Path independence of the additive functionals for stochastic differential equations driven by G-Lévy processes
aa r X i v : . [ m a t h . P R ] M a r PATH INDEPENDENCE OF THE ADDITIVE FUNCTIONALS FORSTOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY G-L ´EVYPROCESSES*
HUIJIE QIAO , AND JIANG-LUN WU
1. School of Mathematics, Southeast UniversityNanjing, Jiangsu 211189, China2. Department of Mathematics, University of Illinois at Urbana-ChampaignUrbana, IL 61801, [email protected]. Department of Mathematics, Computational Foundry, Swansea UniversityBay Campus, Swansea SA1 8EN, [email protected]
Abstract.
In the paper, we consider a type of stochastic differential equations drivenby G-L´evy processes. We prove that a kind of their additive functionals has path inde-pendence and extend some known results. Introduction
Recently, development of mathematical finance forces the appearance of a type ofprocesses-G-Brownian motions([5]). And then the related theory, such as stochastic calcu-lus and stochastic differential equations (SDEs in short) driven by G-Brownian motions,are widely studied([1, 5, 6, 8]). However, in some financial models, volatility uncertaintymakes G-Brownian motions insufficient for simulating these models. One important rea-son lies in the continuity of their paths with respect to the time variable. So, Hu-Peng[2] solved the problem by introducing G-L´evy processes. And the type of processes hasdiscontinuous (right continuous with left limits) paths. Later, Paczka [3] defined the Itˆo-L´evy stochastic integrals, deduced the Itˆo formula, established SDEs driven by G-L´evyprocesses and stated the existence and uniqueness of solutions for these equations underlipschitz conditions. Most recently, under non-lipschitz conditions, Wang-Gao [14] provedwell-definedness of SDEs driven by G-L´evy processes and investigated exponential stabil-ity of their solutions. Here, we follows up the line in [14], define the additive functionalsof SDEs driven by G-L´evy processes and study their path independence.Concretely speaking, we consider the following SDEs on R d :d Y t = b ( t, Y t )d t + h ij ( t, Y t )d h B i , B j i t + σ ( t, Y t )d B t + Z R d \{ } f ( t, Y t , u ) L (d t, d u ) , (1) AMS Subject Classification(2010):
Keywords:
Path independence, additive functionals, G-L´evy processes, stochastic differential equationsdriven by G-L´evy processes.*This work was partly supported by NSF of China (No. 11001051, 11371352, 11671083) and ChinaScholarship Council under Grant No. 201906095034. here B is a G-Brownian motion, h B i , B j i t is the mutual variation process of B i and B j for i, j = 1 , , · · · , d and L (d t, d u ) is a G-random measure (See Subsection 2.4). Thecoefficients b : [0 , T ] × R d R d , h ij = h ji : [0 , T ] × R d R d , σ : [0 , T ] × R d R d × d and f : [0 , T ] × R d × ( R d \ { } ) R d are Borel measurable. Here and hereafter weuse the convention that the repeated indices stand for the summation. Thus, under( H b,h,σ,f )-( H b,h,σ,f ) in Subsection 2.5, by [14, Theorem 3.1], we know that Eq.(1) has aunique solution Y t . And then we introduce the additive functionals of Y t and define pathindependence of these functionals. Finally, we prove that these functionals have pathindependence under some assumption.Next, we say our motivations. First, we mention that Ren-Yang [12] proved pathindependence of additive functionals for SDEs driven by G-Brownian motions. Sincethese equations can not satisfy the actual demand very well, to extend them becomesone of our motivations. Second, by analyzing some special cases, we surprisingly findthat, we can express explicitly these additive functionals. However, in our known results([9, 10, 11]), it is difficult to express explicitly additive functionals. Therefore, this is theother of our motivations.This paper is arranged as follows. In Section 2, we introduce G-L´evy processes, theItˆo-L´evy stochastic integrals, SDEs driven by G-L´evy processes, additive functionals, pathindependence and some related results. The main results and their proofs are placed inSection 3. Moreover, we analysis some special cases and compare our result with someknown results ([9, 10, 11, 12]) in Subsection 3.3.2. Preliminary
In the section, we introduce some concepts and results used in the sequel.2.1.
