Properties of Solutions to Stochastic Set Differential Equations under Non-Lipschitzian Coefficients
A class of stochastic set differential equations (SSDEs) with non-Lipschitzian coefficients is investigated. We first give the preliminaries on the stochastic set differential equations. Then the nonexplosion of solutions to the SSDEs is discussed. Moreover, the existence and uniqueness of the solutions to SSDEs are proven. Finally, the continuous dependence of the solutions to SSDEs is studied.
Set-valued differential equations which were started in 1969 by de Blasi and Lervolino  have been employed in investigations of dynamic systems. The evidence of set differential equations for such areas as control theory, differential inclusions, and fuzzy differential equations can be found in [2–12] and references therein. The set differential equations also are explored in [13–15]. One of the main advantages of investigating deterministic set differential equations is that they can be used as a tool for studying properties of solutions of differential inclusions. On the other hand, the set-valued random processes are first introduced by van Cutsem . Since then the subject has attracted the interest of many mathematicians and further contributions are made from both the theoretical and applied viewpoints (see, e.g., [17–26]). In [27–31], the set-valued random differential equations are studied. The strong solution of Itô type set-valued stochastic differential equation is analyzed in .
As far as we know, there exists a wide literature where attempts have been made to investigate stochastic differential inclusions (see, e.g., [33–40] and references therein). And recently, in , a kind of the SSDEs disturbed by Wiener processes is investigated, where under the Lipschitzian condition the existence and uniqueness of solutions to the SSDEs are proven. Under the non-Lipschitzian condition, the existence and uniqueness of solutions to the stochastic set differential equations are proven in [41, 42]. Moreover, in our present paper, under the non-Lipschitzian condition the nonexplosion and continuous dependence of solutions to the SSDEs are studied. The mathematical tool employed in the paper is the Bihari inequality and the notion of the support function. The work presented here generalizes results obtained both for deterministic and for random set differential equations. Also, it should be noted that the work related to this paper is the discussions of fuzzy-valued processes and stochastic differential equations (see, e.g., [43–51]).
The paper is organized as follows. Section 2 gives an appropriate framework on a set-valued analysis within which the notion of a set-valued stochastic integral is given. In Section 3, moreover, the continuous dependence of the solutions for SSDEs on initial conditions and nonexplosion are discussed. Finally, the conclusions are made in Section 4.
Let be the family of all nonempty compact and convex subsets of . In , we define the Hausdorff metric of two sets as follows:
Throughout this paper, let be complete probability space. is a product -field of . denotes the family of -measurable multifunctions with values in . A multifunction is said to be -integrably bounded, , if there exists such that a.s., where Let us denote
Denote . Let be a complete, filtered probability space where the sub--field family of satisfies the usual conditions. We call a set-valued stochastic process, if for every a mapping is a set-valued random variable. If is -adapted and measurable, then it will be called nonanticipating. Equivalently, the set-valued process is nonanticipating if and only if is measurable with respect to the -algebra , which is defined as follows: where for .
Let and denote the set of all nonanticipating -valued stochastic processes such that . A set-valued stochastic process is called -integrably bounded, if there exists a real-valued stochastic process such that
We define the operation on as follows. For two sets , if there exists such a that , then is the Hukuhara difference of and denoted by . We note that implies that . However, . Indeed, take .
It is well known that for all .
Let is the support function of , where is a unit sphere centered at origin. The support function satisfies the following properties.(i) is bounded on ; that is, .(ii) is Lipschitz continuous in (iii)For all , (iv)For all , .(v)For all , .(vi)For all , if ; then we have .
By we denote the set of nonanticipating and -integrably bounded set-valued stochastic processes. Let . For such and a fixed , by the Fubini Theorem, we can define Aumann’s integral , for . Obviously, for every and the Aumann integral belongs to (see, e.g., [7, 19]).
We say that a set-valued stochastic process is -continuous, if almost all its trajectories, that is, the mappings , are -continuous functions. It is easy to know that if , then the set-valued stochastic process is -continuous (see, e.g., Corollary 1 in ).
In what follows, we state the generalized Bihari inequality (cf., Mao ) which plays an important role in the following section.
Lemma 1 (generalized Bihari inequality). Let be Borel measurable, bounded, nonnegative, and left limit function on and . Let be a continuous nondecreasing function such that for all .
(i) If is a continuous nonnegative nondecreasing function on , then the inequality implies that for all such that where where is the inverse function of and .
(ii) If is a continuous nonpositive nonincreasing function on , then the inequality implies that for all such that
3. Properties of Solutions to SSDEs
In this section, we consider the following stochastic set differential equation (in the integral form): where takes values in , in , is an -dimensional Brownian motion, and is a set-valued random variable. Here, (resp., stands for the family of the continuous functions from the space to the space (resp., to ). In (16), the integral is Aumann’s one, and the integral is a general Itô type stochastic one, whose definition can refer to .
Due to the continuity of , we can know that takes values in . In order to discuss the solutions to (16), by the concept and properties of the support function we consider the following single valued stochastic differential equation, for : We first claim that (16) is equivalent to (17). Indeed, from the properties of the support function and (16) we deduce that, for , which shows that (17) holds. Conversely, (16) can be derived from (17).
