p−1 1 , we obtain In fact, because the function k(•) is concave and increasing, there must exist a positive number L such that Hence, By using the Burkholder-Davis-Gundy inequality and the Hölder inequality, we have a positive real number C p such that the following inequality holds: By the similar arguments, we have Now, we will estimate the fourth term of (4.3).Using the basic inequality |a + b| p ≤ 2 p−1 (|a| p + |b| p ), we have and ds. (4.9) Inserting (4.8) and (4.9) into (4.7), and by Assumption 4.1, we have Similar to (4.4), we have Hence, where where where G(t) =