Notation.
In the subsection, we introduce notations used in the sequel.For convenience, we shall use | · | and k · k for norms of vectors and matrices, re-spectively. Furthermore, let h· , ·i denote the scalar product in R d . Let Q ∗ denote thetranspose of the matrix Q .Let lip ( R n ) be the set of all Lipschitz continuous functions on R n and C b,lip ( R d ) bethe collection of all bounded and Lipschitz continuous functions on R d . Let C b ( R d ) bethe space of bounded and three times continuously differentiable functions with boundedderivatives of all orders less than or equal to 3.2.2. G-L´evy processes.
In the subsection, we introduce G-L´evy processes.(c.f.[2])Let Ω be a given set and H be a linear space of real functions defined on Ω such thatif X , . . . , X n ∈ H , then φ ( X , . . . , X n ) ∈ H for each φ ∈ lip ( R n ). If X ∈ H , we call X asa random variable. Definition 2.1.
If a functional ¯ E : H 7→ R satisfies: for X, Y ∈ H ,(i) X > Y, ¯ E [ X ] > ¯ E [ Y ] ,(ii) ¯ E [ X + Y ] ¯ E [ X ] + ¯ E [ Y ] ,(iii) for all λ > , ¯ E [ λX ] = λ ¯ E [ X ] ,(iv) for all c ∈ R , ¯ E [ X + c ] = ¯ E [ X ] + c ,we call ¯ E as a sublinear expectation on H and (Ω , H , ¯ E ) as a sublinear expectation space. ext, we define the distribution of a random vector on (Ω , H , ¯ E ). For a n -dimensionalrandom vector X = ( X , X , · · · , X n ) for X i ∈ H , i = 1 , , · · · , n , set F X ( φ ) := ¯ E ( φ ( X )) , φ ∈ lip ( R n ) , and then we call F X as the distribution of X . Definition 2.2.
Assume that X , X are two n -dimensional random vectors defined ondifferent sublinear expectation spaces. If for all φ ∈ lip ( R n ) , F X ( φ ) = F X ( φ ) , we say that the distributions of X , X are the same. Definition 2.3.
For two random vectors Y = ( Y , Y , · · · , Y m ) for Y j ∈ H and X =( X , X , · · · , X n ) for X i ∈ H , if for all φ ∈ lip ( R n × R m ) , ¯ E [ φ ( X, Y )] = ¯ E [ ¯ E [ φ ( x, Y )] x = X ] , we say that Y is independent from X . Here, we use two above concepts to define L´evy processes on (Ω , H , ¯ E ). Definition 2.4.
Let X = ( X t ) t > be a d -dimensional c`adl`ag process on (Ω , H , ¯ E ) . If X satisfies(i) X = 0 ;(ii) for t, s > , the increment X s + t − X t is independent from ( X t , X t , · · · , X t n ) , forany n and t < t · · · < t n t ;(iii) the distribution of X s + t − X t does not depend on t ;we call X as a L´evy process. Definition 2.5.
Assume that X is a d -dimensional L´evy process. If there exists a de-composition X t = X ct + X dt for t > , where ( X ct , X dt ) is a d -dimensional L´evy processsatisfying lim t ↓ ¯ E | X ct | t = 0 , ¯ E | X dt | Ct, t > , C > , we call X as a G-L´evy process. In the following, we characterize G-L´evy processes by partial differential equations.
Theorem 2.6.
