Stochastic continuity, irreducibility and non confluence for SDEs with jumps
aa r X i v : . [ m a t h . P R ] J u l Stochastic continuity, irreducibility and nonconfluence for SDEs with jumps
Guangqiang Lan ∗ School of Science, Beijing University of Chemical Technology, Beijing 100029, ChinaEmail: [email protected]
Jiang-Lun Wu
Department of Mathematics, College of Science, Swansea University, Swansea SA2 8PP, UKEmail: [email protected]
Abstract
In this paper, we investigate stochastic continuity (with respect to the initial value),irreducibility and non confluence property of the solutions of stochastic differentialequations with jumps. The conditions we posed are weaker than those relevant con-ditions existing in the literature. We also provide an example to support our newconditions.
MSC 2010:
Key words: stochastic differential equations with jumps; stochastic continuity; irreducibil-ity; non confluence; test function.
Given a probability space (Ω , F , P ) endowed with a complete filtration ( F t ) t ≥ . Let d, m ∈ N be arbitrarily fixed. We are concerned with the following stochastic differential equations(SDEs) with jumps and with random coefficients X t = X + Z t σ ( s, ω, X s ) dB s + Z t b ( s, ω, X s ) ds + Z t +0 Z U f ( s, ω, X s − , u ) ˜ N k ( ds, du ) + Z t +0 Z U f ( s, ω, X s − , u ) N k ( ds, du ) (1.1)where B, N k , ˜ N k denote an m -dimensional ( F t )-Brownian motion, a Poisson random mea-sure and its compensated Poisson martingale measure, respectively, and E ( N k ( ds, du )) = dsν ( du ) with ν being a σ finite measure on a given measurable space ( U, B ( U )), σ : ( t, ω, x ) ∈ ∗ Corresponding author. Supported by China Scholarship Council, National Natural Science Foundationof China (NSFC11026142) and Beijing Higher Education Young Elite Teacher Project (YETP0516). , ∞ ) × Ω × R d σ ( t, ω, x ) ∈ R d ⊗ R m and b : ( t, ω, x ) ∈ [0 , ∞ ) × Ω × R d b ( t, ω, x ) ∈ R d are progressively measurable functions, f i : ( t, ω, x, u ) ∈ [0 , ∞ ) × Ω × R d × U f i ( t, ω, x, u ) ∈ R d , i = 1 , F t ) t ≥ predictable measurable functions withsupp f ( t, ω, x, · ) ∩ supp f ( t, ω, x, · ) = ∅ , ν (supp f ( t, ω, x, · )) < ∞ for all ( t, ω, x ) ∈ [0 , ∞ ) × Ω × R d , and all the four functions are continuous with respect tothe third variable x .In order that the integrals in the definition of the solutions of the equation (1.1) are well-defined, we make the following fundamental assumption which is enforced throughout thepaper E Z T sup | x |≤ R h | b ( s, · , x ) | + || σ ( s, · , x ) || + Z U | f i | j ( s, · , x, u ) ν ( du ) i ds < ∞ (1.2)for all T, R > , i, j = 1 ,
2, where the norm || · || stands for the Hilbert-Schmidt norm || σ || := d P i =1 m P j =1 σ ij for any d × m -matrix σ = ( σ ij ) ∈ R d ⊗ R m and | · | denotes the usualEuclidean norm on R d . As usual, we use < · , · > to denote the Euclidean inner product on R d .Next, we fix R > η R : R + → R + be a differentiable function such that η R (0) = 0 , η ′ R ( x ) ≥ , Z dxη R ( x ) = ∞ . We assume further that the coefficients of SDE (1.1) fulfill the following condition || σ ( t, · , x ) − σ ( t, · , y ) || + 2 h x − y, b ( t, · , x ) − b ( t, · , y ) i + X i =1 Z U | f i ( t, · , x, u ) − f i ( t, · , y, u ) | ν ( du )+ 2 Z U h x − y, f ( t, · , x, u ) − f ( t, · , y, u ) i ν ( du ) ≤ g ( t, · ) η R ( | x − y | ) , t ≥ , | x | ∨ | y | ≤ R, a.s. (1.3)for g : [0 , ∞ ) × Ω → [0 , ∞ ) to be a measurable function satisfying E Z t g ( s, · ) ds < ∞ , ∀ t ≥ . Under the above assumption (1.3), one can show (see e.g. [6]) that there exists a uniquepathwise solution of SDE (1.1) which might blow up in finite time. In order to emphasizethe solutions with different initial values, we use the notation X t ( x ) for t ≥ X = x ∈ R d . Moreover, we denote the explosion timeof the solution X t ( x ) , t ≥
