Criteria for transience and recurrence of regime-switching diffusion ...

arXiv:1403.3135v1 [math.PR] 13 Mar 2014

Criteria for transience and recurrence of regime-switching diffusion processes∗ Jinghai Shao† School of Mathematical Sciences, Beijing Normal University, 100875, Beijing, China March 14, 2014

Abstract We provide some on-off type criteria for recurrence and transience of regime-switching diffusion processes using the theory of M-matrix and the Perron-Frobenius theorem. Stateindependent and state-dependent regime-switching diffusion processes in a finite space and a countable space are both studied. We put forward a finite partition method to deal with switching process in a countable space. As an application, we improve the known criteria for recurrence of linear regime-switching diffusion processes, and provide an on-off type criterion for a kind of nonlinear regime-switching diffusion processes.

AMS subject Classification (2010): 60A10, 60J60, 60J10 Keywords: Regime-switching diffusions, M-matrix, Recurrence, Transience, Ergodicity

1

Introduction

Regime-switching diffusion processes have received much attention lately, and they can provide more realistic formulation for many applications such as biology, mathematical finance, etc. See ∗

Supported in part by NSFC (No.11301030), 985-project and Beijing Higher Education Young Elite Teacher

Project. † Email: [email protected]

1

[5, 7, 8, 16] and references therein for more details on their application. The regime-switching diffusion process (for short, RSDP) studied in this work can be viewed as a number of diffusion processes modulated by a random switching device or as a diffusion process which lives in a random environment. More precisely, RSDP is a two-component process (Xt , Λt ), where (Xt ) describes the continuous dynamics, and (Λt ) describes the random switching device. (Xt ) satisfies the stochastic differential equation (for short, SDE) dXt = b(Xt , Λt )dt + σ(Xt , Λt )dBt , X0 = x ∈ Rd ,

(1.1)

where (Bt ) is a Brownian motion in Rd , d ≥ 1, σ is d × d-matrix, and b is a vector in Rd . While (Λt ) is a continuous time Markov chain on the state space M = {1, 2, . . . , N }, 2 ≤ N ≤ ∞, satisfying

P(Λt+δ = l|Λt = k, Xt = x) =

(

qkl (x)δ + o(δ),

if k 6= l,

1 + qkk (x)δ + o(δ), if k = l,

(1.2)

for δ > 0. The Q-matrix Qx = (qkl (x)) is irreducible and conservative for each x ∈ Rd . If the Q-matrix (qkl (x)) does not depend on x, then (Xt , Λt ) is called a state-independent RSDP ; otherwise, it is called a state-dependent one. When N is finite, namely, (Λt ) is a Markov chain on a finite state space, we call (Xt , Λt ) a RSDP in a finite state space. When N is infinite, we call (Xt , Λt ) a RSDP in an infinite state space. Next, we collect some conditions used later. ¯ > 0 such that (H) There exists constant K (i) x 7→ qij (x) is a bounded continuous function for each pair of i, j ∈ M.

¯ + |x|), (ii) |b(x, i)| + kσ(x, i)k ≤ K(1

x ∈ Rd , i ∈ M.

¯ − y|, (iii) |b(x, i) − b(y, i)| + kσ(x, i) − σ(y, i)k ≤ K|x

x, y ∈ Rd , i ∈ M.

(iv) For each i ∈ M, a(x, i) = σ(x, i)σ(x, i)∗ is uniformly positive definite.

Here and in the sequel, σ ∗ stands for the transpose of matrix σ, and kσk denotes the operator

norm. Hypothesis (Hi),(Hii) and (Hiii) guarantee the existence of a unique nonexplosive solution

of (1.1) and (1.2) (cf. [16, Theorem 2.1]). Hypothesis (Hiv) is used to ensure that (Xt , Λt ) possesses strong Feller property (cf. [15], [17]), which will be used in the study of exponential ergodicity. Corresponding to the process (Xt , Λt ), there is a family of diffusion processes defined by (i)

(i)

(i)

dXt = b(Xt , i)dt + σ(Xt , i)dBt , 2

(1.3)

(i)

for each i ∈ M. These processes (Xt ) (i ∈ M) are the diffusion processes associated with (Xt , Λt ) in each fixed environment. The recurrent behavior of (Xt , Λt ) is intensively connected with its recurrent behavior in each fixed environment. But this connection is rather complicated as having been noted by [11]. In [11], some examples in [0, ∞) with reflecting boundary at 0 (1)

(2)

and M = {1, 2} were constructed. They showed that even when (Xt ) and (Xt ) are both

positive recurrent (transient), (Xt , Λt ) could be transient (positive recurrent, respectively) by

choosing suitable transition rate (qij ) between two states. In view of this complicity, it is a challenging work to determine the recurrent property of a regime-switching diffusion process. There are lots of work having been dedicated to this task. See, for instance, [3, 11, 10, 16, 4] and references therein. Except giving the examples we mentioned above, [11] also studied the reversible state-independent RSDP . In [10], the author provided a theoretically complete characterization of recurrence and transience for a class of state-independent RSDP , which we will state more precisely later. In [4], some on-off type criteria were established to justify the exponential ergodicity of state-independent and state-dependent RSDP in a finite state space. The convergence in total variation norm and in Wasserstein distance were both studied in [4]. The cost function used to define the Wasserstein distance in [4] is bounded. All the previously mentioned work considered only the RSDP in a finite state space. Although the general criteria by the Lyapunov functions for Markov processes still work for RSDP , it is well known that finding a suitable Lyapunov function is a difficult task for RSDP due to the coexistence of generators for diffusion process and jump process. So it is better to provide some easily verifiable criteria in terms of the coefficients of diffusion process (Xt ) and the Q-matrix of (Λt ). In this direction, [17] has provided some criteria for a class of state-dependent RSDP (Xt , Λt ) in a finite state space. Precisely, the continuous component (Xt ) considered in [17] behaves like a linear one and Q-matrix (qij (x)) behaves like a state-independent Q-matrix (ˆ qij ) in a neighborhood of ∞. In addition, the recurrent property for geometric Brownian motion in

a two-state random environment was studied in [12].

