Stochastic Differential Equations with Multi-Markovian Switching

Abstract

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.

1. 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 (𝑡)𝑥(𝑡)(𝑡)𝑥(𝑡)(𝑡)(𝑡)𝑑𝑥=𝑓,𝑡,𝛾𝑑𝑡+𝑔,𝑡,𝛾𝑑𝐵,𝑡₀≤𝑡≤𝑇 (1) with initial conditions 𝑥(𝑡₀)=𝑥₀∈𝐿²_ℱ𝑡0 and 𝛾(𝑡₀)=𝛾₀, where 𝛾(𝑡) is a right-continuous homogenous Markovian chain on the probability space taking values in a finite state space 𝕊={1,2,…,𝑁} and is ℱ_𝑡-adapted but independent of the Brownian motion 𝐵(𝑡), and 𝑓∶ℝ^𝑛×ℝ₊×𝕊→ℝ^𝑛,𝑔∶ℝ^𝑛×ℝ₊×𝕊→ℝ^𝑛+𝑚. (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 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 𝑑𝑋(𝑡)=𝜇𝑋(𝑡)𝑑𝑡+𝜈𝑋(𝑡)𝑑𝐵(𝑡), (3) where 𝜇 is the rate of the return of the underlying assert, 𝜈 is the volatility, and 𝐵(𝑡) is a scalar Brownian motion. Since there is strong evidence to indicate that 𝜇 is not a constant but is a Markovian jump process (see, e.g., [3, 4]), many authors proposed the following model: 𝑑𝑋(𝑡)=𝜇𝛾(𝑡)𝑋(𝑡)𝑑𝑡+𝜈𝛾(𝑡)𝑋(𝑡)𝑑𝐵(𝑡). (4) However, many stochastic factors that affect 𝜇 are different from those that affect 𝜈. Then the following model is more appropriate than model (105) to describe this problem: 𝑑𝑋(𝑡)=𝜇𝛾₁(𝑡)𝑋(𝑡)𝑑𝑡+𝜈𝛾₂(𝑡)𝑋(𝑡)𝑑𝐵(𝑡), (5) where 𝛾_𝑖(𝑡) is a right-continuous homogenous Markovian chain taking values in a finite state space, 𝑖=1,2. 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: 𝑑𝑥₁=𝑥₁𝑟₁₀𝛾(𝑡)−𝑎₁₁𝛾(𝑡)𝑥₁−𝑎₁₂𝛾(𝑡)𝑥₂𝑑𝑡+𝛼₁𝛾(𝑡)𝑥₁𝑑𝐵₁(𝑡),𝑑𝑥₂=𝑥₂𝑟₂₀𝛾(𝑡)−𝑎₂₁𝛾(𝑡)𝑥₁−𝑎₂₂𝛾(𝑡)𝑥₂𝑑𝑡+𝛼₂𝛾(𝑡)𝑥₂𝑑𝐵₂(𝑡), (6) where 𝑥_𝑖 is the size of 𝑖th species at time 𝑡, 𝑟_𝑖0(𝑗) represents the growth rate of 𝑖th species in regime 𝑗 for 𝑖=1,2, 𝑗∈𝕊, and 𝐵₁ and 𝐵₂ 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 𝑥₁ is able to endure a damp weather while 𝑥₂ is fond of a dry environment, then the rain falls will affect 𝑥₂ intensely but have little impact on 𝑥₁. Thus, a more appropriate model is governed by 𝑑𝑥₁=𝑥₁𝑟₁₀𝛾₁₀(𝑡)−𝑎₁₁𝛾₁₁(𝑡)𝑥₁−𝑎₁₂𝛾₁₂(𝑡)𝑥₂𝑑𝑡+𝛼₁𝛾₁(𝑡)𝑥₁𝑑𝐵₁(𝑡),𝑑𝑥₂=𝑥₂𝑟₂₀𝛾₂₀(𝑡)−𝑎₂₁𝛾₂₁(𝑡)𝑥₁−𝑎₂₂𝛾₂₂(𝑡)𝑥₂𝑑𝑡+𝛼₂𝛾₂(𝑡)𝑥₂𝑑𝐵₂(𝑡), (7) where 𝛾_𝑖𝑗(𝑡) and 𝛾_𝑘(𝑡) are right-continuous homogenous Markovian chains taking values in finite state spaces 𝕊_𝑖𝑗 for 𝑖=1,2, 𝑗=0,1,2, and 𝕊_𝑘 for 𝑘=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: 𝑑𝑥(𝑡)=𝑓𝑥(𝑡),𝑡,𝛾₁(𝑡)𝑥(𝑡)𝑑𝑡+𝑔,𝑡,𝛾₂(𝑡)𝑑𝐵(𝑡),𝑡₀≤𝑡≤𝑇 (8) with initial conditions 𝑥(𝑡₀)=𝑥₀∈𝐿²_ℱ0 and 𝛾_𝑖(𝑡₀), where 𝛾_𝑖(𝑡) is a right-continuous homogenous Markovian chain on the probality space taking values in a finite state space 𝕊_𝑖={1,2,…,𝑁_𝑖} and is ℱ_𝑡-adapted but independent of the Brownian motion 𝐵_𝑖(𝑡), 𝑖=1,2, and 𝑓∶ℝ^𝑛×ℝ₊×𝕊₁→ℝ^𝑛,𝑔∶ℝ^𝑛×ℝ₊×𝕊₂→ℝ^𝑛+𝑚. (9) Equation (8) can be regarded as the result of the 𝑁₁×𝑁₂ equations (𝑡)(𝑥(𝑡))𝑥(𝑡)(𝑡)𝑑𝑥=𝑓,𝑡,𝑖𝑑𝑡+𝑔,𝑡,𝑗𝑑𝐵,𝑖∈𝕊₁,𝑗∈𝕊₂ (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 𝑑𝑥(𝑡)=𝑓𝑥(𝑡),𝑡,𝛾₁(𝑡),…,𝛾_𝑛(𝑡)𝑥(𝑡)𝑑𝑡+𝑔,𝑡,𝛾_𝑛+1(𝑡),…,𝛾_𝑛+𝑚(𝑡)𝑑𝐵(𝑡). (11) 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 𝐿^𝑃-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 (Ω,ℱ,{ℱ_𝑡}_𝑡∈𝑅+,𝒫) be a complete probability space. Let 𝐵(𝑡)=(𝐵₁(𝑡),…,𝐵_𝑚(𝑡))^𝑇 be an 𝑚-dimensional Brownian motion defined on the probability space.

In this section, we will consider (8). Let 𝛾(𝑡)=(𝛾₁(𝑡),𝛾₂(𝑡)). We impose a hypothesis.

(H1): 𝛾₁(𝑡) is independent of 𝛾₂(𝑡).

Then 𝛾(𝑡) is a homogenous vector Markovian chain with transition probabilities 𝑃𝛾(𝑠+𝑡)=𝑗₁,𝑗₂∣𝛾(𝑠)=𝑖₁,𝑖₂=𝑃𝛾(𝑡)=𝑗₁,𝑗₂∣𝛾(0)=𝑖₁,𝑖₂=∶𝑃_{𝑖1𝑖2;𝑗1𝑗2}(𝑡), (12) where (𝑖₁,𝑖₂),(𝑗₁,𝑗₂)∈𝕊={(1,1),(1,2),…,(1,𝑁₂),(2,1),…,(𝑁₁,𝑁₂)}.