Assume that X is a d -dimensional G-L´evy process. Then for g ∈ C b ( R d ) with g (0) = 0 , set G X [ g ( · )] := lim t ↓ ¯ E [ g ( X t )] t , and then G X has the following L´evy-Khintchine representation G X [ g ( · )] = sup ( ν,ζ,Q ) ∈U (cid:26)Z R d \{ } g ( u ) ν (d u ) + h ∂ x g (0) , ζ i + 12 tr [ ∂ x g (0) QQ ∗ ] (cid:27) , (2) where U is a subset of M ( R d \ { } ) × R d × R d × d , M ( R d \ { } ) is the collection of allmeasures on ( R d \ { } , B ( R d \ { } )) , R d × d is the set of all d × d matrices and U satisfies sup ( ν,ζ,Q ) ∈U (cid:26)Z R d \{ } | u | ν (d u ) + | ζ | + 12 tr [ QQ ∗ ] (cid:27) < ∞ . (3) heorem 2.7. Suppose that X is a d -dimensional G-L´evy process. Then for φ ∈ C b,lip ( R d ) , v ( t, x ) := ¯ E [ φ ( x + X t )] is the unique viscosity solution of the following partial integro-differential equation: ∂ t v ( t, x ) − G X [ v ( t, x + · ) − v ( t, x )]= ∂ t v ( t, x ) − sup ( ν,ζ,Q ) ∈U (cid:26) Z R d \{ } [ v ( t, x + u ) − v ( t, x )] ν (d u ) + h ∂ x v ( t, x ) , ζ i + 12 tr [ ∂ x v ( t, x ) QQ ∗ ] (cid:27) with the initial condition v (0 , x ) = φ ( x ) . Conversely, if we have a set U satisfying (3), is there a d -dimensional G-L´evy processhaving the L´evy-Khintchine representation (2) with the same set U ? The answer isaffirmed. We take Ω := D ( R + , R d ), where D ( R + , R d ) is the space of all c`adl`ag functions R + ∋ t ω t ∈ R d with ω = 0, equipped with the Skorokhod topology. Theorem 2.8.
Suppose that U satisfies (3). Then there exists a sublinear expectation ¯ E on Ω such that the canonical process X is a d -dimensional G-L´evy process having theL´evy-Khintchine representation (2) with the same set U . A capacity.
In the subsection, we introduce a capacity and related definitions.First of all, fix a set U satisfying (3) and T > T := D ([0 , T ] , R d ) and thesublinear expectation ¯ E in Theorem 2.8. Thus, we know that (Ω T , H , ¯ E ) is a sublinearexpectation space. Here, we work on the space. Let L pG (Ω T ) be the completion of lip (Ω T )under the norm k · k p := ( ¯ E | · | p ) /p , p > V := { ν ∈ M ( R d \ { } ) : ∃ ( ζ , Q ) ∈ R d × R d × d such that ( ν, ζ , Q ) ∈ U } and let G be the set of all the Borel measurable functions g : R d R d with g (0) = 0. Assumption: (i) There exists a measure µ ∈ M ( R d ) such that Z R d \{ } | z | µ (d z ) < ∞ , µ ( { } ) = 0and for all ν ∈ V there exists a function g ν ∈ G satisfying ν ( A ) = µ ( g − ν ( A )) , ∀ A ∈ B ( R d \ { } ) . (ii) There exists 0 < q < ν ∈V Z < | z | < | z | q ν (d z ) < ∞ . (iii) sup ν ∈V ν ( R d \ { } ) < ∞ . Let ( ˜Ω , F , P ) be a probability space supporting a Brownian motion W and a Poissonrandom measure N (d t, d z ) with the intensity measure µ (d z )d t . Let F t := σ (cid:8) W s , N ((0 , s ] , A ) : 0 s t, A ∈ B ( R d \ { } ) (cid:9) ∨N , N := { U ∈ F , P ( U ) = 0 } . We introduce the following