In [13], we studied the ergodicity for RSDP in Wasserstein distance. Both state-independent and state-dependent RSDP in finite and infinite state spaces are studied in [13]. The cost function used in [13] is not necessarily bounded. We put forward some new criteria for ergodicity based on the theory of M-matrix and Perron-Frobenius theorem. Our present work is devoted to studying the recurrent property of RSDP in total variation norm. Compared with [4, 13], the on-off type criteria given there own only one hand “ on ”, that is, if the condition holds, 3

then the process is ergodic. Examples can show these criteria are sharp. In the present work, we shall show that these on-off type criteria can own two hands, both “ on ” and “ off ”, that is, if the condition holds, then the process is recurrent, and if not, then the process is transient. Another contribution of this work is that we put forward a finite partition method to study the transience and recurrence for state-dependent RSDP in an infinite state space. Up to our knowledge, there is few result for the recurrent property of RSDP in an infinite state space. In this work, based on the criteria given by M-matrix theory, we put forward a finite partition approach study the recurrent properties of RSDP in an infinite state space. Its basic idea is to transform the state-dependent RSDP in an infinite state space into a new state-independent RSDP in a finite state space (see Theorem 2.6 for details). As an application of our criteria, we develop the study in [10] and [17]. In [10], the authors considered the state-independent RSDP (Xt , Λt ) in Rd × M with d ≥ 2 and M a finite set. (i)

For each i ∈ M, the associated diffusion (Xt ) has the infinitesimal generator L(i) = 12 ∆ + V ,

where

V (x, i) = |x|δ ˆb(x/|x|, i) · ∇,

δ ∈ [−1, 1).

(1.4)

Let S d−1 denote the d − 1-dimension sphere, and µ be the invariant probability measure for (Λt ). In [10], they studied the process under the condition that ˆb(φ, i) 6≡ 0, ˆb(φ, i) ∈ C 1 (S d−1 ) for each i ∈ M, and

X

i∈M

ˆb(φ, i)µi = 0

for each φ ∈ S d−1 .

(1.5)

Condition (1.5) allows them to transform the problem into studying the recurrent behavior of the generator

2   ˆ = r γ c1 (φ) ∂ + c2 (φ) ∂ + 1 ∂ DS d−1 + 1 LS d−1 , L 2 2 ∂r r ∂r r ∂r r where γ = 0, if −1 ≤ δ ≤ 0, and γ = 2δ, if 0 < δ < 1, c1 (φ) ≥ 0, DS d−1 is a first-order operator

on S d−1 and LS d−1 is a (possible degenerate) diffusion generator on S d−1 . By posing some further conditions on c1 (φ) and c2 (φ), they got a quantity ρ expressed in terms of c1 (φ), c2 (φ) ˆ They showed and the density of invariant probability measure of the process corresponding to L. that (Xt , Λt ) is recurrent or transient according to whether ρ ≤ 0 or ρ > 0. Theoretically, this

result is complete although calculating ρ is a difficult task, which has been pointed out in [10]. P In this work, roughly speaking, we consider the processes corresponding to i∈M µiˆbi (φ, i) 6= 0.

In Section 3, we consider the case δ = 1 and in Section 4, after developing the criteria given in Section 2, we consider the case δ ∈ [−1, 1). Some easily verifiable criteria are provided. 4

The usefulness and sharpness of the criteria established in this work can be seen from the following example. Let (Λt ) be a continuous time Markov chain on {1, 2, . . . , N }, N < ∞, equipped with an irreducible conservative Q-matrix (qij ). Let µ be the invariant probability

measure of (Λt ). Let (Xt ) be a random diffusion on [0, ∞) with reflecting boundary at 0 satisfying dXt = bΛt Xtδ dt + dBt ,

δ ∈ [−1, 1].

P ≤ 0, then (Xt , Λt ) is recurrent; if N i=1 µi bi > 0, then PN (Xt , Λt ) is transient. In the case δ = 1, if i=1 µi bi < 0, then (Xt , Λt ) is exponentially ergodic; P if N i=1 µi bi > 0, then (Xt , Λt ) is transient. In the case δ ∈ [−1, 1), if

PN

i=1 µi bi

This work is organized as follows. We shall provide some new criteria on transience, re-

currence and exponential ergodicity for RSDP in Section 2. In Section 2, we first study the state-independent RSDP in a finite state space, then study the state-dependent RSDP in an infinite state space. In Section 3, we consider the recurrent property of Ornstein-Uhlenbeck process and linear diffusion in random environments. In Section 4, we provide another kind of criteria then apply these criteria to study the recurrent properties of nonlinear regime-switching diffusion processes.

2

Criteria for recurrence and transience: I

Let (Xt , Λt ) be defined by (1.1) and (1.2). We first consider the situation that Q-matrix of (Λt ) is independent of x and N < ∞. For a diffusion process in Rd with generator L = Pd 1 Pd ∂2 ∂ i,j=1 aij (x) ∂xi ∂xj + i=1 bi (x) ∂xi , we write L ∼ (a(x), b(x)) for simplicity, where a(x) = 2 (i)

(aij (x)), b(x) = (bi (x)). For each i ∈ M, (Xt ) is defined by (1.3), its generator L(i) ∼ (a(i) (x), b(i) (x)), where a(i) (x) = σ(x, i)σ(x, i)∗ , b(i) (x) = b(x, i).