Now, we will prepare some lemmas which are important for further study.

Lemma 1. 𝑃_{𝑖1𝑖2;𝑗1𝑗2}(𝑡) has the following properties:(i)𝑃_{𝑖1𝑖2;𝑗1𝑗2}(𝑡)≥0 for (𝑖₁,𝑖₂),(𝑗₁,𝑗₂)∈𝕊;(ii)∑_{(𝑗1,𝑗2)∈𝕊}𝑃_{𝑖1𝑖2;𝑗1𝑗2}(𝑡)=1 for (𝑖₁,𝑖₂)∈𝕊;(iii)𝑃_{𝑖1𝑖2;𝑗1𝑗2}(0)=𝛿_𝑖1𝑗1⋅𝛿_𝑖2𝑗2, where 𝛿_{𝑖𝑘𝑗𝑘}=1 if 𝑖_𝑘=𝑗_𝑘, otherwise 𝛿_{𝑖𝑘𝑗𝑘}=0, 𝑘=1,2;(iv)(Chapman-Kolmogorov equation) For 𝑠,𝑡≥0 and (𝑖₁,𝑖₂),(𝑗₁,𝑗₂)∈𝕊, 𝑃_{𝑖1𝑖2;𝑗1𝑗2}(𝑠+𝑡)=_{𝑘1,𝑘2∈𝕊}𝑃_{𝑖1𝑖2;𝑘1𝑘2}(𝑡)𝑃_{𝑘1𝑘2;𝑗1𝑗2}(𝑠). (13)

Proof. The proofs of (i), (ii), and (iii) are obvious. Now, let us prove (iv): 𝑃_{𝑖1𝑖2;𝑗1𝑗2}()𝛾()=𝑗𝑠+𝑡=𝑃𝑠+𝑡₁,𝑗₂∣𝛾(0)=𝑖₁,𝑖₂=_{(𝑘1,𝑘2)∈𝕊}𝑃𝛾(𝑠+𝑡)=𝑗₁,𝑗₂,𝛾(𝑡)=𝑘₁,𝑘₂∣𝛾(0)=𝑖₁,𝑖₂=_{(𝑘1,𝑘2)∈𝕊}𝑃𝛾(𝑠+𝑡)=𝑗₁,𝑗₂,𝛾(𝑡)=𝑘₁,𝑘₂,𝛾(0)=𝑖₁,𝑖₂×𝑃𝛾(𝑡)=𝑘₁,𝑘₂,𝛾(0)=𝑖₁,𝑖₂⁻¹×𝑃𝛾(𝑡)=𝑘₁,𝑘₂,𝛾(0)=𝑖₁,𝑖₂𝑃𝛾(0)=𝑖₁,𝑖₂=_{(𝑘1,𝑘2)∈𝕊}𝑃𝛾(𝑠+𝑡)=𝑗₁,𝑗₂∣𝛾(𝑡)=𝑘₁,𝑘₂,𝛾(0)=𝑖₁,𝑖₂×𝑃𝛾(𝑡)=𝑘₁,𝑘₂∣𝛾(0)=𝑖₁,𝑖₂=_{𝑘1,𝑘2∈𝕊}𝑃𝛾(𝑠+𝑡)=𝑗₁,𝑗₂∣𝛾(𝑡)=𝑘₁,𝑘₂×𝑃𝛾(𝑡)=𝑘₁,𝑘₂∣𝛾(0)=𝑖₁,𝑖₂=_{𝑘1,𝑘2∈𝕊}𝑃_{𝑘1𝑘2;𝑗1𝑗2}(𝑠)𝑃_{𝑖1𝑖2;𝑘1𝑘2}(𝑡). (14) This completes the proof.

Now, we impose another hypothesis, which is called standard condition.

(H2): lim_𝑡→0𝑃_{𝑖1𝑖2;𝑗1𝑗2}(𝑡)=𝛿_{𝑖1𝑖2;𝑗1𝑗2}.

Lemma 2. Under Assumption (H2), for all 𝑡≥0,(𝑖₁,𝑖₂)∈𝕊, one has 𝑃_{𝑖1𝑖2;𝑖1𝑖2}(𝑡)>0.

Proof. From 𝑃_{𝑖1𝑖2;𝑖1𝑖2}(0)>0 and (H2) we know that, for arbitrary fixed 𝑡>0, we have 𝑃_{𝑖1𝑖2;𝑖1𝑖2}𝑡𝑛>0 (15) for sufficient large 𝑛. Then making use of Chapman-Kolmogorov equation 𝑃_{𝑖1𝑖2;𝑖1𝑖2}(𝑠+𝑡)=_{𝑘1,𝑘2∈𝕊}𝑃_{𝑖1𝑖2;𝑘11𝑘2}(𝑡)𝑃_{𝑘1𝑘2;𝑖1𝑖2}(𝑠)≥𝑃_{𝑖1𝑖2;𝑖1𝑖2}(𝑡)𝑃_{𝑖1𝑖2;𝑖1𝑖2}(𝑠) (16) gives 𝑃_{𝑖1𝑖2;𝑖1𝑖2}(𝑡)≥𝑃_{𝑖1𝑖2;𝑖1𝑖2}𝑡𝑛^𝑛>0, (17) which is the desired assertion.

Lemma 3. Under Assumption (H2), for all (𝑖₁,𝑖₂)∈𝕊, −𝑞_{𝑖1𝑖2;𝑖1𝑖2}∶=lim_𝑡→01−𝑃_{𝑖1𝑖2;𝑖1𝑖2}(𝑡)𝑡 (18) exists (but may be ∞).

Proof. Define 𝜙(𝑡)=−ln𝑃_{𝑖1𝑖2;𝑖1𝑖2}(𝑡)≥0. Then making use of (16) gives 𝜙(𝑠+𝑡)≤𝜙(𝑠)+𝜙(𝑡). (19) Set −𝑞_{𝑖1𝑖2;𝑖1𝑖2}=sup_𝑡>0(𝜙(𝑡)/𝑡). It is easy to see that 0≤−𝑞_{𝑖1𝑖2;𝑖1𝑖2}≤∞,limsup_𝑡→0𝜙(𝑡)𝑡≤−𝑞_{𝑖1𝑖2;𝑖1𝑖2}. (20) Now we will assert liminf_𝑡→0𝜙(𝑡)𝑡≥−𝑞_{𝑖1𝑖2;𝑖1𝑖2}. (21)
In fact, for 0<ℎ<𝑡, ∃𝑛,0≤𝜀≤ℎ such that 𝑡=𝑛ℎ+𝜀. Applying (19) yields 𝜙(𝑡)𝑡≥𝑛ℎ𝜀𝜙(ℎ)ℎ+𝜙(𝜀)𝑡. (22) Note that 𝜀→0, 𝑛ℎ/𝜀→1, 𝜙(𝜀)→0 whenever ℎ→0⁺, then for all 𝑡>0 we have 𝜙(𝑡)𝑡≤liminf_ℎ→0𝜙(ℎ)ℎ. (23) This implies −𝑞_{𝑖1𝑖2;𝑖1𝑖2}≤liminf_𝑡→0(𝜙(𝑡)/𝑡). Thus −𝑞_{𝑖1𝑖2;𝑖1𝑖2}=lim_𝑡→0𝜙(𝑡)𝑡. (24) Using the definition of 𝜙(𝑡) gives lim_𝑡→01−𝑃_{𝑖1𝑖2;𝑖1𝑖2}(𝑡)𝑡=lim_𝑡→01−exp𝜙(𝑡)𝜙(𝑡)𝜙(𝑡)𝑡=−𝑞_{𝑖1𝑖2;𝑖1𝑖2}, (25) which is the required assertion.