set. efinition 2.9. A U ,T is a set of all the processes θ t = ( θ dt , θ ,ct , θ ,ct ) for t ∈ [0 , T ] satisfying(i) ( θ ,ct , θ ,ct ) is a F t -adapted process and θ d is a F t -predictable random field on [0 , T ] × R d ,(ii) For P -a.s. ω and a.e. t ∈ [0 , T ] , ( θ d ( t, · )( ω ) , θ ,ct ( ω ) , θ ,ct ( ω )) ∈ (cid:8) ( g ν , ζ , Q ) ∈ G × R d × R d × d : ( ν, ζ , Q ) ∈ U (cid:9) , (iii) E P (cid:20)Z T (cid:16) | θ ,ct | + k θ ,ct k + Z R d \{ } | θ d ( t, z ) | µ (d z ) (cid:17) d t (cid:21) < ∞ . For θ ∈ A U ,T , set B ,θt := Z t θ ,cs d s + Z t θ ,cs d W s + Z t Z R d \{ } θ d ( s, z ) N (d s, d z ) , t ∈ [0 , T ] , and by Corollary 14 in [3], it holds that for ξ ∈ L G (Ω T )¯ E [ ξ ] = sup θ ∈A U ,T E P θ [ ξ ] , P θ = P ◦ ( B ,θ · ) − . And then define ¯ C ( D ) := sup θ ∈A U ,T P θ ( D ) , D ∈ B (Ω T ) , and ¯ C is a capacity. For D ∈ B (Ω T ), if ¯ C ( D ) = 0, we call D as a polar set. So, if aproperty holds outside a polar set, we say that the property holds quasi-surely (q.s. inshort).2.4. The Itˆo integrals with respect to G-L´evy processes.
In the subsection, weintroduce the Itˆo integrals with respect to G-L´evy processes under the framework of theabove subsection.Let X denote the canonical process on the space, i.e. X t ( ω ) = ω t , t ∈ [0 , T ]. So, X isa d -dimensional G-L´evy process. Although the Itˆo integrals with respect to G-Brownianmotions have been introduced in [8], we need to introduce two related spaces used in thesequel. Take 0 = t < t < · · · < t N = T . Let p > M p, G (0 , T ) := n η t ( ω ) = N X j =1 ξ j − ( ω )1 [ t j − ,t j ) ( t ); ξ j − ( ω ) ∈ L pG (Ω t j − ) o . Let M pG (0 , T ) and H pG (0 , T ) denote the completion of M p, G (0 , T ) under the norm k η k M pG (0 ,T ) = (cid:18)Z T ¯ E | η t | p d t (cid:19) p and k η k H pG (0 ,T ) = ¯ E (cid:18)Z T | η t | d t (cid:19) p ! p , respectively. Let M pG ([0 , T ] , R d ) and H pG ([0 , T ] , R d ) be the collection of all the processes η t = ( η t , η t , · · · , η dt ) , t ∈ [0 , T ] , η i ∈ M pG (0 , T ) and H pG (0 , T ) , respectively. ext, we introduce the Itˆo integrals with respect to random measures. First, define arandom measure: for any 0 t T and A ∈ B ( R d \ { } ), κ t := X t − X t − , L ((0 , t ] , A ) := X
For any f ∈ H SG ([0 , T ] × ( R d \ { } )) , set Z t Z R d \{ } f ( s, u ) L (d s, d u ) := X
In the sub-section, we introduce stochastic differential equations driven by G-L´evy processes andrelated additive functionals.First, we introduce some notations. Let S d be the space of all d × d symmetric matrices.For A ∈ S d , set G ( A ) := 12 sup Q ∈Q tr[ QQ ∗ A ] , where Q is a nonempty, bounded, closed and convex subset of R d × d . And then G : S d R is a monotonic, sublinear and positive homogeneous functional([6]). We choose U ⊂ M ( R d \{ } ) ×{ }×Q satisfying (3) and still work under the framework of Subsection2.3. So, the canonical process X t can be represented as X t = B t + X dt , where B t is a G-Brownian motion associated with Q and X dt is a pure jump G-L´evy process associatedwith M ( R d \ { } ).Next, we consider Eq.