For the vector β = (β1 , . . . , βN )∗ , we use diag(β) = diag(β1 , . . . , βN ) to denote the diagonal matrix generated by vector β as usual. Before stating our results, we introduce some notation and basic properties on M-matrix. We refer the reader to [2] for more discussion on this topic. Let B be a matrix or vector. By B ≥ 0 we mean that all elements of B are non-negative.

By B > 0 we mean that B ≥ 0 and at least one element of B is positive. By B ≫ 0, we mean that all elements of B are positive. B ≪ 0 means that −B ≫ 0. 5

Definition 2.1 (M-matrix) A square matrix A = (aij )n×n is called an M-Matrix if A can be expressed in the form A = sI − B with some B ≥ 0 and s ≥ Ria(B), where I is the n × n identity

matrix, and Ria(B) the spectral radius of B. When s > Ria(B), A is called a nonsingular Mmatrix. We cite some conditions equivalent to that A is a nonsingular M-matrix as follows, and refer to [2] for more discussion on this topic. Proposition 2.2 The following statements are equivalent. 1. A is a nonsingular n × n M-matrix. 2. All of the principal minors of A are positive; that is, a 11 . . . a1k . .. .. . > 0 for every k = 1, 2, . . . , n. a1k . . . akk

3. Every real eigenvalue of A is positive.

4. A is semipositive; that is, there exists x ≫ 0 in Rn such that Ax ≫ 0. 5. There exists x ≫ 0 with Ax > 0 and

Pi

j=1 aij xj

> 0, i = 1, . . . , n.

Next result is our first main result for the recurrent property of state-independent regimeswitching diffusion processes in a finite state space. We will use often the following condition for a function V ∈ C 2 (Rd ). (A1) There exist constants r0 > 0 and βi ∈ R, i ∈ M such that V (x) > 0,

L(i) V (x) ≤ βi V (x),

|x| > r0 .

(2.1)

Here the constant βi could be negative or positive. Theorem 2.3 Assume that (Hi), (Hii), (Hiii) hold and N < ∞. Assume there exists a function
V ∈ C 2 (Rd ) such that condition (A1) is satisfied and the matrix − Q + diag(β) is a nonsin-

gular M-matrix. Then (Xt , Λt ) is positive recurrent if lim|x|→∞ V (x) = ∞, and is transient if lim|x|→∞ V (x) = 0. Assume further that (Hiv) holds. Then (Xt , Λt ) is exponentially ergodic if

lim|x|→∞ V (x) = ∞. 6

Proof. Denote by A the generator of (Xt , Λt ). Due to [14], A f (x, i) = L(i) f (·, i)(x) + Qf (x, ·)(i), where Qg(i) =

P

j6=i qij (gj


− gi ) for g ∈ B(M). As − Q + diag(β) is a nonsingular M-matrix,

by Proposition 2.2, there exists a vector ξ = (ξ1 , . . . , ξN )∗ ≫ 0 such that

Take f (x, i) = V (x)ξi ,


λ = (λ1 , . . . , λN )∗ = − Q + diag(β) ξ ≫ 0.

x ∈ Rd , i ∈ M, then, for |x| > r0 , i ∈ M,

A f (x, i) =Qξ(i)V (x) + ξi L(i) V (x)
≤ Qξ(i) + βi ξi V (x) = −λi V (x) λ  λi i = − f (x, i) ≤ − min f (x, i). 1≤i≤N ξi ξi

(2.2)

As N < ∞, min1≤i≤N (λi /ξi ) > 0, and we have A f (x, i) ≤ 0. If lim|x|→∞ V (x) = ∞, further-

more, we can get A f (x, i) ≤ −ε for some ε > 0 by choosing r0 > 0 large enough. Therefore,

according to the Foster-Lyapunov drift conditions (cf. [10, Section 2] or [16, Chapter 3]), we obtain that (Xt , Λt ) is positive recurrent if lim|x|→∞ V (x) = ∞, and (Xt , Λt ) is transient if

lim|x|→∞ V (x) = 0. According to [15, Theorem 5.1], when (Hiv) holds and lim|x|→∞ V (x) = ∞,

(2.2) yields that (Xt , Λt ) is exponentially ergodic.

Theorem 2.4 Assume that (Hi), (Hii), (Hiii) hold and N < ∞. Suppose that there exists a function V ∈ C 2 (Rd ) such that condition (A1) holds and X µi βi < 0,

(2.3)

i∈M

where µ = (µi )i∈M is the invariant probability measure of (Λt ). Then (Xt , Λt ) is transient if lim|x|→∞ V (x) = 0, and is positive recurrent if lim|x|→∞ V (x) = ∞. Assume further (Hiv) holds, if lim|x|→∞ V (x) = ∞, then (Xt , Λt ) is exponentially ergodic.

Proof. Let Qp = Q + p diag(β), and ηp = −

max

γ∈spec(Qp )

Re γ,

where spec(Qp ) denotes the spectrum of Qp .

Let Q(p,t) = etQp , then the spectral radius Ria(Q(p,t) ) of Q(p,t) equals to e−ηp t . Since all coefficients of Q(p,t) are positive, Perron-Frobenius theorem (see [2, Chapter 2]) yields −ηp is a simple 7

eigenvalue of Qp . Moreover, note that the eigenvector of Q(p,t) corresponding to e−ηp t is also an eigenvector of Qp corresponding to −ηp . Then Perron-Frobenius theorem ensures that there

exists an eigenvector ξ ≫ 0 of Qp associated with the eigenvalue −ηp . Now applying Proposition P 4.2 of [1] (by replacing Ap there with Qp ), if N i=1 µi βi < 0, then there exists some p0 > 0 such

that ηp > 0 for any 0 < p < p0 . Fix a p with 0 < p < min{1, p0 } and an eigenvector ξ ≫ 0, then we obtain

Qp ξ = (Q + p diag(β))ξ = −ηp ξ ≪ 0. Put f (x, i) = V (x)p ξi , x ∈ Rd , i ∈ M. For |x| > r0 , i ∈ M, A f (x, i) = Qξ(i)V (x)p + ξi L(i) V (x)p
≤ Qξ(i) + pβi ξi V (x)p

= −ηp ξi V (x)p = −ηp f (x, i).