Lemma 4. Under Assumption (H2), for (𝑖₁,𝑖₂),(𝑗₁,𝑗₂)∈𝕊,(𝑖₁,𝑖₂)≠(𝑗₁,𝑗₂), 𝑞_{𝑖1𝑖2;𝑗1𝑗2}∶=𝑃^′_{𝑖1𝑖2;𝑗1𝑗2}(0)=lim_𝑡→0𝑃_{𝑖1𝑖2;𝑗1𝑗2}(𝑡)𝑡 (26) exists and is finite.

Proof. By (H2), we note that for all 0<𝜀<1/3, ∃0<𝛿<1, such that 𝑃_{𝑖1𝑖2;𝑖1𝑖2}(𝑡)>1−𝜀,𝑃_{𝑗1𝑗2;𝑗1𝑗2}(𝑡)𝑃>1−𝜀,_{𝑗1𝑗2;𝑖1𝑖2}(𝑡)<𝜀 (27) provided 0<𝑡≤𝛿.
For ∀0≤ℎ<𝑡, set 𝑛=⟨𝑡/ℎ⟩, where ⟨𝑎⟩=max_𝑛≤𝑎{𝑛∈ℤ}. Let 𝑃^(𝑗1𝑗2)_{𝑖1𝑖2;𝑘1𝑘2}(ℎ)=𝑃_{𝑖1𝑖2;𝑘1𝑘2}(ℎ),𝑃^(𝑗1𝑗2)_{𝑖1𝑖2;𝑘1𝑘2}(𝑚ℎ)=_{𝑟1,𝑟2≠𝑗1,𝑗2}𝑃^(𝑗1𝑗2)_{𝑖1𝑖2;𝑟1𝑟2}((𝑚−1)ℎ)𝑃_{𝑟1𝑟2;𝑘1𝑘2}(ℎ), (28) where 𝑃^(𝑗1𝑗2)_{𝑖1𝑖2;𝑘1𝑘2}(𝑚ℎ) means that the probability of the 𝛾(𝑡) will not reach to (𝑗₁,𝑗₂) at times ℎ,2ℎ,…,(𝑚−1)ℎ but will reach to (𝑘₁,𝑘₂) at time 𝑚ℎ. Note that if ℎ≤𝑡≤𝛿, then 𝜀>1−𝑃_{𝑖1𝑖2;𝑖1𝑖2}(𝑡)=_{𝑘1,𝑘2≠𝑖1,𝑖2}𝑃_{𝑖1𝑖2;𝑘1𝑘2}(𝑡)≥𝑃_{𝑖1𝑖2;𝑗1𝑗2}(𝑡)≥^𝑛_𝑚=1𝑃^(𝑗1𝑗2)_{𝑖1𝑖2;𝑗1𝑗2}(𝑚ℎ)𝑃_{𝑗1𝑗2;𝑗1𝑗2}()≥()𝑡−𝑚ℎ1−𝜀^𝑛_𝑚=1𝑃^(𝑗1𝑗2)_{𝑖1𝑖2;𝑗1𝑗2}(𝑚ℎ), (29) which indicates ^𝑛_𝑚=1𝑃^(𝑗1𝑗2)_{𝑖1𝑖2;𝑗1𝑗2}()≤𝜀𝑚ℎ.1−𝜀 (30) Then making use of 𝑃_{𝑖1𝑖2;𝑖1𝑖2}(𝑚ℎ)=𝑃^(𝑗1𝑗2)_{𝑖1𝑖2;𝑖1𝑖2}()+𝑚ℎ_{𝑚−1𝑙=1}𝑃^(𝑗1𝑗2)_{𝑖1𝑖2;𝑗1𝑗2}(𝑙ℎ)𝑃_{𝑗1𝑗2;𝑖1𝑖2}((𝑚−𝑙)ℎ), (31) we obtain 𝑃^(𝑗1𝑗2)_{𝑖1𝑖2;𝑖1𝑖2}(𝑚ℎ)≥𝑃_{𝑖1𝑖2;𝑖1𝑖2}(𝑚ℎ)−^𝑚−1_𝑙=1𝑃^(𝑗1𝑗2)_{𝑖1𝑖2;𝑗1𝑗2}()𝜀𝑙ℎ≥1−𝜀−.1−𝜀 (32) Consequently, 𝑃_{𝑖1𝑖2;𝑗1𝑗2}(𝑡)>^𝑛_𝑚=1𝑃^(𝑗1𝑗2)_{𝑖1𝑖2;𝑖1𝑖2}((𝑚−1)ℎ)𝑃_{𝑖1𝑖2;𝑗1𝑗2}(ℎ)𝑃_{𝑗1𝑗2;𝑗1𝑗2}()𝜀𝑡−𝑚ℎ≥𝑛1−𝜀−𝑃1−𝜀^{𝑖1𝑖2;𝑗1𝑗2}(ℎ)()𝑃)(1−𝜀≥𝑛1−3𝜀_{𝑖1𝑖2;𝑗1𝑗2}(ℎ). (33) Dividing both sides of the above inequality by ℎ and noting 𝑛ℎ→𝑡 whenever ℎ→0 yield limsup_ℎ→0𝑃_{𝑖1𝑖2;𝑗1𝑗2}(ℎ)ℎ≤1𝑃1−3𝜀^{𝑖1𝑖2;𝑗1𝑗2}(𝑡)𝑡<∞. (34) Then letting 𝑡→0 gives limsup_ℎ→0𝑃_{𝑖1𝑖2;𝑗1𝑗2}(ℎ)ℎ≤11−3𝜀liminf_𝑡→0𝑃_{𝑖1𝑖2;𝑗1𝑗2}(𝑡)𝑡, (35) and the required assertion follows immediately by letting 𝜀→0. This completes the proof.

Set 𝛾(𝑡)=(𝛾₁(𝑡),𝛾₂(𝑡)), then it is easy to see that almost every sample path of 𝛾(𝑡) is a right continuous step function. Now letting 𝐏(𝑡)=(𝑃_{𝑖1𝑖2;𝑗1𝑗2}(𝑡))_{𝑁1𝑁2×𝑁1𝑁2}, 𝐐=(𝑞_{𝑖1𝑖2;𝑗1𝑗2})_{𝑁1𝑁2×𝑁1𝑁2}=𝐏^′(0). Then by Chapman-Kolmogorov equation 𝐏(𝑡+ℎ)=𝐏(𝑡)𝐏(ℎ)=𝐏(ℎ)𝐏(𝑡), (36) we have 𝐏(𝑡+ℎ)−𝐏(𝑡)ℎ(𝑡𝐏(ℎ)=𝐏)−𝐈ℎ=𝐏(ℎ)−𝐈ℎ𝐏(𝑡). (37) Letting ℎ→0 and taking limits give 𝐏^′(𝑡)(𝑡)𝐏=𝐏𝐐,_′(𝑡)=𝐐𝐏(𝑡). (38) Note that 𝐏(0)=𝐈. (39) Then by solving the ordinary differential equations (38) and (39), we obtain the following lemma.