0, by ζ x := inf { t > | X t ( x ) | = + ∞} . Our first main result concerns the uniformly stochastic continuity of the solution of SDE(1.1). We have the following 2 heorem 1.1
Assume that the condition (1.3) holds. Let X t ( x ) and X t ( y ) be the solutionsof SDE (1.1) starting from x, y ∈ R d , respectively. Then for any ε > , lim y → x P ( sup s
Note that when the solutions are not global, the supremum must be taken with s < t ∧ ζ x ∧ ζ y , otherwise it will be absurd since | X s ( x ) − X s ( y ) | = ∞ in this case. Next, we consider the following SDE with deterministic coefficients (i.e., all coefficientsare independent of ω ∈ Ω) X t = x + Z t σ ( s, X s ) dB s + Z t b ( s, X s ) ds + Z t +0 Z U f ( s, X s − , u ) ˜ N k ( ds, du ) + Z t +0 Z U f ( s, X s − , u ) N k ( ds, du ) . (1.4)Under the above condition (1.3), there exists a unique solution of equation (1.4). Similarto Theorem 2.9.1 of [4], one can show that the solution is a Markov process. For f ∈ C b ( R d ),we define operator P s,t f ( x ) := E s,x f ( X t ) := Z y ∈ R d f ( y ) p s,t ( x, dy ) , ≤ s ≤ t, x ∈ R d where p s,t ( x, A ) := P ( X t ∈ A | X s = x ) , ≤ s ≤ t, x ∈ R d , A ∈ B ( R d )is the transition probability measure of the Markov process. The operator family { P s,t } ≤ s ≤ t is the Markov semigroup associated with the solution. Furthermore, our Theorem 1.1 ensuresthat the Markov semigroup { P s,t } is Fellerian. That is, for any f ∈ C b ( R d ), we have P s,t f ∈ C b ( R d ) . We are going to study the irreducibility of the transition probability measure p s,t for any0 ≤ s ≤ t . We say that the family { p s,t } ≤ s ≤ t is irreducible if for any 0 ≤ s ≤ t and x ∈ R d , p s,t ( x, A ) > A ⊂ R d . Similar to [8] and [5], we introduce the following monotonicity and growth conditions.Suppose that || σ ( t, x ) − σ ( t, y ) || + h x − y, b ( t, x ) − b ( t, y ) i + X i =1 Z U | f i ( t, x, u ) − f i ( t, y, u ) | ν ( du )+ Z U h x − y, f ( t, x, u ) − f ( t, y, u ) i ν ( du ) ≤ g ( t ) η ( | x − y | ) , t ≥ , a.s. (1.5)3olds with g ≥ , R t g ( s ) ds < ∞ , ∀ t > ,η ( x ) := (cid:26) x log x , x ≤ r < e ,r log r + (log r − x − r ) , r < x, || σ ( t, x ) || + 2 h x, b ( t, x ) i + X i =1 Z U | f i ( t, x, u ) | ν ( du )+ 2 Z U h x, f ( t, x, u ) i ν ( du ) ≤ f ( t )( | x | + 1) (1.6)and | σ T ( t, x ) x | + X i =1 Z U (2 h x, f i ( t, x, u ) i + | f i ( t, x, u ) | ) ν ( du ) ≤ f ( t )( | x | + 1) (1.7)holds for certain measurable function f : [0 , ∞ ) → [0 , ∞ ) with R t f ( s ) ds < ∞ , ∀ t > p s,t , we assume that m ≥ d ,and we need the following so called strong ellipticity condition on the coefficient σ , that is,there exists λ > || σ − ( t, x ) || ≤ λ, t > , x ∈ R d , (1.8)where σ − stands for the left inverse of matrix σ. Our second main result is the following
Theorem 1.3
Assume that the conditions (1.5), (1.6), (1.7) and (1.8) hold. If there exists ≤ p < such that || σ ( t, x ) || + X i =1 Z U | f i ( t, x, u ) | ν ( du ) + ( Z U | f ( t, x, u ) | ν ( du )) ≤ f ( t )( | x | p + 1) (1.9) hold with f : [0 , ∞ ) → [0 , ∞ ) being a measurable function satisfying R t f p ( s ) ds < ∞ , ∀ t ≥ ,then { p s,t } is irreducible. Remark 1.4
It is worthwhile mentioning that here we do not need the assumption that || σ ( t, x ) || is linear growth. The linear growth condition on the coefficient σ was required inboth [8, 5] while [8] even only deals with SDE without jumps. Our conditions (1.6), (1.7)and (1.9) are weaker than the linear growth condition on the coefficient σ (see Section 4) in[8, 5], even for relatively simpler SDEs without jumps in [8]. Our final task of the present paper concerns the non confluence property of the time-homogeneous SDE (1 .
4) in which the coefficients are independent of t . We say that thesolution X t of equation (1.4) has non confluence property , if for any initial values x = y ,P ( X t ( x ) = X t ( y ) , ∀ t >
0) = 1 .
