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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 357869, 11 pages
Stochastic Differential Equations with Multi-Markovian Switching
1School of Mathematical Science, Huaiyin Normal University, Huai'an 223300, China
2Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China
Received 31 August 2012; Revised 2 March 2013; Accepted 5 March 2013
Academic Editor: Jose L. Gracia
Copyright © 2013 Meng Liu and Ke Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper is concerned with stochastic differential equations (SDEs) with multi-Markovian switching. The existence and uniqueness of solution are investigated, and the pth moment of the solution is estimated. The classical theory of SDEs with single Markovian switching is extended.
Stochastic modeling has played an important role in many branches of industry and science. SDEs with single continuous-time Markovian chain have been used to model many practical systems where they may experience abrupt changes in their parameters and structure caused by phenomena such as abrupt environment disturbances. SDEs with single Markovian switching can be denoted by with initial conditions and , where is a right-continuous homogenous Markovian chain on the probability space taking values in a finite state space and is -adapted but independent of the Brownian motion , and
Owing to their theoretical and practical significance, (1) has received great attention and has been recently studied extensively, and we here mention Skorokhod  and Mao and Yuan  among many others.
However, in the real world, the condition that coefficients and in (1) are perturbed by the same Markovian chain is too restrictive. For example, in the classical Black-Scholes model, the asset price is given by a geometric Brownian motion where is the rate of the return of the underlying assert, is the volatility, and is a scalar Brownian motion. Since there is strong evidence to indicate that is not a constant but is a Markovian jump process (see, e.g., [3, 4]), many authors proposed the following model: However, many stochastic factors that affect are different from those that affect . Then the following model is more appropriate than model (105) to describe this problem: where is a right-continuous homogenous Markovian chain taking values in a finite state space, . Another example is the stochastic Lotka-Volterra model with single Markovian switching which has received great attention and has been studied extensively recently (see, e.g., [5–12]). For the sake of convenience, we take the following two-dimensional competitive model as an example: where is the size of th species at time , represents the growth rate of th species in regime for , , and and are independent standard Brownian motions. However, there are many stochastic factors that affect some coefficients intensely but have little impact on other coefficients in (6). For example, suppose that the stochastic factor is rain falls and is able to endure a damp weather while is fond of a dry environment, then the rain falls will affect intensely but have little impact on . Thus, a more appropriate model is governed by where and are right-continuous homogenous Markovian chains taking values in finite state spaces for , , and for , respectively.
Thus the above examples show that the study of the following SDEs with multi-Markovian switchings is essential and is of great importance from both theoretical and practical points: with initial conditions and , where is a right-continuous homogenous Markovian chain on the probality space taking values in a finite state space and is -adapted but independent of the Brownian motion , , and Equation (8) can be regarded as the result of the equations switching among each other according to the movement of the Markovian chains. It is important for us to discover the properties of the system (8) and to find out whether the presence of two Markovian switchings affects some known results. The first step and the foundation of those studies are to establish the theorems for the existence and uniqueness of the solution to system (8). So in this paper, we will give some theorems for the existence and uniqueness of the solution to system (8) and study some properties of this solution. The theory developed in this paper is the foundation for further study and can be applied in many different and complicated situations, and hence the importance of the results in this paper is clear.
It should be pointed out that the theory developed in this paper can be generalized to cope with the more general SDEs with more Markovian chains The reason we concentrate on (8) rather than (11) is to avoid the notations becoming too complicated. Once the theory developed in this paper is established, the reader should be able to cope with the more general (11) without any difficulty.
The remaining part of this paper is as follows. In Section 2, the sufficient criteria for existence and uniqueness of solution, local solution, and maximal local solution will be established, respectively. In Section 3, the -estimates of the solution will be given. In Section 4, we will introduce an example to illustrate our main result. Finally, we will close the paper with conclusions in Section 5.
2. SDEs with Markovian Chains
Throughout this paper, let be a complete probability space. Let be an -dimensional Brownian motion defined on the probability space.
In this section, we will consider (8). Let . We impose a hypothesis.
(H1): is independent of .
Then is a homogenous vector Markovian chain with transition probabilities where .
Now, we will prepare some lemmas which are important for further study.