Lemma 5. For 𝐏(𝑡) and 𝐐 one has 𝐏(𝑡)=exp{𝐐𝑡}. (40)

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^𝑛-valued stochastic process {𝑥(𝑡)}_{𝑡0≤𝑡≤𝑇} is called a solution of (8) if it has the following properties:(i){𝑥(𝑡)}_{𝑡0≤𝑡≤𝑇} is continuous and ℱ_𝑡-adapted;(ii){𝑓(𝑥(𝑡),𝑡,𝛾₁(𝑡))}_{𝑡0≤𝑡≤𝑇}∈ℒ¹([𝑡₀,𝑇];ℝ^𝑛) while {𝑔(𝑥(𝑡),𝑡,𝛾₂(𝑡))}_{𝑡0≤𝑡≤𝑇}∈ℒ²([𝑡₀,𝑇];ℝ^𝑛×𝑚);(iii)for any 𝑡∈[𝑡₀,𝑇], equation 𝑥(𝑡)=𝑥𝑡₀+^𝑡_𝑡0𝑓𝑥(𝑠),𝑠,𝛾₁(𝑠)+𝑑𝑠_𝑡𝑡0𝑔𝑥(𝑠),𝑠,𝛾₂(𝑠)𝑑𝐵(𝑠) (41)holds with probability 1.

A solution {𝑥(𝑡)}_{𝑡0≤𝑡≤𝑇} is said to be unique if any other solution {𝑥^∼(𝑡)}_{𝑡0≤𝑡≤𝑇} is indistinguishable from {𝑥(𝑡)}_{𝑡0≤𝑡≤𝑇}.

Now we can give our main results in this section.

Theorem 7. Assume that there exist two positive constants 𝐾 and 𝐾 such that.
(Lipschitz condition) for all 𝑥,𝑦∈ℝ^𝑛, 𝑡∈[𝑡₀,𝑇] and (𝑖₁,𝑖₂)∈𝕊𝑓𝑥,𝑡,𝑖₁−𝑓𝑦,𝑡,𝑖₁²∨𝑔𝑥,𝑡,𝑖₂−𝑔𝑦,𝑡,𝑖₂²≤𝐾𝑥−𝑦². (42) (Linear growth condition) for all (𝑥,𝑡,(𝑖₁,𝑖₂))∈ℝ^𝑛×[𝑡₀,𝑇]×𝕊𝑓𝑥,𝑡,𝑖₁²∨𝑔𝑥,𝑡,𝑖₂²≤𝐾(1+|𝑥|)². (43) Then there exists a unique solution 𝑥(𝑡) to (8) and, moreover, the solution obeys 𝐸sup_{𝑡0≤𝑡≤𝑇}|𝑥(𝑡)|²≤1+3𝐸𝑥₀²exp3𝐾𝑇−𝑡₀𝑇−𝑡₀+4. (44)

Proof. Recall that almost every sample path of 𝛾(𝑡)=(𝛾₁(𝑡),𝛾₂(𝑡)) is a right continuous step function with a finite number of jumps on [𝑡₀,𝑇]. Thus there exists a sequence of stopping times {𝜏_𝑘}_𝑘≥0 such that (i)for almost every 𝜔∈Ω there is a finite 𝑘_𝜔 for 𝑡₀=𝜏₀<𝜏₁<⋯<𝜏_𝑘=𝑇 and 𝜏_𝑘=𝑇 if 𝑘>𝑘_𝜔; (ii)both 𝛾₁(⋅) and 𝛾₂(⋅) in 𝛾(⋅) are constants on interval [[𝜏_𝑘,𝜏_𝑘+1[[, namely, 𝛾₁(𝑡)=𝛾₁𝜏_𝑘on𝜏_𝑘≤𝑡≤𝜏_𝑘+1𝛾∀𝑘≥0,₂(𝑡)=𝛾₂𝜏_𝑘on𝜏_𝑘≤𝑡≤𝜏_𝑘+1∀𝑘≥0. (45)First of all, let us consider (8) on 𝑡∈[[𝜏₀,𝜏₁[[, then (8) becomes 𝑑𝑥(𝑡)=𝑓𝑥(𝑡),𝑡,𝛾₁𝑡₀𝑥(𝑡)𝑑𝑡+𝑔,𝑡,𝛾₂𝑡₀𝑑𝐵(𝑡) (46) with initial conditions 𝑥(𝑡₀)=𝑥₀,𝛾(𝑡₀)=(𝛾₁(𝑡₀),𝛾₁(𝑡₀)). Then by the theory of SDEs, we obtain that (46) has a unique solution which obeys 𝑥(𝜏₁)∈𝐿²_ℱ𝜏1(Ω;ℝ^𝑛). We next consider (8) on 𝑡∈[[𝜏₁,𝜏₂[[ which becomes 𝑑𝑥(𝑡)=𝑓𝑥(𝑡),𝑡,𝛾₁𝜏₁𝑥(𝑡)𝑑𝑡+𝑔,𝑡,𝛾₂𝜏₁𝑑𝐵(𝑡). (47) Again by the theory of SDEs, (47) has a unique solution which obeys 𝑥(𝜏₁)∈𝐿²_ℱ𝜏2(Ω;ℝ^𝑛). Repeating this procedure, we conclude that (8) has a unique solution 𝑥(𝑡) on [𝑡₀,𝑇].
Now, let us prove (44). For every 𝑘≥1, define the stopping time 𝜏_𝑘=𝑇∧inf𝑡∈𝑡₀,𝑇∶|𝑥(𝑡)|≥𝑘. (48) It is obvious that 𝜏_𝑘↑𝑇 a.s. Set 𝑥_𝑘(𝑡)=𝑥(𝑡∧𝜏_𝑘) for 𝑡∈[𝑡₀,𝑇]. Then 𝑥_𝑘(𝑡) obeys the equation 𝑥_𝑘(𝑡)=𝑥₀+^𝑡_𝑡0𝑓𝑥_𝑘(𝑠),𝑠,𝛾₁(𝑠)𝐼_{[[𝑡0,𝜏𝑘]]}(𝑠)+𝑑𝑠_𝑡𝑡0𝑔𝑥_𝑘(𝑠),𝑠,𝛾₂(𝑠)𝐼_{[[𝑡0,𝜏𝑘]]}(𝑠)𝑑𝐵(𝑠). (49) Making use of the elementary inequality |𝑎+𝑏+𝑐|²≤3(|𝑎|²+|𝑏|²+|𝑐|²), the Hölder inequality, and (43), we can see that 𝑥_𝑘(𝑡)²=3𝑥₀²+3𝐾𝑇−𝑡₀^𝑡_𝑡01+𝑥_𝑘(𝑠)²𝑑𝑠+3^𝑡_𝑡0𝑔𝑥_𝑘(𝑠),𝑠,𝛾₂(𝑠)𝐼_{[[𝑡0,𝜏𝑘]]}(𝑠)𝑑𝐵(𝑠)². (50) Thus, applying the Doob martingale inequality and (43), we can further show that 𝐸sup_{𝑡0≤𝑠≤𝑡}𝑥_𝑘(𝑠)²=3𝐸𝑥₀²+3𝐾𝑇−𝑡₀^𝑡_𝑡01+𝐸𝑥_𝑘(𝑠)²𝑑𝑠+12𝐸^𝑡_𝑡0𝑔𝑥_𝑘(𝑠),𝑠,𝛾₂(𝑠)²𝐼_{[[𝑡0,𝜏𝑘]]}(𝑠)𝑥𝑑𝑠≤3𝐸₀²+3𝐾𝑇−𝑡₀+4^𝑡_𝑡01+𝐸𝑥_𝑘(𝑠)²𝑑𝑠. (51) That is to say, 1+𝐸sup_{𝑡0≤𝑠≤𝑡}𝑥_𝑘(𝑠)²≤1+3𝐸𝑥₀²+3𝐾𝑇−𝑡₀×+4_𝑡𝑡01+𝐸sup_{𝑡0≤𝑟≤𝑠}𝑥_𝑘(𝑟)²𝑑𝑠. (52) Using the Gronwall inequality leads to 1+𝐸sup_{𝑡0≤𝑠≤𝑡}𝑥_𝑘(𝑠)²≤1+3𝐸𝑥₀²exp3𝐾𝑇−𝑡₀𝑇−𝑡₀+4. (53) Consequently, 1+𝐸sup_{𝑡0≤𝑡≤𝜏𝑘}|𝑥(𝑡)|²≤1+3𝐸𝑥₀²exp3𝐾𝑇−𝑡₀𝑇−𝑡₀+4. (54) Then the required inequality (44) follows immediately by letting 𝑘→∞.
Condition (42) indicates that the coefficients 𝑓(𝑥,𝑡,𝑖₁) and 𝑔(𝑥,𝑡,𝑖₂) do not change faster than a linear function of 𝑥 as change in 𝑥. This means in particular the continuity of 𝑓(𝑥,𝑡,𝑖₁) and 𝑔(𝑥,𝑡,𝑖₂) in 𝑥 for all 𝑡∈[𝑡₀,𝑇]. Then functions that are discontinuous with respect to 𝑥 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 𝜎_∞ be a stopping time such that 𝑡₀≤𝜎_∞≤𝑇 a.s. An ℝ^𝑛-valued ℱ_𝑡-adapted continuous stochastic process {𝑥(𝑡)}_{𝑡0≤𝑡<𝜎∞} is called a local solution of (8) if 𝑥(𝑡₀)=𝑥₀ and, moreover, there is a nondecreasing sequence {𝜎_𝑘}_𝑘≥1 such that 𝑡₀≤{𝜎_𝑘}↑𝜎_∞ a.s. and 𝑥(𝑡)=𝑥𝑡₀+^{𝑡∧𝜎𝑘}_𝑡0𝑓𝑥(𝑠),𝑠,𝛾₁(𝑠)+𝑑𝑠_{𝑡∧𝜎𝑘𝑡0}𝑔𝑥(𝑠),𝑠,𝛾₂(𝑠)𝑑𝐵(𝑠) (55) holds for any 𝑡∈[𝑡₀,𝑇) and 𝑘≥1 with probability one. If, furthermore, limsup_{𝑡→+𝜎∞}|𝑥(𝑡)|=∞whenever𝜎_∞<𝑇, (56) then it is called a maximal local solution and 𝜎_∞ is called the explosion time. A maximal local solution {𝑥(𝑡)∶𝑡₀≤𝑡<𝜎_∞} is said to be unique if any other maximal local solution {𝑥(𝑡)∶𝑡₀≤𝑡<𝜎_∞} is indistinguishable from it, namely, 𝜎_∞=𝜎_∞ and 𝑥(𝑡)=𝑥(𝑡) for 𝑡₀≤𝑡<𝜎_∞ with probability one.