4n an early work [1], Emery studied such kind of non confluence property for generalstochastic differential equations without jumps under Lipschitzian coefficients. Yamadaand Ogura considered in [7] for SDEs without jumps with non-Lipschitz coefficients. Weaim to give a new sufficient condition for the non confluence property of the solution X t ofthe equation (1.4).Fix R > γ R : R + → R + be a differentiable function such that γ R (0) = 0 , Z dxγ R ( x ) = ∞ and x ( γ ′ R ( x ) + 1) γ R ( x ) ≤ K, ∀ x ∈ [0 , ∞ )for some constant K > which is independent of x and R .We have the following Theorem 1.5
Assume that (1.3) holds with all the coefficients independent of t and ω . Let K be given as above. If for any | x | ∨ | y | ≤ R || σ ( x ) − σ ( y ) || − K − h x − y, b ( x ) − b ( y ) i + 12 K − h Z U | f ( x, u ) − f ( y, u ) | − | f ( x, u ) − f ( y, u ) | + 2 h x − y, f ( x, u ) − f ( y, u ) i i ν ( du ) ≤ η R ( | x − y | ) (1.10) and | f ( x, u ) − f ( y, u ) | + 2 h x − y, f ( x, u ) − f ( y, u ) i ≥ , (1.11) then the unique solution of the time-homogeneous SDE (1 . has non confluence property. Remark 1.6
When there is no jumps, that is f i ≡ , i = 1 , , in [3], the authors showedthat in one-dimensional case, (1.10) also implies that the solution is stochastic monotonic.However, in the present case, there is no stochastic monotonicity of the solution. Actually,we can conclude that if conditions (1.3) and (1.10) are satisfied, then the process is stochasticmonotonic between any two successive jumps. The rest of the paper is organized as follows. In next section, Section 2, we show theuniformly stochastic continuity with respect to initial value of the solution. Section 3 isdevoted to the proof of irreducibility of the transition probability { p s,t } ≤ s ≤ t . In Section 4,we present an example to illustrate that our conditions in Theorem 1.3 is indeed weakerthan those relevant known conditions in the literature. Finally in Section 5, we verify thenon confluence property of solution of the time-homogeneous equation (1.4). Proof of Theorem 1.1 ε > , let x, y ∈ R d be such that | y − x | < ε. Denote ξ t := | η t | := | X t ( y ) − X t ( x ) | , τ ( x, y ) := inf { t > , ξ t > ε } . Define the function ϕ δ : [0 , ∞ ) → [0 , ∞ ) by ϕ δ ( x ) := Z x dsγ R ( s ) + δ . Then ϕ ′′ δ ( x ) ≤ , x > . We can extend ϕ δ to the real line (denoted by ϕ δ again) suchthat ϕ ′′ δ ( x ) ≤ , x ∈ R . Denote τ R ( x, y ) := inf { t, | X t ( x ) | ∨ | X t ( y ) | > R } ,h t := b ( t, · , X t ( x )) − b ( t, · , X t ( y )) , e t := σ ( t, · , X t ( x )) − σ ( t, · , X t ( y ))and k i ( t − , u ) := f i ( t, · , X t − ( x ) , u ) − f i ( t, · , X t − ( y ) , u ) , i = 1 , . It’s clear that τ R ( x, y ) → ζ x ∧ ζ y as R → ∞ . By Itˆo’s formula, we have ϕ δ ( ξ t ∧ τ ( x,y ) ∧ τ R ) = ϕ δ ( | x − y | ) + M t + Z t ∧ τ ( x,y ) ∧ τ R ( x,y )0 ϕ ′ δ ( ξ s )[2 h η s , h s i + || e s || + Z U | k ( s, u ) | ν ( du )] ds + 2 Z t ∧ τ ( x,y ) ∧ τ R ( x,y )0 ϕ ′′ δ ( ξ s ) | e Ts η s | ds + Z t ∧ τ ( x,y ) ∧ τ R ( x,y )0 Z U [ ϕ δ ( | η s + k ( s, u ) | ) − ϕ δ ( ξ s )] ν ( du ) ds + Z t ∧ τ ( x,y ) ∧ τ R ( x,y )0 Z