(1) and assume: H b,h,σ,f ) There exists a constant C > t ∈ [0 , T ] and x, y ∈ R d , | b ( t, x ) − b ( t, y ) | + | h ij ( t, x ) − h ij ( t, y ) | + k σ ( t, x ) − σ ( t, y ) k + sup ν ∈V Z R d \{ } | f ( t, x, u ) − f ( t, y, u ) | ν (d u ) C ρ ( | x − y | ) , where ρ : (0 , + ∞ ) (0 , + ∞ ) is a continuous, increasing and concave functionso that ρ (0+) = 0 , Z drρ ( r ) = + ∞ . ( H b,h,σ,f ) There exists a constant C > t ∈ [0 , T ] | b ( t, | + | h ij ( t, | + k σ ( t, k + sup ν ∈V Z R d \{ } | f ( t, , u ) | ν (d u ) C . By [14, Theorem 3.1], we know that under ( H b,h,σ,f )-( H b,h,σ,f ), Eq.(1) has a uniquesolution Y t with ¯ E " sup t ∈ [0 ,T ] | Y t | < ∞ . (5)And then we introduce the following additive functional F s,t := α Z ts G ( g )( r, Y r )d r + β Z ts g ij ( r, Y r )d h B i , B j i r + Z ts h g ( r, Y r ) , d B r i + Z ts Z R d \{ } g ( r, Y r , u ) L (d r, d u ) + Z ts sup ν ∈V Z R d \{ } γg ( r, Y r , u ) ν (d u )d r, s < t T, (6)where α, β, γ ∈ R are three constants and g : [0 , T ] × R d R d × d , g ij = g ji ,g : [0 , T ] × R d R d ,g : [0 , T ] × R d × ( R d \ { } ) R , are Borel measurable so that F s,t is well-defined. Definition 2.11.
The additive functional F s,t is called path independent, if there exists afunction V : [0 , T ] × R d R , such that for any s ∈ [0 , T ] and Y s ∈ L G (Ω T ) , the solution ( Y t ) t ∈ [ s,T ] of Eq.(1) satisfies F s,t = V ( t, Y t ) − V ( s, Y s ) . (7)3. Main results and their proofs
In the section, we state and prove the main results under the framework of Subsection2.5. And then we analysis some special cases and compare our result with some knownresults. .1. Main results.
In the subsection, we state and prove the main results. Let us beginwith a key lemma.
Lemma 3.1.
Assume that Q is bounded away from and Z t is a -dimensional G-Itˆo-L´evy process, i.e. Z t = Z t Γ s d s + Z t Φ ij ( s )d h B i , B j i s + Z t h Ψ s , d B s i + Z t Z R d \{ } K ( s, u ) L (d s, d u ) , (8) where Γ ∈ M G (0 , T ) , Φ ij ∈ M G (0 , T ) , Φ ij = Φ ji , i, j = 1 , , · · · , d, Ψ ∈ H G ([0 , T ] , R d ) , K ∈H G ([0 , T ] × ( R d \ { } )) . Then Z t = 0 for all t ∈ [0 , T ] q.s. if and only if Γ t = 0 , Φ ij ( t ) =0 , Ψ t = 0 a.e. × q.s. on [0 , T ] × Ω T and K ( t, u ) = 0 a.e. × a.e. × q.s. on [0 , T ] × ( R d \{ } ) × Ω T .Proof. Sufficiency is direct if one inserts Γ t = 0 , Φ ij ( t ) = 0 , Ψ t = 0 , K ( t, u ) = 0 into (8).Let us prove necessity. If Z t = 0 for any t ∈ [0 , T ], we get that0 = Z t Γ s d s + Z t Φ ij ( s )d h B i , B j i s + Z t Ψ ∗ s d B s + Z t Z R d \{ } K ( s, u ) L (d s, d u ) . (9)By taking the quadratic process with R t h Ψ s , d B s i on two