Definition 9 (local Lipschitz condition). For every integer 𝑘≥1, there exists a positive constant ℎ_𝑘 such that, for all 𝑡∈[𝑡₀,𝑇], 𝑖=(𝑖₁,𝑖₂)∈𝕊 and those 𝑥,𝑦∈ℝ^𝑛 with |𝑥|∨|𝑦|≤𝑘𝑓𝑥,𝑡,𝑖₁−𝑓𝑦,𝑡,𝑖₁²∨𝑔𝑥,𝑡,𝑖₂−𝑔𝑦,𝑡,𝑖₂²≤ℎ_𝑘𝑥−𝑦². (57)
The following theorem shows the existence of unique maximal local solution under the local Lipschitz condition without the linear growth condition.

Theorem 10. Under condition (57), there exists a unique maximal local solution of (8).

Proof. Define functions 𝑓_𝑘𝑥,𝑡,𝑖₁=𝑓𝑥,𝑡,𝑖₁|𝑥|𝑓,if≤𝑘,𝑘𝑥|𝑥|,𝑡,𝑖₁|𝑥|𝑔,if>𝑘,_𝑘𝑥,𝑡,𝑖₂=𝑔𝑥,𝑡,𝑖₂|𝑥|𝑔,if≤𝑘,𝑘𝑥|𝑥|,𝑡,𝑖₂,if|𝑥|>𝑘. (58) Then 𝑓_𝑘 and 𝑔_𝑘 satisfy the Lipschitz condition and the linear growth condition. Thus by Theorem 7, there is a unique solution 𝑥_𝑘(𝑡) of the equation 𝑑𝑥_𝑘(𝑡)=𝑓_𝑘𝑥_𝑘(𝑡),𝑡,𝛾₁(𝑡)𝑑𝑡+𝑔_𝑘𝑥_𝑘(𝑡),𝑡,𝛾₂(𝑡)𝑑𝐵(𝑡) (59) with the initial conditions 𝑥_𝑘(𝑡₀)=𝑥₀ and 𝛾(𝑡₀)=(𝛾₁(𝑡₀),𝛾₂(𝑡₀)). Define the stopping times 𝜎_𝑘=𝑇∧inf𝑡∈𝑡₀,𝑇∶𝑥_𝑘(𝑡)≥𝑘. (60) Clearly, if 𝑡₀≤𝑡≤𝜎_𝑘, 𝑥_𝑘(𝑡)=𝑥_𝑘+1(𝑡), (61) which indicates that 𝜎_𝑘 is increasing so 𝜎_𝑘 has its limit 𝜎_∞=lim_𝑘→+∞𝜎_𝑘. Define {𝑥(𝑡)∶𝑡₀≤𝑡<𝜎_∞} by 𝑥(𝑡)=𝑥_𝑘(𝑡),𝑡∈𝜎_𝑘−1,𝜎_𝑘,𝑘≥1, (62) where 𝜎₀=𝑡₀. Applying (61), one can show that 𝑥(𝑡∧𝜎_𝑘)=𝑥_𝑘(𝑡∧𝜎_𝑘). It then follows from (59) that 𝑥𝑡∧𝜎_𝑘=𝑥₀+^{𝑡∧𝜎𝑘}_𝑡0𝑓_𝑘𝑥(𝑠),𝑠,𝛾₁(𝑠)+𝑑𝑠_{𝑡∧𝜎𝑘𝑡0}𝑔_𝑘𝑥(𝑠),𝑠,𝛾₂(𝑠)𝑑𝐵(𝑠)=𝑥₀+^{𝑡∧𝜎𝑘}_𝑡0𝑓𝑥(𝑠),𝑠,𝛾₁(𝑠)+𝑑𝑠_{𝑡∧𝜎𝑘𝑡0}𝑔𝑥(𝑠),𝑠,𝛾₂(𝑠)𝑑𝐵(𝑠), (63) for any 𝑡∈[𝑡₀,𝑇) and 𝑘≥1. It is easy to see that if 𝜎_∞≤𝑇, then limsup_{𝑡→𝜎∞}|𝑥(𝑡)|≥limsup_𝑘→+∞𝑥𝜎_𝑘=limsup_𝑘→+∞𝑥_𝑘𝜎_𝑘=∞. (64) Therefore {𝑥(𝑡)∶𝑡₀≤𝑡<𝜎_∞} is a maximal local solution.
Now, we will prove the uniqueness. Let {𝑥(𝑡)∶𝑡₀≤𝑡<𝜎_∞} be another maximal local solution. Define 𝜎_𝑘=𝜎_∞∧inf𝑡∈𝑡₀,𝜎_∞∶𝑥(𝑡)≥𝑘. (65) Then 𝜎_𝑘→𝜎_∞ a.s. and 𝑃𝑥(𝑡)=𝑥(𝑡),𝑡∈𝑡₀,𝜎_𝑘∧𝜎_𝑘=1∀𝑘≥1. (66) Letting 𝑘→∞ gives 𝑃𝑥(𝑡)=𝑥(𝑡),𝑡∈𝑡₀,𝜎_∞∧𝜎_∞=1. (67) In order to complete the proof, we need only to show that 𝜎_∞=𝜎_∞ a.s. In fact, for almost every 𝜔∈{𝜎_∞<𝜎_∞}, we have 𝑥𝜎_∞,𝜔=lim_𝑘→+∞𝑥𝜎_𝑘,𝜔=lim_𝑘→+∞𝑥𝜎_𝑘,𝜔=∞, (68) which contradicts the fact that 𝑥(𝑡,𝜔) is continuous on 𝑡∈[𝑡₀,𝜎_∞(𝜔)). This implies 𝜎_∞≥𝜎_∞ a.s. In the same way, one can show 𝜎_∞≤𝜎_∞ a.s. Thus we must have 𝜎_∞=𝜎_∞ a.s. This completes the proof.