Then analogous to the argument of Theorem 2.3, we can conclude the proof. Remark 2.5 We give a heuristic explanation of the condition (2.3) in previous theorem. As µ is the invariant probability measure of (Λt ), µi represents in some sense the time ratio spent (i)

by (Λt ) in the state i. βi represents the recurrent behavior of (Xt ). Therefore, the quantity P (i) i∈M µi βi averages the recurrent behavior of (Xt ) with respect to µ, which determine the

recurrent behavior of (Xt , Λt ) according to previous theorem.

Now we go to consider the state-dependent RSDP with (Λt ) being a Markov chain in a countable space M. Namely, the Q-matrix (qij (x)) is dependent on x and N = ∞. Let

V ∈ C 2 (Rd ) such that (A1) holds and M = supi∈M βi < ∞. As the M-matrix theory is about

matrices with finite size, we shall put forward a finite partition method to transform the RSDP in an infinite state space into a new RSDP in a finite state space. Let Γ = {−∞ = k0 < k1 < . . . < km−1 < km = M }

be a finite partition of (−∞, M ]. Corresponding to Γ, there exists a finite partition F = {F1 , . . . , Fm } of M defined by Fi = {j ∈ M; βj ∈ (ki−1 , ki ]},

8

i = 1, 2, . . . , m.

We assume each Fi is nonempty, otherwise, we can delete some points in the partition Γ. Set βiF = sup βj , j∈Fi

F qik

=

(

Then

qiiF = −

X

F , qik

(2.4)

k6=i

P supx∈Rd supr∈Fi j∈Fk qrj (x), P inf x∈Rd inf r∈Fi j∈Fk qrj (x),

for k < i, for k > i.

(2.5)

F βj ≤ βiF , ∀ j ∈ Fi , and βi−1 < βiF , i = 2, . . . , m.

After doing these preparation, we can get the following result. Theorem 2.6 Assume (Hi), (Hii), (Hiii) hold and N = ∞. Let V ∈ C 2 (Rd ) such that (A1) is

satisfied and M = supi∈M βi < ∞. Define the partition Γ and the corresponding vector (βiF ),
F ) + QF H finite matrix QF as above. Suppose that the m × m matrix − diag(β1F , . . . , βm m is a nonsingular M-matrix, where



Hm

1 1 1 ···

1



 1   1  ..  .  1

 0 1 1 · · ·   = 0 0 1 · · · . . .  .. .. .. · · ·  0 0 0 ···

.

(2.6)

m×m

Then (Xt , Λt ) is recurrent if lim|x|→∞ V (x) = ∞, and is transient if lim|x|→∞ V (x) = 0.
F ) H is a nonsingular M-matrix, by Proposition 2.2, there Proof. As − QF + diag(β1F , . . . , βm m F )∗ ≫ 0 such that exists a vector η F = (η1F , . . . , ηm


F λF = (λF1 , . . . , λFm )∗ = − QF + diag(β1F , . . . , βm ) Hm η F ≫ 0.

¯ := max1≤i≤m λF > 0. Set ξ F = Hm η F . Then Hence, λ i F ξiF = ηm + · · · + ηiF ,

i = 1, . . . , m,

F which implies that ξi+1 < ξiF for i = 1, . . . , m − 1, and ξ F ≫ 0. For each j ∈ M, we define

ξj = ξiF if j ∈ Fi , which is reasonable as (Fi ) is a finite partition of M. Via this method, we get

a vector ξ = (ξ1 , ξ2 , . . .)∗ from ξ F .

9

P

Let I : M → {1, 2, . . . , m} be a map defined by I (j) = k if j ∈ Fk . Let Qx g(i) =

j6=i qij (x)(gj F (βi ) and QF ,

− gi ) for g ∈ B(M). Set f (x, r) = V (x)ξr , x ∈ Rd , r ∈ M. By the definition of

we have, for r ∈ Fi , Qx ξ(r) =

X j6=r

=

X X ki

j∈Fk

F F qik (ξk − ξiF ) = QF ξ F (I (r)).

Moreover, A f (x, r) = Qx ξ(r)V (x) + ξr L(r) V (x)
F F ≤ QF ξ F (I (r)) + βI (r) ξI (r) V (x)

= −λI (r) V (x) ≤ 0. Let

 τ = inf t > 0; (Xt , Λt ) ∈ {x ∈ Rd ; |x| ≤ r0 }×{1, 2, . . . , m0 } ,

(2.7)

where m0 < ∞ is a fixed number. Applying Itˆo’s formula to (Xt , Λt ) with X0 = x, Λ0 = l, and

|x| > r0 , l > m0 (cf. [14]),

Ef (Xt∧τ , Λt∧τ ) = f (x, l) + E

Z

0

t∧τ

A f (Xs , Λs )ds ≤ f (x, l) = V (x)ξl .