Lemma 1. has the following properties:(i) for ;(ii) for ;(iii), where if , otherwise , ;(iv)(Chapman-Kolmogorov equation) For and ,
Proof. The proofs of (i), (ii), and (iii) are obvious. Now, let us prove (iv): This completes the proof.
Now, we impose another hypothesis, which is called standard condition.
Lemma 2. Under Assumption (H2), for all , one has .
Proof. From and (H2) we know that, for arbitrary fixed , we have for sufficient large . Then making use of Chapman-Kolmogorov equation gives which is the desired assertion.
Lemma 3. Under Assumption (H2), for all , exists (but may be ).
Proof. Define . Then making use of (16) gives
Set . It is easy to see that
Now we will assert
In fact, for , such that . Applying (19) yields Note that , , whenever , then for all we have This implies . Thus Using the definition of gives which is the required assertion.
Lemma 4. Under Assumption (H2), for , exists and is finite.
Proof. By (H2), we note that for all , , such that
For , set , where . Let where means that the probability of the will not reach to at times but will reach to at time . Note that if , then which indicates Then making use of we obtain Consequently, Dividing both sides of the above inequality by and noting whenever yield Then letting gives and the required assertion follows immediately by letting . This completes the proof.
Set , then it is easy to see that almost every sample path of is a right continuous step function. Now letting , . Then by Chapman-Kolmogorov equation we have Letting and taking limits give Note that Then by solving the ordinary differential equations (38) and (39), we obtain the following lemma.
Lemma 5. For and one has
We are now in the position to give the sufficient conditions for the existence and uniqueness of the solution of (8). For this end, let us first give the definition of the solution.
Definition 6. An -valued stochastic process is called a solution of (8) if it has the following properties:(i) is continuous and -adapted;(ii) while ;(iii)for any , equation holds with probability 1.
A solution is said to be unique if any other solution is indistinguishable from .
Now we can give our main results in this section.
Theorem 7. Assume that there exist two positive constants and such that.
(Lipschitz condition) for all , and (Linear growth condition) for all Then there exists a unique solution to (8) and, moreover, the solution obeys
Proof. Recall that almost every sample path of is a right continuous step function with a finite number of jumps on . Thus there exists a sequence of stopping times such that (i)for almost every there is a finite for and if ; (ii)both and in are constants on interval , namely,
First of all, let us consider (8) on , then (8) becomes
with initial conditions . Then by the theory of SDEs, we obtain that (46) has a unique solution which obeys . We next consider (8) on which becomes
Again by the theory of SDEs, (47) has a unique solution which obeys . Repeating this procedure, we conclude that (8) has a unique solution on .
Now, let us prove (44). For every , define the stopping time It is obvious that a.s. Set for . Then obeys the equation Making use of the elementary inequality , the Hölder inequality, and (43), we can see that Thus, applying the Doob martingale inequality and (43), we can further show that That is to say, Using the Gronwall inequality leads to Consequently, Then the required inequality (44) follows immediately by letting .
Condition (42) indicates that the coefficients and do not change faster than a linear function of as change in . This means in particular the continuity of and in for all . Then functions that are discontinuous with respect to are excluded as the coefficients. Besides, there are many functions that do not satisfy the Lipschitz condition. These imply that the Lipschitz condition is too restrictive. To improve this Lipschitz condition let us introduce the concept of local solution.
Definition 8. Let be a stopping time such that a.s. An -valued -adapted continuous stochastic process is called a local solution of (8) if and, moreover, there is a nondecreasing sequence such that a.s. and holds for any and with probability one. If, furthermore, then it is called a maximal local solution and is called the explosion time. A maximal local solution is said to be unique if any other maximal local solution is indistinguishable from it, namely, and for with probability one.
Definition 9 (local Lipschitz condition). For every integer , there exists a positive constant such that, for all , and those with
The following theorem shows the existence of unique maximal local solution under the local Lipschitz condition without the linear growth condition.
Proof. Define functions
Then and satisfy the Lipschitz condition and the linear growth condition. Thus by Theorem 7, there is a unique solution of the equation
with the initial conditions and . Define the stopping times
Clearly, if ,
which indicates that is increasing so has its limit . Define by
where . Applying (61), one can show that . It then follows from (59) that
for any and . It is easy to see that if , then
Therefore is a maximal local solution.