In many situations, we often consider an SDE on [𝑡₀,∞)𝑑𝑥(𝑡)=𝑓𝑥(𝑡),𝑡,𝛾₁(𝑡)𝑥(𝑡)𝑑𝑡+𝑔,𝑡,𝛾₂(𝑡)𝑑𝐵(𝑡),𝑡₀≤𝑡<∞ (69) with initial data 𝑥(𝑡₀)=𝑥₀ and 𝛾(𝑡₀). If the assumption of the existence-and-uniqueness theorem holds on every finite subinterval [𝑡₀,𝑇] of [𝑡₀,∞), then (69) has a unique solution 𝑥(𝑡) on the entire interval [𝑡₀,∞). 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 𝐿𝑉 from ℝ^𝑛×ℝ₊×𝕊 to ℝ which is given by 𝐿𝑉𝑥,𝑡,𝑖₁,𝑖₂=𝑉_𝑡𝑥,𝑡,𝑖₁,𝑖₂+𝑉_𝑥𝑥,𝑡,𝑖₁,𝑖₂𝑓𝑥,𝑡,𝑖₁+0.5trace𝑔^𝑇𝑥,𝑡,𝑖₂𝑉_𝑥𝑥𝑥,𝑡,𝑖₁,𝑖₂𝑔𝑥,𝑡,𝑖₂+_{𝑗1,𝑗2∈𝕊}𝑞_{𝑖1𝑖2;𝑗1𝑗2}𝑉𝑥,𝑡,𝑗₁,𝑗₂, (70) where 𝑉(𝑥,𝑡,(𝑖₁,𝑖₂))∈𝐶^2,1(ℝ^𝑛×ℝ₊×𝕊;ℝ₊) and 𝑉_𝑡𝑥,𝑡,𝑖₁,𝑖₂=𝜕𝑉𝑥,𝑡,𝑖₁,𝑖₂;𝑉𝜕𝑡_𝑥𝑥,𝑡,𝑖₁,𝑖₂=𝜕𝑉𝑥,𝑡,𝑖₁,𝑖₂𝜕𝑥₁𝑖,…,𝜕𝑉𝑥,𝑡,₁,𝑖₂𝜕𝑥_𝑛;𝑉_𝑥𝑥𝑥,𝑡,𝑖₁,𝑖₂=𝜕²𝑉𝑥,𝑡,𝑖₁,𝑖₂𝜕𝑥_𝑖𝜕𝑥_{𝑗𝑛×𝑛}. (71)

Theorem 11. Assume that the local Lipschitz condition (57) holds. Assume also that there is a function 𝑉(𝑥,𝑡,(𝑖₁,𝑖₂))∈𝐶^2,1(ℝ^𝑛×ℝ₊×𝕊;ℝ₊) and a constant 𝜃>0 such that lim_|𝑥|→∞inf_{𝑖𝑡,1,𝑖2∈ℝ+×𝕊}𝑉𝑥,𝑡,𝑖₁,𝑖₂𝑖=∞,𝐿𝑉𝑥,𝑡,₁,𝑖₂≤𝜃1+𝑉𝑥,𝑡,𝑖₁,𝑖₂,∀𝑥,𝑡,𝑖₁,𝑖₂∈ℝ^𝑛×ℝ₊×𝕊. (72)(73) Then there exists a unique global solution 𝑥(𝑡) to (69).

Proof. We need only to prove the theorem for any initial condition 𝑥₀∈ℝ^𝑛 and (𝛾₁(𝑡₀),𝛾₂(𝑡₀))∈𝕊. From Theorem 10, we know that the local Lipschitz condition guarantees the existence of the unique maximal solution 𝑥(𝑡) on [𝑡₀,𝜎_∞), where 𝜎_∞ is the explosion time. We need only to show 𝜎_∞=∞ a.s. If this is not true, then we can find a pair of positive constants 𝜀 and 𝑇 such that 𝑃𝜎_∞≤𝑇>2𝜀. (74) For each integer 𝑘≥1, define the stopping time 𝜎_𝑘=inf𝑡≥𝑡₀∶|𝑥(𝑡)|≥𝑘. (75) Since 𝜎_𝑘→𝜎_∞ almost surely, we can find a sufficiently large integer 𝑘₀ for 𝑃𝜎_𝑘≤𝑇>𝜀,∀𝑘≥𝑘₀. (76) Fix any 𝑘≥𝑘₀, then for any 𝑡₀≤𝑡≤𝑇, by virtue of the generalized Itô formula (see, e.g., [1]) 𝐸𝑉𝑥𝑡∧𝜎_𝑘,𝑡∧𝜎_𝑘,𝛾𝑡∧𝜎_𝑘=𝑉𝑥₀,𝑡₀,𝛾₀+𝐸^{𝑡∧𝜎𝑘}_𝑡0𝑥(𝑠)(𝑠)𝑥𝐿𝑉,𝑠,𝛾𝑑𝑠≤𝑉₀,𝑡₀,𝛾₀+𝜃𝑇−𝑡₀+𝜃^𝑡_𝑡0𝐸𝑉𝑥𝑠∧𝜎_𝑘,𝑠∧𝜎_𝑘,𝛾𝑠∧𝜎_𝑘𝑑𝑠. (77) Making use of the Gronwall inequality gives 𝐸𝑉𝑥𝑡∧𝜎_𝑘,𝑡∧𝜎_𝑘,𝛾𝑡∧𝜎_𝑘≤𝑉𝑥₀,𝑡₀,𝛾₀+𝜃𝑇−𝑡₀exp𝜃𝑇−𝑡₀. (78) Therefore 𝐸𝐼_{{𝜎𝑘≤𝑇}}𝑉𝑥𝜎_𝑘,𝜎_𝑘,𝛾𝜎_𝑘≤𝑉𝑥₀,𝑡₀,𝛾₀+𝜃𝑇−𝑡₀exp𝜃𝑇−𝑡₀. (79) At the same time, set 𝑔_𝑘=inf𝑉𝑥,𝑡,𝑖₁,𝑖₂∶|𝑥|≥𝑘,𝑡∈𝑡₀,𝑇,𝑖₁,𝑖₂∈𝕊. (80) Then (72) means 𝑔_𝑘→∞. It follows from (76) and (79) that 𝑉𝑥₀,𝑡₀,𝛾₀+𝜃𝑇−𝑡₀exp𝜃𝑇−𝑡₀≥𝑔_𝑘𝑃𝜎_𝑘≤𝑇≥𝜀𝑔_𝑘. (81) Letting 𝑘→∞ yields a contradiction, that is to say, 𝜎_∞=∞. The proof is complete.