(2.8)

(1) When lim|x|→∞ V (x) = 0, if P(τ < ∞) = 1, then passing t to ∞ in (2.8), we get inf

{y:|y|≤r0 }

V (y) ≤ EV (Xτ ) ≤ max i,k

 ξF  i

ξkF

V (x),

as |Xτ | ≤ r0 . We get inf {y:|y|≤r0 } V (y) > 0 by the compactness of set {y; |y| ≤ r0 } and positive-

ness of function V . So letting |x| tend to ∞ in previous inequality, the right hand goes to 0,

but the left hand is strictly bigger than a positive constant, which is a contradiction. Therefore, P(τ < ∞) > 0, and the process (Xt , Λt ) is transient. (2) Consider the case lim|x|→∞ V (x) = ∞. Introduce another stopping time τK = inf{t > 0; |Xt | ≥ K}. 10

As the process (Xt , Λt ) is nonexplosive, τK increases to ∞ almost surely as K → ∞. Itˆo’s

formula also yields that

E[V (Xt∧τK ∧τ )ξΛt∧τK ∧τ ] ≤ V (x)ξl . Letting t → ∞, Fatou’s lemma implies that  ξF   E V (XτK ∧τ )] ≤ max iF V (x), i,k ξk

and hence,

P(τ ≥ τK ) ≤ max i,k

 ξF  i ξkF

V (x) inf {y;|y|=K} V (y)

.

Since lim|x|→∞ V (x) = ∞, letting K → ∞ in the previous inequality, we obtain that P(τ =

∞) ≤ 0. We have completed the proof till now.

As an application of Theorem 2.6, we construct an example of state-independent RSDP in an infinite state space. Example 2.1 Let (Λt ) be a birth-death process on M = {1, 2, . . .} with bi ≡ b > 0 and ai ≡ a > 0. Let Xt be a random diffusion process on [0, ∞) with reflecting boundary at 0 and

satisfies

dXt = βΛt Xt dt +

√

2dBt ,

where βi = κ − i−1 for i ≥ 1. First, set V (x) = x. Let us take the finite partition F = {F1 , F2 } F = b and q F = a . Then to be F1 = {1} and F2 = {2, 3, . . .}. It is easy to see that q12 1 2 21

L(i) V (x) = βi V (x),

x > 1, i ≥ 1.

So β1F = κ − 1, β2F = κ, and
− QF + diag(β1F , β2F ) H2 =

b1 − β1F −a2

−β1F

−β2F

!

.

By Proposition 2.2, previous matrix is a nonsingular M-matrix if and only if κ < b + 1,

and κ2 − (b + a + 1)κ + a > 0.

Therefore, according to Theorem 2.6, if (2.9) holds, the process (Xt , Λt ) is recurrent.

11

(2.9)

Second, set V (x) = x−1 . For m ≥ 2, we take F1 = {1, 2, . . . , m−1} and F2 = {m, m+1, . . .},

so F = {F1 , F2 } is a finite partition of M. Then

L(i) V (x) = (−βi + 2x−2 )x−1 ≤ (−βi + 2r0−2 )x−1 ,

for x > r0 .

1 2 m + 2r0 . diag(β1F , β2F ))H2 is

Therefore, in this case, β1F = −κ + 1 + 2r02 and β2F = −κ + r0 > 0 and m ≥ 2, we get that the matrix − QF +

By the arbitrariness of a nonsingular M-matrix

if

κ + b − 1 > 0 and

κ2 + (b + a − 1)κ − a > 0.

(2.10)

Therefore, if (2.10) holds, then (Xt , Λt ) is transient. By this example, we also want to show that when (Λt ) is a Markov chain on a countable set, the process (Λt ) and (Xt , Λt ) may have very different recurrent property. More precisely, √ if we take b = 2 and a = 1, then (Λt ) is transient, but for κ < 2 − 3, (2.9) holds and hence

(Xt , Λt ) is recurrent. If we take b = 1 and a = 2, then (Λt ) is exponentially ergodic, but for √ κ > 3 − 1, (2.10) holds and hence (Xt , Λt ) is transient.

3

Recurrent property of Ornstein-Uhlenbeck process and linear diffusion in random environments

In this section, we first consider the Ornstein-Uhlenbeck type process in random environment, that is, the process (Xt , Λt ) satisfies: dXt = bΛt Xt dt + σΛt dBt ,

X0 = x ∈ R,

(3.1)

where (Bt ) is a Brownian motion in R, and (Λt ) is a continuous Markov chain on the space M = {1, . . . , N } with N < ∞. (Λt ), (Bt ) are mutually independent. The Q-matrix (qij ) of

(Λt ) is independent of (Xt ), and is irreducible and conservative. We assume that d × d matrix

σi is positive definite for every i ∈ M. Let µ = (µi ) be the invariant probability measure of P (qij ). In [9], the authors showed that when i∈M µi bi < 0, the process (Xt , Λt ) is ergodic in

weak topology, that is, the distribution of (Xt , Λt ) converges weakly to a probability measure ν. In [6, 1], the tail behavior of ν was studied. Using the criteria in Section 2, we can get the following result. 12

X

Proposition 3.1 Let (Xt , Λt ) be defined as above. If

µi bi < 0, then (Xt , Λt ) is exponen-

i∈M

X

tially ergodic. If

µi bi > 0, then (Xt , Λt ) is transient.

i∈M (i)

Proof. (1) By (3.1), we get the generator L(i) of (Xt ) is given by L(i) =

d d X ∂ 1 X (i) ∂ 2 bi x k akl + , 2 ∂xk ∂xl ∂xk k=1

k,l=1

where a(i) = σi σi∗ . Take V (x) = |x|, then for each i ∈ M, L(i) V (x) = bi |x|, for |x| > 1. As lim |x| = ∞, by Theorem 2.4, we get (Xt , Λt ) is exponentially ergodic if |x|→∞

X

µi bi < 0.

i∈M

(2) Now we take V (x) = |x|−γ with γ > 0. We have

γ(γ + 2) X (i) −γ−4 γ X (i)
−γ−2 xk xl − akl |x| − γbi |x|−γ akk |x| 2 2 k,l k  X (i) X (i)  γ γ(γ + 2) akl xk xl − |x|−2 akk , |x|−4 = |x|−γ − γbi + 2 2

L(i) V (x) =

k,l

k

for |x| > r0 > 0. When r0 is sufficiently large, it is easy to see that X (i) γ(γ + 2) −4 X (i) γ 1 |x| akl xk xl − |x|−2 akk ≤ , 2 2 r0 k,l

k

Therefore, we get L(i) V (x) ≤ (−γbi +

1 )V (x), r0

∀ |x| > r0 .

|x| > r0 .