U h ϕ δ ( | η s + k ( s, u ) | ) − ϕ δ ( ξ s ) − ϕ ′ δ ( ξ s )( | k ( s, u ) | + 2 h η s , k ( s, u ) i i ν ( du ) ds. where k i ( s, u ) is defined similar to k i ( s − , u ) with X s − replaced by X s . Since ϕ ′′ δ ≤ , bycondition (1.11), we have Z t ∧ τ ( x,y ) ∧ τ R ( x,y )0 Z U [ ϕ δ ( | η s + k ( s, u ) | ) − ϕ δ ( ξ s )] ν ( du ) ds ≤ Z t ∧ τ ( x,y ) ∧ τ R ( x,y )0 Z U ϕ ′ δ ( ξ s )( | k ( s, u ) | + 2 h η s , k ( s, u ) i ν ( du ) ds and Z t ∧ τ ( x,y ) ∧ τ R ( x,y )0 Z U h ϕ δ ( | η s + k ( s, u ) | ) − ϕ δ ( ξ s ) − ϕ ′ δ ( ξ s )( | k ( s, u ) | + 2 h η s , k ( s, u ) i ) i ν ( du ) ds ≤ . Thus E ϕ δ ( ξ t ∧ τ ( x,y ) ∧ τ R ( x,y ) ) ≤ ϕ δ ( | x − y | ) + E Z t g ( s, · ) ds. δ = | x − y | in the above inequality, we have E ϕ δ ( ξ t ∧ τ ( x,y ) ∧ τ R ( x,y ) ) ≤ δ + E Z t g ( s, · ) ds. Hence P ( τ ( x, y ) < t ∧ τ R ( x, y )) ϕ δ ( ε ) ≤ E ϕ δ ( ξ t ∧ τ ( x,y ) ∧ τ R ( x,y ) ) ≤ δ + C t , where C t = E R t g ( s, · ) ds. It follows that P ( sup ≤ s
Assume (1.6) and (1.7) hold. Then for any ≤ p < , the maximalprocess Y t := sup s ≤ t | X s | , t ≥ satisfies E ( Y tp ) ≤ C t,p , t ≥ . To prove Proposition 3.1, we need the following lemma.
Lemma 3.2
Suppose the assumptions of Proposition 3.1 hold. Let M ct := 2 Z t h X s , σ ( s, X s ) dB s i ,M dt := X i =1 Z t +0 Z U (cid:16) | f i ( s, X s − , u ) | + 2 h X s − , f i ( s, X s − , u ) i (cid:17) ˜ N k ( ds, du ) and M c ∗ t := sup s ≤ t | M cs | , M d ∗ t := sup s ≤ t | M ds | . Then for any ≤ p < , there exist K > and L > such that E (( M c ∗ t ) p ) ≤ C p (cid:16) K E (( Y pt + 1)) + K t p − Z t f p ( s ) E ( Y ps + 1) ds (cid:17) (3.1) and E (( M d ∗ t ) p ) ≤ C ′ p (cid:16) − p L / (4 − p ) E ( Y pt + 1) + pL /p Z t f ( s ) E ( Y ps + 1) ds (cid:17) . (3.2)7 roof By Burkholder-Davis-Gundy inequality (for continuous martingales), E (( M c ∗ t ) p ) ≤ C p E [( Z t | σ T ( s, X s ) X s | ds ) p ] ≤ C p E (cid:16) ( Y t + 1) p (cid:0) Z t | σ T ( s, X s ) X s | ( | X s | + 1) ds (cid:1) p (cid:17) ≤ C p (cid:16) K E (( Y t + 1) p ) + K E (cid:2)(cid:0) Z t | σ T ( s, X s ) X s | ( | X s | + 1) ds (cid:1) p (cid:3)(cid:17) ≤ C p (cid:16) K E (( Y pt + 1)) + K t p − Z t f p ( s ) E ( Y ps + 1) ds (cid:17) . We have used Young’s inequality in the last second derivation and H¨older inequality inthe last derivation.On the other hand, by Burkholder-Davis-Gundy inequality for c´adl´ag martingales (seee.g., [5]), it follows that E (( M d ∗ t ) p ) ≤ C ′ p X i =1 E (cid:16) Z t +0 Z U F i ( s − , u ) N k ( ds, du ) (cid:17) p ≤ C ′ p X i =1 E (cid:16) ( Y t + 1) rp ( Z t +0 Z U F i ( s − , u ) ( | X s − | + 1) r N k ( ds, du )) p (cid:17) where F i ( s − , u ) := | f i ( s, X s − , u ) | + 2 h X s − , f i ( s, X s − , u ) i , (3.3) r > E (( M d ∗ t ) p ) ≤ C ′ p X i =1 E (cid:16) aL a ( Y t + 1) arp + L b b (cid:16) Z t +0 Z U F i ( s − , u ) ( | X s − | + 1) r N k ( ds, du ) (cid:17) bp (cid:17) , where a, b > a + b = 1 . Take a, b, r such that1 a + 1 b = 1 , bp arp p. (3.4)Then we have b = p > , a = − p and r = − p . Thus, by condition (1.7), it follows that E (( M d ∗ t ) q ) ≤ C ′ p X i =1 E (cid:16) − p L / (4 − p ) ( Y pt + 