3. 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 𝑥(𝑡), 𝑡₀≤𝑡≤𝑇 be the unique solution of (8) with initial conditions 𝑥(𝑡₀)=𝑥₀ and 𝛾(𝑡₀), and we will estimate the 𝑝th moment of the solution.

Theorem 12. Assume that there is a function 𝑉(𝑥,𝑡,(𝑖₁,𝑖₂))∈𝐶^2,1(ℝ^𝑛×ℝ₊×𝕊;ℝ₊) and positive constants 𝑝,𝜂,𝜃 such that for all (𝑥,𝑡,(𝑖₁,𝑖₂))∈ℝ^𝑛×ℝ₊×𝕊, 𝜂|𝑥|^𝑝≤𝑉𝑥,𝑡,𝑖₁,𝑖₂,𝐿𝑉𝑥,𝑡,𝑖₁,𝑖₂≤𝜃𝑉𝑥,𝑡,𝑖₁,𝑖₂. (82)(83) Assume also the initial condition 𝑥(𝑡₀)=𝑥₀ and 𝛾(𝑡₀) obeys 𝐸𝑉(𝑥₀,𝑡₀,𝛾(𝑡₀))<∞, then one has 𝐸|𝑥(𝑡)|^𝑝≤𝐸𝑉𝑥₀,𝑡₀,𝛾𝑡₀exp𝜃𝑡−𝑡₀𝜂,∀𝑡∈𝑡₀,𝑇. (84)

Proof. For each integer 𝑘≥1, define the stopping time 𝜎_𝑘=𝑇∧inf𝑡≥𝑡₀∶|𝑥(𝑡)|≥𝑘. (85) Thus 𝜎_𝑘→𝑇 a.s. Using the generalized Itô's formula and (83), we obtain that for 𝑡∈[𝑡₀,𝑇]𝐸𝑉𝑥𝑡∧𝜎_𝑘,𝑡∧𝜎_𝑘,𝛾𝑡∧𝜎_𝑘≤𝐸𝑉𝑥₀,𝑡₀,𝛾𝑡₀+𝜃^𝑡_𝑡0𝐸𝑉𝑥𝑠∧𝜎_𝑘,𝑠∧𝜎_𝑘,𝛾𝑠∧𝜎_𝑘𝑑𝑠. (86) Then the Gronwall inequality indicates 𝐸𝑉𝑥𝑡∧𝜎_𝑘,𝑡∧𝜎_𝑘,𝛾𝑡∧𝜎_𝑘≤𝐸𝑉𝑥₀,𝑡₀,𝛾𝑡₀exp𝜃𝑡−𝑡₀ (87) for all 𝑡∈[𝑡₀,𝑇]. By virtue of condition (83) we obtain the required assertion (84).

Corollary 13. Assume 𝑝≥2 and 𝑥₀∈𝐿^𝑝_ℱ𝑡0(Ω;ℝ^𝑛). Assume also that there exists a constant 𝜃>0 such that, for all (𝑥,𝑡,(𝑖₁,𝑖₂))∈ℝ^𝑛×ℝ₊×𝕊, 𝑥^𝑇𝑓𝑥,𝑡,𝑖₁+𝑝−12𝑔𝑥,𝑡,𝑖₂²≤𝜃1+|𝑥|². (88) Then one has 𝐸|𝑥(𝑡)|^𝑝≤2^(𝑝−2)/21+𝐸𝑥₀^𝑝exp𝑝𝜃𝑡−𝑡₀∀𝑡∈𝑡₀,𝑇. (89)

Proof. Define 𝑉(𝑥,𝑡,(𝑖₁,𝑖₂))=(1+|𝑥|²)^𝑝/2. Making use of (88) yields 𝐿𝑉𝑥,𝑡,𝑖₁,𝑖₂=𝑝1+|𝑥|^{2(𝑝−2)/2}𝑥^𝑇𝑓𝑥,𝑡,𝑖₁+𝑝21+|𝑥|^{2(𝑝−2)/2}𝑔𝑥,𝑡,𝑖₂²+𝑝𝑝−221+|𝑥|^{2(𝑝−4)/2}𝑥^𝑇𝑔𝑥,𝑡,𝑖₂²≤𝑝1+|𝑥|^{2(𝑝−2)/2}𝑥^𝑇𝑓𝑥,𝑡,𝑖₁+𝑝−12𝑔𝑥,𝑡,𝑖₂²≤𝑝𝜃1+|𝑥|^2𝑝/2. (90) Then by Theorem 12, we get 𝐸1+|𝑥(𝑡)|^2𝑝/2≤𝐸1+𝑥₀^2𝑝/2exp𝑝𝜃𝑡−𝑡₀ (91) 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 𝜃=√𝐾+𝐾(𝑝−1)/2. Now, we will show the other important properties of the solution.

Theorem 14. Let 𝑝≥2 and 𝑥₀∈𝐿^𝑝_ℱ𝑡0(Ω;ℝ^𝑛). Assume also that the linear growth condition (43) holds. Then one has 𝐸|𝑥(𝑡)−𝑥(𝑠)|^𝑝≤𝐶(𝑡−𝑠)^𝑝/2, (92) where 𝐶=2^𝑝−21+𝐸𝑥₀^𝑝exp𝑝𝜃𝑇−𝑡₀×2𝑇−𝑡₀^𝑝/2+𝑝𝑝−1^𝑝/2 (93) and 𝜃=√𝐾+𝐾(𝑝−1)/2. Particularly, the 𝑝th moment of the solution is continuous on [𝑡₀,𝑇].