(3.2)

By Theorem 2.4, as lim |x|−γ = 0, if |x|→∞

N X

N

µi (−γbi +

X 1 1 µ i bi + ) = −γ ≤ 0, r0 r0 i=1

i=1

PN

then (Xt , Λt ) is transient. When i=1 µi bi > 0, we can always find a constant r0 > 0 sufficiently P PN 1 large such that −γ N i=1 µi bi + r0 < 0. Hence, when i=1 µi bi > 0, (Xt , Λt ) is transient. 13

Now we consider the linear diffusion in random environments. Let (Xt , Λt ) satisfy dXt = b(Λt )Xt dt + σ(Λt )Xt dBt ,

X0 = x ∈ Rd ,

(3.3)

where b(i) ∈ Rd × Rd , σ(i) = (σ1 (i), . . . , σd (i)) and each σj (i) ∈ Rd × Rd , i ∈ M, (Bt ) is a

Brownian motion in Rd , (Λt ) is a continuous time Markov chain on a finite state space M = {1, . . . , N }. We first consider the state-independent switching, and in this case (Λt ), (Bt ) are

mutually independent. The Q-matrix (qij ) of (Λt ) is irreducible and conservative. Rewrite (3.3) in component form, dXj (t) =

d X

bjk (Λt )Xk (t)dt +

k=1

d X d X


σl (Λt ) jk Xk (t)dBl (t),

l=1 k=1

j = 1, . . . , d.

Therefore, for i ∈ M, the generator L(i) is given by L

(i)

d X d d  ∂ X 1 X ∂2 = bjk (i)xk ajm (i) + , 2 ∂xj ∂xm ∂xj j=1

j,m=1

where ajm (i) = every p ∈ R,

d X

l=1 ∂|x|p ∂xj =

(3.4)

k=1


P

∗ σl (i)x)j (σl (i)x)m or in matrix form ajm (i) = dl=1 σl (i)x σl (i)x . For p|x|p−2 xj and

By direct calculation,

∂ 2 |x|p ∂xj ∂xm

= p|x|p−2 δjm + p(p − 2)|x|p−4 xm xj , j, m = 1, . . . , d.

d

d

l=1

l=1

X

p(p − 2) p−4 X ∗ p x σl (i)x x∗ σl (i)∗ x + p|x|p−2 x∗ b(i)x. x∗ σl (i)∗ σl (i)x + |x| L |x| = |x|p−2 2 2 (i)

p

Set

λ(i) max (A (i) λmin (A

+ b) = max

φ∈S d−1

+ b) = min

φ∈S d−1

and (i) ϑmax

= max

φ∈S d−1

d X l=1

Then for 0 < p < 1,

∗



φ σl (i)φ φ∗ σl (i)∗ φ ,

d X

l=1 d X l=1

φ∗


σl (i)∗ σl (i) + b(i) φ, 2


σl (i)∗ σl (i) + b(i) φ, φ 2

(i) ϑmin

= min

φ∈S d−1

d X l=1



φ∗ σl (i)φ φ∗ σl (i)∗ φ .

 p − 2 (i)  p L(i) |x|p ≤ p λ(i) (A + b) + ϑmin |x| . max 2 14

(3.5)

∗

(3.6)

(3.7)

For p < 0,

 p − 2 (i)  p (i) L(i) |x|p ≤ p λmin (A + b) + ϑmax |x| . 2

(3.8)

Consequently, according to Theorem 2.4, and letting p → 0+ or p → 0− , we obtain the following result.

  (i) (A + b) − ϑ µi λ(i) max min < 0, then (Xt , Λt ) defined by (3.3) is positive i∈M  X  (i) recurrent. If µi λmin (A + b) − ϑ(i) max > 0, then (Xt , Λt ) is transient. Theorem 3.2 If

X

i∈M

Remark 3.3 It is easy to see that Theorem 3.2 improves the result of [17, Theorem 5.1], where   P P they showed that if i∈M µi λmax b(i) + b(i)∗ + dj=1 σj (i)σj (i)∗ < 0, then (Xt , Λt ) is positive

recurrent.

Corollary 3.4 When d = 1, i.e. (Xt ) satisfies dXt = bΛt Xt dt + σΛt Xt dBt ,

X0 = x > 0,

where bi , σi are constants for i ∈ M. Then (Xt , Λt ) is positive recurrent if P 1 2 and is transient if i∈M µi (bi − 2 σi ) > 0.

P

1 2 i∈M µi (bi − 2 σi )

k, inf x qjk jl (x), if l > k, q˜kl = ; qˆkl = supx qik il (x), if l < k, supx qjk jl (x), if l < k.