1) + pL /p Z t +0 Z U F i ( s − , u ) ( | X s − | + 1) (4 − p ) / N k ( ds, du ) (cid:17) ≤ C ′ p X i =1 E (cid:16) − p L / (4 − p ) ( Y pt + 1) + pL /p Z t ds Z U F i ( s, u ) ( | X s | + 1) (4 − p ) / ν ( du ) (cid:17) ≤ C ′ p (cid:16) − p L / (4 − p ) E ( Y pt + 1) + pL /p Z t f ( s ) E ( Y ps + 1) ds (cid:17) . We complete the proof. (cid:3) roof of Proposition 3.1 By Itˆo’s formula, we have | X t | = | x | + Z t (cid:16) h X s , b ( s, X s ) i + || σ ( s, X s ) || (cid:17) ds + Z t (cid:16) Z U ( | f | ( s, X s , u ) + F ( s, u ) (cid:17) ν ( du ) ds + 2 Z t h X s , σ ( s, X s ) dB s i + X i =1 Z t +0 Z U F i ( s − , u ) ˜ N k ( ds, du ) , (3.5)where F i ( s − , u ) is defined by (3.3), as in Lemma 3.2, and F i ( s, u ) is defined in the sameway with X s − replaced by X s . Thus, by (1.6), Y t ≤ | x | + Z t f ( s )( Y s + 1) ds + M c ∗ t + M d ∗ t . (3.6)Then we have E ( Y pt ) ≤ C p (cid:16) | x | p + ( Z t f ( s ) E ( Y ps + 1) ds ) + E (( M c ∗ t ) p ) + E (( M d ∗ t ) p ) (cid:17) (3.7)By Lemma 3.2 and (3.6), we have E ( Y pt ) ≤ C ′ p n | x | p + t p − ( Z t f p ( s ) E ( Y ps + 1) ds )+ (cid:16) K E ( Y pt + 1) + K t p − Z t f p ( s ) E ( Y ps + 1) ds (cid:17) + (cid:16) − p L / (4 − p ) E ( Y pt + 1) + pL /p Z t f ( s ) E ( Y ps + 1) ds (cid:17)o . (3.8)Set C ′ p K = (4 − p ) C ′ p L / (4 − p ) = 14 . We have K = 2 C ′ p , L = ( C ′ p (8 − p )) (4 − p ) / . It follows that E ( Y pt + 1) ≤ A + B Z t ( f p ( s ) + f ( s )) E ( Y ps + 1) ds o (3.9)where A = 1 + 2 C ′ p | x | p , B = C ′ p (( C ′ p + 1) t p − + p ( C ′ p (8 − p )) (4 − p ) /p ). We then complete theproof by using Gronwall’s lemma. (cid:3) In what follows, we consider the irreducibility of p s,t . For any T > , let us fix t ∈ (0 , T ) , whose value will be determined below. For any ε >
0, define X εt := X t · {| X t |≤ ε } . Then by Proposition 3.1, for any 2 ≤ p < , lim ε ↓ E | X εt − X t | p = 0 . t ∈ [ t , T ] and y ∈ R d , define Y εt := T − tT − t X εt + t − t T − t y and h εt := y − X εt T − t − b ( t, Y εt ) . Then Y εt = X εt , Y εT = y and Y εt = X εt + Z tt b ( s, Y εs ) ds + Z tt h εs ds, t ∈ [ t , T ] . Consider the following SDE on [ t , T ]: Y t = X t + Z tt b ( s, Y s ) ds + Z tt h εs ds + Z tt σ ( s, Y s ) dB s + Z t + t Z U f ( s, Y s − , u ) ˜ N k ( ds, du ) + Z t + t Z U f ( s, Y s − , u ) N k ( ds, du ) . (3.10)We have the following Proposition 3.3
Suppose b, σ and f i satisfy (1.5), (1.6), (1.7) and (1.9). Then for any T > , E | Y T − y | ≤ C ( t , T, p ) e − R Tt g ( s )+1) ds . (3.11) Proof
Set Z εt := Y t − Y εt . Since the coefficient b is continuous with respect to x , and h εt is independent of Y t bydefinition, then conditions (1.5) and (1.6) still hold when b ( t, x ) is replaced by b ( t, x ) + h εt .Thus, the SDE (3.10) has a unique non explosive solution on [ t , T ] . By Itˆo’s formula and condition (1.5) we have E | Z εt | = E | X εt − X t | + E n Z tt (cid:16) h Z εs , b ( s, Y s ) − b ( s, Y εs ) i + || σ ( s, Y s ) || (cid:17) ds + Z tt Z U ( X i =1 Z U | f i ( s, Y s , u ) | + 2 h Z εs , f ( s, Y s , u ) i ) ν ( du ) ds o ≤ E | X εt − X t | + E n Z tt g ( s ) η ( | Z εs | ) ds + 2 Z tt || σ ( s, Y εs ) || ds + 2 Z tt ( Z U ( X i =1 Z U | f i ( s, Y εs , u ) | + | Z εs || f ( s, Y εs , u ) | ν ( du )) ds o . (3.12)10n the other hand, since2 Z tt Z U | Z εs || f ( s, Y εs , u ) | ν ( du ) ds ≤ Z tt | Z εs | ds + Z tt ( Z U | f ( s, Y εs , u ) | ν ( du )) ds, and η is concave and η ( x ) ≥ x for r small enough by definition, by condition (1.9) and thefact that | Y εs | p ≤ C p (sup s ≤ T | X s | p + | y | p ) , we have E | Z εt | ≤ E | X εt − X t | + Z tt [ g ( s ) η ( E ( | Z εs | )) + E ( | Z εs | )] ds + 2( C p ( E sup s ≤ T | X s | p + | y | p ) + 1) Z tt f ( s ) ds ≤ C ( t , T, p ) + 2 Z tt ( g ( s ) + 1) η ( E ( | Z εs | )) ds, where C ( t , T, p ) = E | X εt − X t | + 2( C p ( E sup s ≤ T | X s | p + | y | p ) + 1) Z Tt f ( s ) ds. (3.13)Now, by utilising Bihari’s inequality (see Lemma 2.1 of [8]), we could ensure that E | Z εt | ≤ C ( t , T, p ) e − R Tt g ( s )+1) ds holds for all t ∈ [ t , T ] . The proof is thus completed. (cid:3)
Now we are in a position to prove Theorem 1.3.
Proof of Theorem 1.3
Define Y t := X t , t ∈ [0 , t ] . Then for any t ∈ [0 , T ], Y t = x + Z t b ( s, Y s ) ds + Z t s>t h εs ds + Z t σ ( s, Y s ) dB s + Z t +0 Z U f ( s, Y s − , u ) ˜ N k ( ds, du ) + Z t +0 Z U f ( s, Y s − , u ) N k ( ds, du ) . (3.14)Define ¯ B t := B t + Z t s>t σ − ( s, Y εs ) h εs ds and R εT := exp h − Z T s>t σ − ( s, Y εs ) h εs dB s − Z T s>t | σ − ( s, Y εs ) h εs | ds i . Then by (1.8), the definition of h εs and the continuity of b with respect to x, we knowthat R εT · P is a probability measure which is equivalent to P , and ¯ B t is a R εT · P Brownian11otion. On the other hand, by [6] Theorem 124 ˜ N k is still a Poisson martingale measurewith the same compensator ν ( du ) ds under the new probability measure R εT · P . By (3.14),we have Y t = x + Z t b ( s, Y s ) ds + Z t σ ( s, Y s ) d ¯ B s + Z t +0 Z U f ( s, Y s − , u ) ˜ N k ( ds, du ) + Z t +0 Z U f ( s, Y s − , u ) N k ( ds, du ) . (3.15)By the pathwise uniqueness of SDE (1.4) (hence the uniqueness in law), Y · ( x ) has thesame law as X · ( x ) on [0 , T ] for any T >
0. Thus we only need to prove that for each0 ≤ s < t, x ∈ R d , P ( | Y t ( x ) − Y s ( x ) | > a ) < a > R εT · P and P are equivalent. Now P ( | Y t ( x ) − Y s ( x ) | > a ) ≤ P ( | Y t ( x ) − y | > a P ( | Y s ( x ) − y | > a ≤ a ( E ( | Y t ( x ) − y | ) + E ( | Y s ( x ) − y | )) . According to Proposition 3.3, it follows that E ( | Y t ( x ) − y | ) + E ( | Y s ( x ) − y | ) ≤ C (˜ t , t, p ) e − R t ˜ t g ( r )+1) dr + C (˜ t , s, p ) e − R s ˜ t g ( r )+1) dr where ˜ t ≤ t, ˜ t ≤ s. Now let ε to be sufficiently small, ˜ t close to t and ˜ t close to s . Wehave P ( | Y t ( x ) − Y s ( x ) | > a ) < . This completes the proof. (cid:3)
As pointed out in Remark 1.4 in Section 1, our assumption on the coefficient σ is weakerthan those relevant conditions carried out in [8, 5]. Here let us give an example to supportour conditions. We create an example in the manner that it does satisfy our conditions(1.6), (1.7) and (1.9) but it neither fulfill the condition ( H ) of Theorem 1.1 in [8] nor thecondition ( H ) of Theorem 1.3 in [5]. Thus our example indicates that our conditions areindeed weaker than those known conditions existing in the literare. Example