Proof. Applying the elementary inequality |𝑎+𝑏|^𝑝≤2^𝑝−1(|𝑎|^𝑝+|𝑏|^𝑝), the Hölder inequality, and the linear growth condition, we can derive that 𝐸|𝑥(𝑡)−𝑥(𝑠)|^𝑝≤2^𝑝−1𝐸^𝑡_𝑠𝑓𝑥(𝑢),𝑢,𝛾₁(𝑢)𝑑𝑢^𝑝+2^𝑝−1𝐸^𝑡_𝑠𝑔𝑥(𝑢),𝑢,𝛾₂(𝑢)𝑑𝐵(𝑢)^𝑝≤[2(𝑡−𝑠)]^𝑝−1𝐸^𝑡_𝑠𝑓𝑥(𝑢),𝑢,𝛾₁(𝑢)^𝑝𝑑𝑢+0.52𝑝𝑝−1^𝑝/2(𝑡−𝑠)^(𝑝−2)/2𝐸×^𝑡_𝑠𝑔𝑥(𝑢),𝑢,𝛾₂(𝑢)^𝑝𝑑𝑢≤𝐶₁(𝑡−𝑠)^{(𝑝−2)/2𝑡}_𝑠𝐸1+|𝑥(𝑢)|^2𝑝/2𝑑𝑢, (94) where 𝐶₁=2^(𝑝−2)/2𝐾^𝑝/2([2(𝑇−𝑡₀)]^𝑝/2+[𝑝(𝑝−1)]^𝑝/2). Using (91) yields 𝐸|𝑥(𝑡)−𝑥(𝑠)|^𝑝≤𝐶₁(𝑡−𝑠)^(𝑝−2)/2×^𝑡_𝑠2^(𝑝−2)/21+𝐸𝑥₀^𝑝exp𝑝𝜃𝑢−𝑡₀𝑑𝑢≤𝐶₁2^(𝑝−2)/21+𝐸𝑥₀^𝑝×exp𝑝𝜃𝑇−𝑡₀(𝑡−𝑠)^(𝑝/2), (95) which is the desired inequality.

Theorem 15. Let 𝑝≥2 and 𝑥₀∈𝐿^𝑝_ℱ𝑡0(Ω;ℝ^𝑛). Assume that there is a 𝐾>0 such that for all (𝑥,𝑡,(𝑖₁,𝑖₂))∈ℝ^𝑛×[𝑡₀,𝑇]×𝕊𝑥^𝑇𝑓𝑥,𝑡,𝑖₁∨𝑔𝑥,𝑡,𝑖₂²≤𝐾(1+|𝑥|)². (96) Then 𝐸sup_{𝑡0≤𝑡≤𝑇}|𝑥(𝑡)|^𝑝≤1+2𝐸𝑥₀^𝑝exp2𝑝10𝑝+1𝐾𝑇−𝑡₀. (97)

Proof. Making use of the generalized Itô's formula and condition (96), we have |𝑥(𝑡)|^𝑝≤𝑥₀^𝑝(𝑡)++𝑀_𝑡𝑡0𝑝|𝑥(𝑠)|^𝑝−2𝑥^𝑇(𝑠)𝑓𝑥(𝑠),𝑠,𝛾₁(𝑠)+𝑝−12𝑔𝑥(𝑠),𝑠,𝛾₂(𝑠)²≤𝑥𝑑𝑠_0𝑝(𝑡)𝐾×+𝑀+0.5𝑝𝑝+1_𝑡𝑡0|𝑥(𝑠)|^𝑝−21+|𝑥(𝑠)|²≤𝑥𝑑𝑠_0𝑝+𝑀(𝑡)+𝑝𝑝+1𝐾^𝑡_𝑡01+|𝑥(𝑠)|^𝑝𝑑𝑠, (98) where 𝑀(𝑡)=^𝑡_𝑡0𝑝|𝑥(𝑠)|^𝑝−2𝑥^𝑇(𝑠)𝑔𝑥(𝑠),𝑠,𝛾₂(𝑠)𝑑𝐵(𝑠). (99) Therefore, for any 𝑡₁∈[𝑡₀,𝑇], 𝐸sup_{𝑡0≤𝑡≤𝑡1}|𝑥(𝑡)|^𝑝≤𝐸𝑥₀^𝑝+𝐸sup_{𝑡0≤𝑡≤𝑡1}|𝑀(𝑡)|+𝑝𝑝+1𝐾𝐸^𝑡1_𝑡01+|𝑥(𝑠)|^𝑝𝑑𝑠. (100) At the same time, applying the well-known Burkholder-Davis-Gundy inequality (see, e.g., [2]) gives 𝐸sup_{𝑡0≤𝑡≤𝑡1}|𝑀(𝑡)|≤3𝐸^𝑡1_𝑡0𝑝²|𝑥(𝑠)|^2𝑝−2𝑔𝑥(𝑠),𝑠,𝛾₂(𝑠)²𝑑𝑠^0.5≤3𝐸sup_{𝑡0≤𝑡≤𝑡1}|𝑥(𝑡)|^𝑝×^𝑡1_𝑡0𝑝²|𝑥(𝑠)|^𝑝−2𝑔𝑥(𝑠),𝑠,𝛾₂(𝑠)²𝑑𝑠^0.5≤0.5𝐸sup_{𝑡0≤𝑡≤𝑡1}|𝑥(𝑡)|^𝑝×+4.5𝐸_𝑡1𝑡0𝑝²𝐾|𝑥(𝑠)|^𝑝−21+|𝑥(𝑠)|²𝑑𝑠≤0.5𝐸sup_{𝑡0≤𝑡≤𝑡1}|𝑥(𝑡)|^𝑝+9𝑝²𝐾𝐸^𝑡1_𝑡01+|𝑥(𝑠)|^𝑝𝑑𝑠. (101) Substituting the above inequality into (100) gives 𝐸sup_{𝑡0≤𝑡≤𝑡1}|𝑥(𝑡)|^𝑝≤2𝐸𝑥₀^𝑝+2𝑝10𝑝+1𝐾^𝑡1_𝑡01+|𝑥(𝑠)|^𝑝𝑑𝑠. (102) Thus 1+𝐸sup_{𝑡0≤𝑡≤𝑡1}|𝑥(𝑡)|^𝑝≤1+2𝐸𝑥₀^𝑝𝐾×+2𝑝10𝑝+1_𝑡1𝑡01+sup_{𝑡0≤𝑡≤𝑠}|𝑥(𝑡)|^𝑝𝑑𝑠. (103) Then the required assertion follows from the Gronwall inequality.
Up to now, we have discussed the 𝐿^𝑝-estimates for the solution in the case when 𝑝≥2. As for 0<𝑝<2, the similar results can be given without any difficulty as long as we note that the Hölder inequality implies 𝐸|𝑥(𝑡)|^𝑝≤𝐸|𝑥(𝑡)|^20.5𝑝. (104)

4. Example

Consider the following Black-Scholes model: 𝑑𝑋(𝑡)=𝜇𝛾₁(𝑡)𝑋(𝑡)𝑑𝑡+𝜈𝛾₂(𝑡)𝑋(𝑡)𝑑𝐵(𝑡), (105) where 𝛾₁(𝑡) is a right-continuous homogenous Markovian chain taking values in finite state spaces 𝕊₁={1,2} and 𝛾₂(𝑡) is a right-continuous homogenous Markovian chain taking values in finite state spaces 𝕊₂={1,2,3}, 𝜇(𝑖)=𝑖, 𝑖=1,2, 𝜈(𝑗)=𝑗+1, 𝑗=1,2,3. Taking 𝐾=16, 𝐾=9, 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 𝑝th 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 𝛾₁(𝑡) and 𝛾₂(𝑡) are right-continuous homogenous Markovian chains such that 𝛾(𝑡)=(𝛾₁(𝑡),𝛾₂(𝑡)) 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 𝛾₁(𝑡)≡𝛾₂(𝑡) and 𝛾₁(𝑡) is a right-continuous homogenous Markovian chain, then (H1)′ is fulfilled immediately. At the same time, if 𝛾₁(𝑡)≡𝛾₂(𝑡), (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.

Acknowledgments

The authors thank the editor and referees for their very important and helpful comments and suggestions. The authors also thank Dr. C. Zhang for helping them to improve the English exposition. This research is supported by the NSFC of China (nos. 11171081 and 11171056).