Nonexplosion and Pathwise Uniqueness of Stochastic Differential Equation Driven by Continuous Semimartingale with Non-Lipschitz Coefficients

We study a class of stochastic differential equations driven by semimartingale with non-Lipschitz coefficients. New sufficient conditions on the strong uniqueness and the nonexplosion are derived for -dimensional stochastic differential equations on with non-Lipschitz coefficients, which extend and improve Fei’s results.

Recently, many studies have focused on the strong uniqueness and the nonexplosion of stochastic differential equations with the coefficients satisfying the local Lipchitz condition (see Stroock and Varadhan [2] and Krylov [3]).However, the results on the strong uniqueness of stochastic differential equations are still very few when the coefficients satisfy non-Lipschitz condition, except the one-dimensional case (for details, see Ikeda and Watanabe [4] and Revuz and Yor [5]).Let   ( 0 , ) be the solution of the stochastic differential equations (1).If the driving process (, ) is only the Brownian motion and local characteristic is locally Lipschitzian, Protter proved that the solution admitted a continuous version X ( 0 , ) (see [6]).When the drift term is independent of , satisfying local   −  integrated condition, and the diffusion term is identity matrix, Krylov and Röckner proved the existence and path uniqueness for stochastic differential equations on a given area (see [7]).From the viewpoint of application, to what extent the condition on coefficients should be weakened is an important problem.Fang and Zhang discussed the pathwise uniqueness and the nonexplosion for a class of stochastic differential equations driven by Brownian motion with non-Lipschitz coefficients (see [8]).Using Zvonkin's transformation, Zhang studied the homeomorphic property of solutions of multidimensional stochastic differential equations with non-Lipschitz coefficients (see [9]).When the diffusion coefficient is uniformly nondegenerate and non-Lipschitz and drift coefficient is locally integrable, Zhang also proved the existence of a unique strong solution up to the explosion time for a stochastic differential equation.Moreover, two nonexplosion conditions are given (see [10]).Davie proved the uniqueness of solutions of stochastic differential equations when the drift term is bounded Borel function and the diffusion term is identity matrix (see [11]).Using Gronwall lemma, Fei studied the existence and uniqueness of solutions for fuzzy random differential equations with non-Lipschitz coefficients and discussed the dependence of fuzzy random differential equations on initial values (see [12]).Luo studied the behaviors of small subsets under the flows generated by the ordinary and stochastic differential equations whose coefficients satisfy certain non-Lipschitz conditions (see [13]).Further, for the stochastic differential equations with non-Lipschitz coefficients driven by Brown motion, Lan proved the pathwise uniqueness and nonexplosion, deriving the new sufficient condition (see [14]).Note that there are very few results about the case of semimartingale.Recently, Fei proved that the solution of the stochastic differential equations (1) was not exploding, when the local characterized  and  satisfy the following conditions: where  > 0 is a constant and  ∈  1 ([1, +∞]) is a strictly positive function which satisfies the following conditions: And the pathwise uniqueness of the stochastic differential equations ( 1) is also proved, under the following conditions:  (, , ) − 2 (, , ) +  (, , ) where  : [0,  0 ] → R + is a strictly positive  1 function satisfying the conditions For details, see [15].Ulteriorly, Cao proved the existence and pathwise uniqueness of solutions of SDE driven by class of special semimartingale when the coefficient of martingale term is non-Lipschitz and satisfies Taniguchi's condition (see [16]) and Yang established the pathwise uniqueness of SPDE with non-Lipschitz coefficients (see [17]).
In this paper, we derive the new sufficient conditions on the strong uniqueness and the nonexplosion for dimensional stochastic differential equations (1) driven by semimartingale with non-Lipschitz coefficients on R  ( > 2), and the new conditions are sharp in a sense.The theory of related martingale would be found in [4,5,18].This paper is organized as follows.In Section 2, we derive the new sufficient conditions and prove the nonexplosion for stochastic differential equations (1) on R  ( > 2).In Section 3, we prove the pathwise uniqueness under some new sufficient conditions and stochastic continuity.

No Explosion of Solutions
In this section, we prove the following nonexplosion result of stochastic differential equations (1).Let   () be the solution of the stochastic differential equations (1) and is the lifetime of solution process   .

Pathwise Uniqueness of Solutions
In this section, we will prove the following pathwise uniqueness result; when the driving process is Brownian motion, such kind of properties was studied for non-Lipschitzian coefficients in [8].
Proof.Let   =   −  and   =  2  , where (, ) and (, ) are the solutions of stochastic differential equations (1) with the same initial value.Let  > 0; define the function Γ  : [0, 1] → R + as follows: It is obvious that, for any 0 <  < 1, we have Because of we get the conclusion that Γ  is a concave function on [0, ).Letting then Γ   is a nonpositive random measure in the distributive sense.
By the above pathwise uniqueness, we can also obtain the following continuity result.Theorem 4. Suppose that the coefficients  and  satisfy assumptions (7)  as  = | − | → 0. The proof is completed.
Remark 5. We should point out that the results (Theorems 1 and 2) of [15] are the special case of this paper result (Theorems 1 and 2).In fact, let () = (), () = () (where (), () are the corresponding function of [15]); it is clear that (), () satisfy conditions ( 7) and (25), respectively.Hence, the result of this paper includes the result of [15].What is more,  only need continuity in Theorem 2; however,  must be differential function in [8,15], and our proof is more directive.