(3.9)

Applying Theorem 2.6, we obtain the following result.
˜ HN is a nonsingular M-matrix, then (Xt , Λt ) is Proposition 3.5 If − diag(β˜i1 , . . . , β˜iN ) + Q
ˆ HN is a nonsingular M-matrix, then (Xt , Λt ) is transient. recurrent. If − diag(βˆj1 , . . . , βˆjN )+ Q

4

Criteria for transience and recurrence: II

According to Foster-Lyapunov drift condition for diffusion processes, if there exists a V ∈ (i)

C 2 (Rd ) satisfying (A1) with βi ≤ 0 and lim|x|→∞ V (x) = ∞, then the diffusion process (Xt ) (i)

is exponentially ergodic. When there is no diffusion process (Xt ), i ∈ M, being exponentially

ergodic, we can not find suitable function V ∈ C 2 (Rd ) satisfying (A1), so the criteria introduced

in Section 2 are useless for this kind of RSDP . For example, the diffusion process corresponding to L(i) = 1 ∆ + |x|δ ˆb(x/|x|, i) · ∇ with δ ∈ [0, 1) is not exponentially ergodic. Therefore, to deal 2

with this kind of processes, we need to extend our method introduced in Section 2. Let (Xt , Λt ) (i)

be defined by (1.1) and (1.2) and (Xt ) be the corresponding diffusion process in the fixed environment i ∈ M with the generator L(i) . Instead of finding a function V satisfying condition

(A1), we need to find two functions h, g ∈ C 2 (Rd ) satisfying the following condition: (A2) There exists some constant r0 > 0 such that for each i ∈ M, h(x), g(x) > 0,

L(i) h(x) ≤ βi g(x),

g(x) = 0, |x|→∞ h(x) lim

|x| > r0 ,

L(i) g(x) = 0. |x|→∞ g(x) lim

Theorem 4.1 Let (Xt , Λt ) be state-independent RSDP defined by (1.1) and (1.2) with N < ∞.

Assume that (Hi), (Hii), (Hiii) hold. Let µ be the invariant probability measure of the process (Λt ). Suppose that there exist two functions h, g ∈ C 2 (Rd ) such that (A2) holds and N X

µi βi < 0.

i=1

Then (Xt , Λt ) is recurrent if

lim h(x) = ∞, is transient if lim h(x) = 0.

|x|→∞

|x|→∞

16

Proof. As

PN

i=1 µi βi

< 0, by the Fredholm alternative we obtain that there exist a constant

κ > 0 and a vector ξ such that Qξ(i) = −κ − βi ,

i ∈ M.

Set f (x, i) = h(x) + ξi g(x). We obtain A f (x, i) = L(i) h(x) + ξi L(i) g(x) + Qξ(i)g(x)  L(i) g(x)  ≤ βi + Qξ(i) + ξi g(x). g(x)

By (4.1) and condition (A2), we get  L(i) g(x)  g(x) ≤ 0, A f (x, i) ≤ − κ + ξi g(x) As N < ∞, ξ is bounded. Since lim|x|→∞

g(x) h(x)

as |x| → ∞.

(4.1)

(4.2)

g(x) = 0 and f (x, i) = (1 + ξi h(x) )h(x) for |x| > r0 ,

it is easy to see that there exists r1 > 0 such that f (x, i) > 0 for |x| > r1 . In addition, if

lim|x|→∞ h(x) = ∞, then lim|x|→∞ f (x, i) = ∞; if lim|x|→∞ h(x) = 0, then lim|x|→∞ f (x, i) =

0. By the method of Lyapunov function, inequality (4.2) yields that (Xt , Λt ) is recurrent if lim|x|→∞ h(x) = ∞ and is transient if lim|x|→∞ h(x) = 0. Now we consider the following state-independent RSDP in a finite state space, dXt = |Xt |δ ˆb(Xt /|Xt |, Λt )dt + σ(Xt , Λt )dBt , X0 = x ∈ Rd , d ≥ 1,

(4.3)

where δ ∈ [−1, 1), ˆb(·, ·) : S d−1 × M → Rd , σ(·, ·) : Rd × M → Rd×d , and (Bt ) is a d-dimensional

Brownian motion. Let (Λt ) be a continuous time Markov chain on M with irreducible conservative Q-matrix (qij ), which is also independent of (Bt ). Let µ be the invariant probability

measure of (Λt ). Set a(i) (x) = σ(x, i)σ(x, i)∗ . Suppose conditions (Hi), (Hii), (Hiii) are satisfied. In [10], the authors considered the recurrent property of (Xt , Λt ) under the condition X µiˆb(φ, i) = 0, ∀ φ ∈ S d−1 . i∈M

In this section, we shall study the case

P

ˆ

i∈M µi b(φ, i)

6= 0.

Theorem 4.2 Assume that ka(i) (·)k is bounded on Rd for every i ∈ M. Let  d P  ˆbk ( x , i) xk ,  if δ ∈ (−1, 1), lim sup  |x| |x|   |x|→∞ k=1 d P βi = (i) Pd ˆ x akl xk xl   P   (i) k=1 bk ( |x| ,i)xk k,l=1  lim sup 1 d a (x) −  , if δ = −1, +  2 k=1 kk 2 |x| 2|x| |x|→∞

17

(4.4)

and

β˜i =

 d P   ˆb ( x , i) xk ,  lim inf   |x|→∞ k=1 k |x| |x|

 P   (i)  d 1   lim inf 2 k=1 akk (x) −

d P

P

i∈M µi βi

(i)

akl xk xl

k,l=1

+

2|x|2

|x|→∞

If

if δ ∈ (−1, 1),

P

< 0, then (Xt , Λt ) is recurrent. If

Pd

 ˆ x k=1 bk ( |x| ,i)xk

i∈M

(4.5)

, if δ = −1.

|x|

µi β˜i > 0, then (Xt , Λt ) is transient.