For simplicity, we only consider the time-homogeneous case with f = f ≡ d = m = 2. For any 2 < p < , define the 2 × σ ( x ) and thedrift vector coefficient b ( x ), respectively, by σ ( x ) := (cid:18) x | x | x | x | − (1 + | x | p − ) x (1 + | x | p − ) x (cid:19) (4.1)and b ( x ) := − K (1 + | x | p − ) x, with constant K ≥ . (4.2)12hen, we have || σ ( x ) || + 2 h x, b ( x ) i = | x | (1 + | x | ) + (1 + | x | p − ) | x | − K (1 + | x | p − ) | x | ≤ | x | p − ) | x | − K (1 + | x | p − ) | x | ≤ ≤ | x | + 1and || σ ( x ) || = | x | (1 + | x | ) + (1 + | x | p − ) | x | ≤ | x | p ) . (4.3)On the other hand, it is clear that σ ( x ) x = (cid:18) x | x | x | x | − (1 + | x | p − ) x (1 + | x | p − ) x (cid:19) (cid:18) x x (cid:19) = | x | | x | ! . So | σ ( x ) x | ≤ | x | ≤ | x | . (4.4)Thus conditions (1.6), (1.7) and (1.9) hold in this case. We now show that condition (1.5)holds also. We have indeed || σ ( x ) − σ ( y ) || = X i =1 ( x i | x | − y i | y | ) + X i =1 [(1 + | x | p − ) x i − (1 + | y | p − ) y i ] ≤ | x − y | + 2 | x − y | + 2( | x | p + | y | p − | x || y | ) p − h x, y i )and h x − y, b ( x ) − b ( y ) i = K (2 + | x | p − + | y | p − ) h x, y i − K ( | x | + | y | + | x | p + | y | p ) ≤ K ( | x | p − + | y | p − ) | x || y | − K ( | x | p + | y | p ) . Note that( | x | p − + | y | p − ) | x || y | − ( | x | p + | y | p ) = − ( | x | − | y | )( | x | p − − | y | p − ) ≤ | x | p + | y | p − | x || y | ) p − h x, y i ) − (( | x | p + | y | p ) − ( | x | p − + | y | p − ) h x, y i ) ≤ | x || y | ( | x | p − − | y | p − ) . We have then || σ ( x ) − σ ( y ) || + h x − y, b ( x ) − b ( y ) ≤ | x − y | − ( K − | x | − | y | )( | x | p − − | y | p − )+ 2 | x || y | ( | x | p − − | y | p − ) . On the other hand, if | x | > | y | , then | x || y | ( | x | p − − | y | p − ) − ( | x | − | y | )( | x | p − − | y | p − ) ≤ | x || y | ( | x | p − − | y | p − ) − | y | ( | x | − | y | )( | x | p − − | y | p − )= | y | ( | x | p − − | y | p − )[ | x | ( | x | p − − | y | p − ) − ( | x | − | y | )( | x | p − + | y | p − )]= | y | ( | x | p − − | y | p − )[ | y | p − ( | y | − | x | ) + | x || y | ( | x | p − − | y | p − )] ≤ < p < x and y , we know that | x || y | ( | x | p − − | y | p − ) − ( | x | − | y | )( | x | p − − | y | p − ) ≤ , ∀ x, y ∈ R . Hence || σ ( x ) − σ ( y ) || + h x − y, b ( x ) − b ( y ) ≤ | x − y | − ( K − | x | − | y | )( | x | p − − | y | p − ) ≤ | x − y | . (4.5)We have thus verified the condition (1.5).Since σ ≥ I, the condition (1.8) also holds. By Theorem 1.3, the transition probability p s,t is irreducible. However, it is clear to see that there is no K > || σ ( x ) || ≤ K (1 + | x | )and | b ( x ) | ≤ K ( | x | + 1)indicating that neither ( H ) of Theorem 1.1 in [8] nor the condition ( H ) of Theorem 1.3 in[5] are fulfilled by our example. Let us fix x , y ∈ R d with x = y . For 0 < ε < | x − y | , we defineˆ τ ε := inf { t > , | X t ( x ) − X t ( y ) | ≤ ε } , ˆ τ := inf { t > , X t ( x ) = X t ( y ) } . (5.1)It’s obvious that ˆ τ ε → ˆ τ , almost surely as ε → . Next, we denote τ := inf { t > , | X t ( x ) − X t ( y ) | ≥ | x − y |} . (5.2)Set the function ϕ δ ( x ) := exp Z c x dsγ R ( s ) + δ . Then ϕ ′ δ ( x ) = − ϕ δ ( x ) γ R ( x ) + δ ≤ , ϕ ′′ δ ( x ) = ϕ δ ( x )(1 + γ ′ R ( x ))( γ R ( x ) + δ ) . By Itˆo’s formula, we have 14 δ ( ξ t ∧ τ ∧ τ R ) = ϕ δ ( | x − y | ) + M t + Z t ∧ τ ∧ τ R ϕ ′ δ ( ξ s )[2 h η s , h s i + || e s || + Z U | k ( s, u ) | ν ( du )] ds + 2 Z t ∧ τ ∧ τ R ϕ ′′ δ ( ξ s ) | e Ts η s | ds + Z t ∧ τ ∧ τ R Z U [ ϕ δ ( | η s + k ( s − , u ) | ) − ϕ δ ( ξ s )] ν ( du ) ds + Z t ∧ τ ∧ τ R Z U h ϕ δ ( | η s + k ( s, u ) | ) − ϕ δ ( ξ s ) − ϕ ′ δ ( ξ s )( | k ( s, u ) | + 2 h η s , k ( s, u )) i ν ( du ) ds, where τ R = inf { t > , | X t ( x ) | ∨ | X t ( y ) | > R } . By the definition of ϕ δ and condition (1.10), ϕ δ ( ξ t ∧ τ ∧ τ R ) ≤ ϕ δ ( | x − y | ) + M t + Z t ∧ τ ∧ τ R ϕ δ ( ξ s ) γ R ( ξ s ) + δ h γ ′ R ( ξ s )) γ R ( ξ s ) + δ | e Ts η s | − h η s , h s i − || e s || + Z U ( | k ( s, u ) | − | k ( s, u ) | + 2 h η s , k ( s, u )) ν ( du ) i ds ≤ ϕ δ ( | x − y | ) + (2 K − Z t ∧ τ ∧ τ R ϕ δ ( ξ s ) γ R ( ξ s ) + δ h || e s || − K − h η s , h s i + 12 K − Z U ( | k ( s, u ) | − | k ( s, u ) | + 2 h η s , k ( s, u )) ν ( du ) i ds + M t ≤ ϕ δ ( | x − y | ) + M t + (2 K − Z t ϕ δ ( ξ s ) ds, where M t = 2 Z t ∧ τ ∧ τ R h η s , e s dB s i + X i =1 Z t ∧ τ ∧ τ R +0 Z U (cid:16) | k i ( s − , u ) | + 2 h η s − , k i ( s − , u ) (cid:17) ˜ N k ( ds, du )is a real martingale. Take expectation on both sides. By Gronwall’s lemma, we have E ( ϕ δ ( ξ t ∧ τ ∧ ˆ τ ε ∧ τ R )) ≤ ϕ δ ( | x − y | ) e (2 K − t . On the other hand, E (Φ δ ( | X t ∧ ˆ τ ε ∧ τ ∧ τ R ( x ) − X t ∧ ˆ τ ε ∧ τ ∧ τ R ( y ) | )) ≥ E (Φ δ ( | X t ∧ ˆ τ ε ∧ τ ∧ τ R ( x ) − X t ∧ ˆ τ ε ∧ τ ∧ τ R ( y ) | )1 ˆ τ ε ≤ t ∧ τ ∧ τ R )= Φ δ ( ε ) P (ˆ τ ε ≤ t ∧ τ ∧ τ R ) . Thus, P (ˆ τ ε ≤ t ∧ τ ∧ τ R ) ≤ C t exp ( − Z ξ ε dsγ ( s ) + δ ) , (5.3)where the constant C t is independent of R. Let R → ∞ , δ → , ε → t,P (ˆ τ ≤ t ∧ τ ∧ ζ x ∧ ζ y ) = 0 . t → ∞ , it follows that P (ˆ τ ≤ τ ∧ ζ x ∧ ζ y ) = 0. Therefore, ξ · is positive almost surelyon the interval [0 , τ ∧ ζ x ∧ ζ y ] . Now we define T := 0 , T := τ ∧ ζ x ∧ ζ y ,T := inf { t > T , | X t ( x ) − X t ( y ) | ≤ | x − y |} ∧ ζ x ∧ ζ y , (5.4)and generally T n := inf { t > T n − , | X t ( x ) − X t ( y ) | ≤ | x − y |} ∧ ζ x ∧ ζ y ,T n +1 := inf { t > T n , | X t ( x ) − X t ( y ) | ≥ | x − y |} ∧ ζ x ∧ ζ y . (5.5)Due to Fang and Zhang [2], it is obvious that T n → ζ x ∧ ζ y , a.s. as n → ∞ . Thus, ξ · . T n − , T n ] . By Theorem 1.1, X t ( x ) is stochastic continuous withrespect to the initial value x , thus the solution process X t ( x ) is a Feller process. Furthermore, { X t } t ≥ has the strong Markovian property since the process is right continuous withleft limit. Starting from T n and applying the same arguments as in the first part of theproof, ξ · is also positive almost surely on the interval [ T n , T n +1 ]. We complete the proof. (cid:3) Acknowledgement
The authors would like to thank Professor Feng-Yu Wang for usefuldiscussions.