Stochastic Differential Equations with Multi-Markovian Switching

Owing to their theoretical and practical significance, (1) has received great attention and has been recently studied extensively, andwe heremention Skorokhod [1] andMao and Yuan [2] among many others. However, in the real world, the condition that coefficients f and g 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


Introduction
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  () =  ( () , ,  ())  +  ( () , ,  ())  () ,  0 ≤  ≤  (1) with initial conditions ( 0 ) =  0 ∈  2 and ( 0 ) =  0 , where () is a right-continuous homogenous Markovian chain on the probability space taking values in a finite state space S = {1, 2, . . ., } and is F  -adapted but independent of the Brownian motion (), and  : R  × R + × S → R  ,  : R  × R + × S → R + . (2) Owing to their theoretical and practical significance, (1) has received great attention and has been recently studied extensively, and we here mention Skorokhod [1] and Mao and Yuan [2] 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:  () =  ( ())  ()  + ] ( ())  ()  () .
For the sake of convenience, we take the following twodimensional competitive model as an example: where   is the size of th species at time ,  0 () represents the growth rate of th species in regime  for  = 1,2,  ∈ S, and  1 and  2 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  1 is able to endure a damp weather while  2 is fond of a dry environment, then the rain falls will affect  2 intensely but have little impact on  1 .Thus, a more appropriate model is governed by where   () and   () are right-continuous homogenous Markovian chains taking values in finite state spaces S  for  = 1, 2,  = 0, 1, 2, and S  for  = 1, 2, 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 ( 0 ) =  0 ∈  2 F 0 and   ( 0 ), where   () is a right-continuous homogenous Markovian chain on the probality space taking values in a finite state space S  = {1, 2, . . .,   } and is F  -adapted but independent of the Brownian motion   (),  = 1, 2, and Equation ( 8) can be regarded as the result of the  1 ×  2 equations  () =  ( () , , )  +  ( () , , )  () ,  ∈ S 1 ,  ∈ S 2 (10) 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  () =  ( () , ,  1 () , . . .,   ())  +  ( () , ,  +1 () , . . .,  + ())  () . ( 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.
Proof.From   1  2 ; 1  2 (0) > 0 and (H2) we know that, for arbitrary fixed  > 0, we have for sufficient large .Then making use of Chapman-Kolmogorov equation gives which is the desired assertion.
Proof.By (H2), we note that for all 0 <  < 1/3, ∃0 <  < 1, such that where (ℎ) means that the probability of the () will not reach to ( 1 ,  2 ) at times ℎ, 2ℎ, . . ., ( − 1)ℎ but will reach to ( 1 ,  2 ) at time ℎ.Note that if ℎ ≤  ≤ , then which indicates Then making use of we obtain Consequently, Dividing both sides of the above inequality by ℎ and noting ℎ →  whenever ℎ → 0 yield lim sup Then letting  → 0 gives lim sup and the required assertion follows immediately by letting  → 0. This completes the proof.
Set () = ( 1 (),  2 ()), 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 ℎ → 0 and taking limits give Note that Then by solving the ordinary differential equations ( 38) and (39), we obtain the following lemma.
Lemma 5.For P() and Q 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 R  -valued stochastic process {()}  0 ≤≤ is called a solution of (8) if it has the following properties: First of all, let us consider (8) on  ∈ [[ 0 ,  1 [[, then (8) becomes with initial conditions ( 0 ) =  0 , ( 0 ) = ( 1 ( 0 ),  1 ( 0 )).
Then by the theory of SDEs, we obtain that (46) has a unique solution which obeys ( 1 ) ∈  2 Again by the theory of SDEs, (47) has a unique solution which obeys ( 1 ) ∈  2 (Ω; R  ).Repeating this procedure, we conclude that (8) has a unique solution () on [ 0 , ].
Now, let us prove (44).For every  ≥ 1, define the stopping time It is obvious that Then   () obeys the equation Consequently, Then the required inequality (44) follows immediately by letting  → ∞.Condition (42) indicates that the coefficients (, ,  1 ) and (, ,  2 ) do not change faster than a linear function of  as change in .This means in particular the continuity of (, ,  1 ) and (, ,  2 ) in  for all  ∈ [ 0 , ].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.
Proof.We need only to prove the theorem for any initial condition  0 ∈ R  and ( 1 ( 0 ),  2 ( 0 )) ∈ S. 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  ≥ 1, define the stopping time Since   →  ∞ almost surely, we can find a sufficiently large integer  0 for Fix any  ≥  0 , then for any  0 ≤  ≤ , by virtue of the generalized Itô formula (see, e.g., [1]) 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.

𝐿 𝑃 -Estimates
In the previous section, we have investigated the existence and uniqueness of the solution to (8).In this section, as above, let (),  0 ≤  ≤  be the unique solution of ( 8) with initial conditions ( 0 ) =  0 and ( 0 ), and we will estimate the th moment of the solution.
Up to now, we have discussed the   -estimates for the solution in the case when  ≥ 2. As for 0 <  < 2, the similar results can be given without any difficulty as long as we note that the Hölder inequality implies

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.
Under hypothesis (H1)  , the results given in this paper can be established similarly.It is easy to see that if  1 () ≡  2 () and  1 () is a right-continuous homogenous Markovian chain, then (H1)  is fulfilled immediately.At the same time, if  1 () ≡  2 (), (8) will reduce to the classical SDEs with single Markovian chain; that is to say, the classical theory about SDEs with single Markovian chain is a special case of our theory.On the other hand, many theorems in this paper will play important roles in further study.For example, Theorem 15 will be useful when one studies the approximate solutions.
Some important and interesting questions can be further investigated using the results in this paper.For example, approximate solutions, boundedness and stability, stochastic functional differential equations with vector Markovian switching and their applications.In particular, the stability of ( 8) is one of the most important and interesting topics, and those investigations are in progress.