Proof. (1) Set h(x) = |x|γ , γ > 0, and g(x) = |x|γ+δ−1 . Then it holds that L(i) g(x) = 0. |x|→∞ g(x)

g(x) = 0, |x|→∞ h(x) lim

By direct calculation we get Pd h

(i) k,l=1 akl (x)xk xl 2|x|δ+3

(i)

L h(x) = (γ − 1)

lim

+

(i) k=1 akk (x) 2|x|δ+1

Pd

+

ˆ x i k=1 bk ( |x| , i)xk

Pd

|x|

γg(x).

(4.6)

When δ ∈ (−1, 1), h

(i)

−δ−1

L h(x) = O(|x| which implies that if

P

X

i∈M

i∈M µi βi

)+

ˆ x i k=1 bk ( |x| , i)xk

Pd

|x|

γg(x),

< 0, then there exists r0 > 0 such that

 µi O(|x|−δ−1 ) +

ˆ x  k=1 bk ( |x| , i)xk

Pd

|x|

< 0,

for |x| > r0 .

Applying Theorem 4.1, as γ > 0, we obtain that (Xt , Λt ) is recurrent if When δ = −1, it holds Pd lim sup lim(γ − 1) |x|→∞ δ↓0

Therefore, if

P

i∈M µi βi

(i) k,l=1 akl (x)xk xl 2|x|δ+3

+

(i) k=1 akk (x) 2|x|δ+1

Pd

+

P

i∈M

µi βi < 0.

ˆ x k=1 bk ( |x| , i)xk

Pd

|x|

= βi .

< 0, by choosing γ > 0 sufficiently small and r0 > 0 sufficiently large,

we can use Theorem 4.1 to show that (Xt , Λt ) is recurrent. (2) Now we set h(x) = |x|−γ and g(x) = |x|−γ+δ−1 for γ > 0. Then it still holds g(x) = 0, |x|→∞ h(x) lim

L(i) g(x) = 0, |x|→∞ g(x) lim

18

and (i)

h

L h(x) = − (γ + 1)

(i) k,l=1 akl (x)xk xl 2|x|δ+3

Pd

+

Pd

(i)

k=1 akk (x) + 2|x|δ+1

ˆ x i k=1 bk ( |x| , i)xk

Pd

|x|

(−γ)g(x). (4.7)

Note that it is −γ < 0 before g(x) in above equality. Similar to the argument in step (1), we can conclude the proof.

When the dimension d is equal to 1, we can obtain more explicit and complete criteria presented as follows. Corollary 4.3 Let (Xt , Λt ) be a regime-switching diffusion on [0, ∞) with reflecting boundary at 0. (Xt ) satisfies

dXt = bΛt Xtδ dt + σΛt dBt ,

δ ∈ [−1, 1),

where bi , σi are constants for i in a finite set M. (Λt ) is a continuous time Markov chain on P M independent of (Bt ). Then (Xt , Λt ) is recurrent if and only if i∈M µi bi ≤ 0.

Proof. By taking h(x) and g(x) as in the Theorem 4.2, it is easy to check that βi = β˜i = P bi . So according to Theorem 4.2, (Xt , Λt ) is recurrent if i∈M µi bi < 0 and is transient if P P i∈M µi bi > 0. Therefore, we only need to consider the case i∈M µi bi = 0. To deal with this

situation, we have to consider it separatively according to the range of δ. Note that it holds P P −1 i∈M µi bi (Q b)(i) < 0 as i∈M µi bi = 0 (cf. [10]). Case 1: δ ∈ (0, 1). For p > 0, set

f (x, i) = xp − p(Q−1 b)(i)xp−1+δ + ci xp−2+2δ , where the vector (ci ) would be determined later. By noting that δ ∈ (0, 1), we obtain   A f (x, i) = − p(p − 1 + δ)bi (Qb)(i) + Qc(i) xp−2+2δ + o(xp−2+2δ ).

Take p ∈ (0, 1− δ), then

P

i∈M

p(p − 1+ δ)µi bi (Q−1 b)(i) > 0. By the Fredholm alternative, there

exist a constant β > 0 and a vector (ci ) such that Qc(i) = p(p − 1 + δ)bi (Q−1 b)(i) − β. Choosing

these p and (ci ), we have A f (x, i) = −βxp−2+2δ + o(xp−2+2δ ). As lim|x|→∞ f (x, i) = ∞ for each P i ∈ M, we obtain that (Xt , Λt ) is recurrent when i∈M µi bi = 0 and δ ∈ (0, 1).

19

Case 2: δ ∈ [−1, 0). In this situation, we take f (x, i) = xp − p(Q−1 b)(i)xp−1+δ . Then 1  1 2 σi p(p−1)xp−2 −p(Q−1 b)(i) (p−1+δ)(p−2+δ)σi2 xp−3+δ +bi (p−1+δ)xp−2+2δ 2 2 1 2 = σi p(p − 1)xp−2 + o(xp−2 ). 2

A f (x, i) =

By setting p ∈ (0, 1), we have limx→∞ f (x, i) = ∞ and A f (x, i) ≤ 0. Hence, (Xt , Λt ) is recurrent.

Case 3: δ = 0. We take f (x, i) = xp − p(Q−1 b)(i)xp−1 + ci xp−2 . Then 1 2  σi p(p − 1) − p(p − 1)bi (Q−1 b)(i) + Qc(i) xp−2 + o(xp−2 ). 2
P Putting p ∈ (0, 1), as p(p − 1) i∈M µi σi2 − bi (Q−1 b)(i) < 0, there exist a vector (ci ) and a A f (x, i) =

positive constant β such that

1 Qc(i) + σi2 p(p − 1) − p(p − 1)bi (Q−1 b)(i) = −β < 0. 2 Therefore, we get A f (x, i) ≤ 0 and limx→∞ f (x, i) = ∞, which implies that (Xt , Λt ) is recurrent.

We complete the proof.

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