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.
Consider the following stochastic differential equations driven by a nonlinear integrator (or semimartingale) of the formequivalentlywhere is a probability space, is fields, is a -dimensional process of finite variation and adapted function, is a -dimensional continuous local martingale, and the pair is the local characteristic of (for details, see Mao ).
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  and Krylov ). 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  and Revuz and Yor ). Let 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 (see ). 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 ). 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 ). Using Zvonkin’s transformation, Zhang studied the homeomorphic property of solutions of multidimensional stochastic differential equations with non-Lipschitz coefficients (see ). 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 ). 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 ). 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 ). 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 ). 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 ). 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 is a constant and is a strictly positive function which satisfies the following conditions: (i).(ii).(iii).And the pathwise uniqueness of the stochastic differential equations (1) is also proved, under the following conditions:where is a strictly positive function satisfying the conditions (i);(ii);(iii).For details, see . 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 ) and Yang established the pathwise uniqueness of SPDE with non-Lipschitz coefficients (see ).
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 , 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 . In Section 3, we prove the pathwise uniqueness under some new sufficient conditions and stochastic continuity.
2. 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 .
Theorem 1. Let be a function which satisfies the following conditions:(i).(ii).(iii)for any , there exists such that , . is a positive and progressive measurable function which satisfies If the condition holds, then the stochastic differential equations (1) have no explosion .
Proof. DefineAccording to the continuity of and , there exists such that condition (7) holds with respect to for any . Let ; then , Due to (iii) of condition (7), we know that there exists such that for or . It is obvious that the function is not continuous at the point ; hence the Itô formula is not applied for . Now take such that , and In fact, exists for any ; we can take and , .
Let ; define then as . LetIf , then we havewhere . Note thatand the stochastic covariation is given byBy the Itô formula and (15) and (16), we haveTaking expectation for (17) and according to (7) and (14), we haveFor , due to (14), we have For , noting that for or , we haveCombining (19) and (20), it holds thatHence, Letting , by the Fatou lemma, we have If , then there exists such that ; take in (23); we have , which is impossible, because of .
3. 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 .
Theorem 2. Let , , be a positive and continuous function on some intersection , and and is a positive and progressive measurable function satisfying If and satisfy the condition then the stochastic differential equations (1) have the pathwise uniqueness.
Proof. Let and , where and are the solutions of stochastic differential equations (1) with the same initial value. Let ; define the function as follows: It is obvious that, for any , we have Because of we get the conclusion that is a concave function on . Lettingthen is a nonpositive random measure in the distributive sense.
Let ; by Itô formula, we have Notice that , ; applying hypothesis (25) and taking expectation for (30), we get Letting in inequality (31), we have . It means that, for any given, If , then there exists some big enough such that . By (32), for all , . It follows that, on , , which is absurd with the definition of . Therefore almost surely and, for any given, almost surely. By the continuity of samples, two solutions are indistinguishable.
By the above pathwise uniqueness, we can also obtain the following continuity result.
Proof. Consider and satisfying ; let Define ; let be the function defined in Theorem 2. By Itô formula, we have where is the same as in (31). Let in the above inequality; it holds that soThen we can obtainas . The proof is completed.
Remark 5. We should point out that the results (Theorems 1 and 2) of  are the special case of this paper result (Theorems 1 and 2). In fact, let , (where are the corresponding function of ); it is clear that satisfy conditions (7) and (25), respectively. Hence, the result of this paper includes the result of . What is more, only need continuity in Theorem 2; however, must be differential function in [8, 15], and our proof is more directive.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author deeply thanks the anonymous referees for their very kind and useful suggestions, which have improved the presentation of the paper. This work is supported by Ph.D. Research Start-Up Foundation (Grant no. 109-400211411) of Xi’an University of Technology. The author gratefully acknowledges the support.
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