It is known that (17) has a solution up to a lifetime which depends on . Set We call the lifetime of solution to (16). Obviously, . Here, by the concepts of explosion time and lifetime time from Pages 158 and 191 in , the lifetime of solution to (16) or (17) is the same as the explosion time of the solution of (16) or (17).
If (17) has the pathwise uniqueness, then we show the existence and uniqueness of the solution to the SSDE (16). So the study of pathwise uniqueness is of great interest. It is a classical result that, under the Lipschitz coefficients, the pathwise uniqueness holds and the solution of (17) can be constructed by using Picard iteration; moreover, the solution depends on the initial values continuously. However, under non-Lipschitzian condition, Fei  presents the existence and uniqueness of solutions to (16); hence, the existence and uniqueness of solutions to (17) also are proven.
In what follows, we discuss the nonexplosion of the solutions to (16) under non-Lipschitzian condition. Our idea is to derive an inequality so that the generalized Bihari inequality (Lemma 1) can be applied.
Theorem 2. Let be a continuous function such that(i) is nondecreasing and concave;(ii).
Assume that, for some constant , If , then the lifetime of the solution to (16) is infinite: . Moreover,
Proof. Let , where is a solution to (16). Hence, we have
Since , we get . Set ; it follows that from (20) and property (i) of the support function
where we have utilized the concavity of the functions and .
Denote . Noticing , we get where takes the maximum at 1 on the interval .
It is easy to see that which deduce that Thus we have In virtue of (23), we have Set By condition (ii) it is easy to show that is strictly increasing, as and as .
From (28), the generalized Bihari inequality (Lemma 1(i)), and we obtain which proves that .
On the other hand, from (20) and property (i) of the support function we have Due to inequality (24), we have By Lemma 1(ii), we get which shows that by the property of the function in (29). Thus, the proof of the theorem is complete.
Theorem 3. Let be a continuous function such that(i) is nondecreasing and concave;(ii).
Assume that for some constant ,, Then SSDE (16) has a unique solution.
Proof. By constructing the sequence of set-valued random variables as that in , the existence of the solutions to (16) is similarly proven. Next, we prove the uniqueness of the solutions to (16).
Let and be two solutions of (16). Set . Hence, by Itô formula we have Since , we get . Set .
From condition (34) and the property of the support function, we get which deduces where we have utilized the concavity of the functions .
Denote . In virtue of (37), we have
Let We easily show that is strictly increasing, as and as .
By Lemma 1 (i), we obtain which shows that . Thus we complete the proof.
Note that function is a typical example satisfying conditions (i) and (ii).
Next, we will study the dependence of the solutions to the SSDE (16) on initial data. For the mapping , we call mean square continuous on , uniformly with respect to if as on any compact subset of , where the limit of is in sense of the metric .
Proof. Take . Consider a small parameter . Assume such that . For , let
By (17) and Itô formula, we have
From condition (34) and the property of the support function, we get which deduces where we have utilized the concavity of the functions .
Setting , from (45) we have Define We easily have that is strictly increasing, as and as .
By Lemma 1 (i), we have For arbitrary and given , it is easy to deduce that there exists with , which shows , such that
Since is increasing, we have that which shows that Thus, we show is mean square continuous on , with respect to in any compact subset. Therefore, we complete the proof.
Definition 5. The two solutions and of SSDE (16) with initial value and , respectively, if for any , , can be called nonconfluence.
The following theorem gives the sufficient condition.
Theorem 6. Suppose that the conditions in Theorem 3 hold. For , we have that and are nonconfluence.
From condition (34) and the property of the support function, for , we get Thus, similar to the discussion in the proof of Theorem 4, we obtain which shows that Take as in the proof of Theorem 4.
By Lemma 1 (ii), we have Since , from the property of , we obtain which means that . Thus, the proof is complete.
Theorem 7. Let condition (34) hold. Then for any defines a flow of homeomorphisms of .
In many real dynamic systems, we are often faced with random experiments whose outcomes might be multivalued. Moreover, the stochastic set differential equations may be employed in characterizing a large class of physically important dynamic systems which can be applied in such areas as control, economics, and finance. In this paper, we study the behavior of solutions to SSDEs disturbed by a Wiener process with the non-Lipschitzian coefficients. First, the nonexplosion theorem of the Itô type SSDEs is proven. Then the existence and uniqueness theorem of solutions to SSDEs is given. Moreover, the continuous dependence of solutions to the SSDEs is investigated. Main mathematical tool is the notion of the support function and the generalized Bihari inequality. Besides, the present case can be extended to the SSDEs driven by a multidimensional semimartingale in future.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The author wishes to express his many thanks to the referees for many valuable suggestions and comments on this paper. This paper is supported by the National Natural Science Foundation of China (71171003; 71271003), Anhui Natural Science Foundation (10040606003), and Anhui Natural Science Foundation of Universities (KJ2012B019, KJ2013B023).
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