Now, we will prove the uniqueness. Let be another maximal local solution. Define Then a.s. and Letting gives In order to complete the proof, we need only to show that a.s. In fact, for almost every , we have which contradicts the fact that is continuous on . This implies a.s. In the same way, one can show a.s. Thus we must have a.s. This completes the proof.
In many situations, we often consider an SDE on with initial data and . If the assumption of the existence-and-uniqueness theorem holds on every finite subinterval of , then (69) has a unique solution on the entire interval . Such a solution is called a global solution. To establish a more general result about global solution, we need more notations. To this end, we introduce an operator from to which is given by where and
Proof. We need only to prove the theorem for any initial condition and . From Theorem 10, we know that the local Lipschitz condition guarantees the existence of the unique maximal solution on , where is the explosion time. We need only to show a.s. If this is not true, then we can find a pair of positive constants and such that For each integer , define the stopping time Since almost surely, we can find a sufficiently large integer for Fix any , then for any , by virtue of the generalized Itô formula (see, e.g., ) Making use of the Gronwall inequality gives Therefore At the same time, set Then (72) means . It follows from (76) and (79) that Letting yields a contradiction, that is to say, . The proof is complete.
In the previous section, we have investigated the existence and uniqueness of the solution to (8). In this section, as above, let , be the unique solution of (8) with initial conditions and , and we will estimate the th moment of the solution.
Theorem 12. Assume that there is a function and positive constants such that for all , Assume also the initial condition and obeys , then one has
Proof. For each integer , define the stopping time Thus a.s. Using the generalized Itô's formula and (83), we obtain that for Then the Gronwall inequality indicates for all . By virtue of condition (83) we obtain the required assertion (84).
Corollary 13. Assume and . Assume also that there exists a constant such that, for all , Then one has
Proof. Define . Making use of (88) yields
Then by Theorem 12, we get
and the required assertion (89) follows.
It is useful to point out that if the linear growth condition (43) is satisfied, then (88) is fulfilled with . Now, we will show the other important properties of the solution.
Theorem 14. Let and . Assume also that the linear growth condition (43) holds. Then one has where and . Particularly, the th moment of the solution is continuous on .
Proof. Applying the elementary inequality , the Hölder inequality, and the linear growth condition, we can derive that where . Using (91) yields which is the desired inequality.
Theorem 15. Let and . Assume that there is a such that for all Then
Proof. Making use of the generalized Itô's formula and condition (96), we have
Therefore, for any ,
At the same time, applying the well-known Burkholder-Davis-Gundy inequality (see, e.g., ) gives
Substituting the above inequality into (100) gives
Then the required assertion follows from the Gronwall inequality.
Up to now, we have discussed the -estimates for the solution in the case when . As for , the similar results can be given without any difficulty as long as we note that the Hölder inequality implies
Consider the following Black-Scholes model: where is a right-continuous homogenous Markovian chain taking values in finite state spaces and is a right-continuous homogenous Markovian chain taking values in finite state spaces , , , , . Taking , , then (42) and (43) hold. Therefore, by Theorem 7, (105) has a unique solution.
5. Conclusions and Further Research
This paper is devoted to studying the existence and uniqueness of solution of SDEs with multi-Markovian switchings and estimating the th moment of the solution. We have used two continuous-time Markovian chains to model the SDEs. This area is becoming increasingly useful in engineering, economics, communication theory, active networking, and so forth. The sufficient criteria for existence and uniqueness of solution, local solution, and maximal local solution were established. Those results indicate that (8) keeps many properties that (89) owns. At the same time, although the hypothesis (H1) is used in this paper, we want to point out that this hypothesis is not essential. In fact, (H1) can be replaced by the following generalized hypothesis.
(H1)′: both and are right-continuous homogenous Markovian chains such that is a homogenous vector chain.
Under hypothesis (H1)′, the results given in this paper can be established similarly. It is easy to see that if and is a right-continuous homogenous Markovian chain, then (H1)′ is fulfilled immediately. At the same time, if