sides of (9), it holds that0 = h Z · Ψ ∗ s d B s , Z · Ψ ∗ s d B s i t = Z t Ψ is Ψ js d h B i , B j i s = Z t tr(Ψ s Ψ ∗ s d h B i s )= Z t tr(d h B i s Ψ s Ψ ∗ s ) = Z t h d h B i s Ψ s , Ψ s i , where h B i := h B , B i h B , B i · · · h B , B d i ... ... ... h B d , B i h B d , B i · · · h B d , B d i . Note that Q is bounded away from 0. Thus, there exists a constant ι > h B i s > ιsI d and then 0 = Z t h d h B i s Ψ s , Ψ s i > ι Z t h Ψ s , Ψ s i d s. From this, we know that Ψ t = 0 a.e. × q.s.. So, (9) becomes0 = Z t Γ s d s + Z t Φ ij ( s )d h B i , B j i s + Z t Z R d \{ } K ( s, u ) L (d s, d u ) . Next, set τ = 0 , τ n := inf { t > τ n − : κ t = 0 } , n = 1 , , · · · , and { τ n } is a stopping time sequence with respect to ( B t ) t > and τ n ↑ ∞ as n → ∞ q.s.(c.f. [3, Proposition 16]). So, by (4) it holds that for t ∈ [0 , τ ∧ T ),0 = Z τ ∧ T Γ s d s + Z τ ∧ T Φ ij ( s )d h B i , B j i s , i.e. − Z τ ∧ T Γ s d s = Z τ ∧ T Φ ij ( s )d h B i , B j i s . hus, by the similar deduction to that in [12, Corollary 1] one can have that¯ E Z τ ∧ T (tr[Φ s Φ s ]) / d s = ¯ E Z τ ∧ T | Γ s | d s = 0 . Based on this, we know that Φ t = 0 , Γ t = 0 for t ∈ [0 , τ ∧ T ). If τ > T , the proof is over;if τ < T , we continue. For t = τ , (9) goes toΦ t = 0 , Γ t = 0 , K ( t, κ t ) = 0 . For t ∈ [ τ , τ ∧ T ), by the same means to the above for t ∈ [0 , τ ∧ T ), we get thatΦ t = 0 , Γ t = 0 for t ∈ [ τ , τ ∧ T ). If τ > T , the proof is over; if τ < T , we continuetill T τ n . Thus, we obtain that Γ t = 0 , Φ ij ( t ) = 0 , Ψ t = 0 a.e. × q.s. on [0 , T ] × Ω T and K ( t, u ) = 0 a.e. × a.e. × q.s. on [0 , T ] × ( R d \ { } ) × Ω T . The proof is complete. (cid:3) The main result in the section is the following theorem.
Theorem 3.2.
Assume that Q is bounded away from and b, h, σ, f satisfy ( H b,h,σ,f )-( H b,h,σ,f ). Then for V ∈ C , b ([0 , T ] × R d ) , F s,t is path independent in the sense of (7) ifand only if ( V, g , g , g ) satisfies the partial integral-differential equation ∂ t V ( t, x ) + h ∂ x V ( t, x ) , b ( t, x ) i = αG ( g )( t, x ) + sup ν ∈V R R d \{ } γg ( t, x, u ) ν (d u ) , h ∂ x V ( t, x ) , h ij ( t, x ) i + h ∂ x V ( t, x ) σ i ( t, x ) , σ j ( t, x ) i = βg ij ( t, x ) , ( σ T ∂ x V )( t, x ) = g ( t, x ) ,V (cid:16) t, x + f ( t, x, u ) (cid:17) − V ( t, x ) = g ( t, x, u ) ,t ∈ [0 , T ] , x ∈ R d , u ∈ R d \ { } . (10) Proof.
First, we prove necessity. On one hand, since F s,t is path independent in the senseof (7), by Definition 2.11 it holds that V ( t, Y t ) − V ( s, Y s ) = α Z ts G ( g )( r, Y r )d r + β Z ts g ij ( r, Y r )d h B i , B j i r + Z ts h g ( r, Y r ) , d B r i + Z ts Z R d \{ } g ( r, Y r , u ) L (d r, d u ) + Z ts sup ν ∈V Z R d \{ } γg ( r, Y r , u ) ν (d u )d r. (11)On the other hand, applying the Itˆo formula for G-Itˆo-L´evy processes ([3, Theorem 32])to V ( t, Y t ), one can obtain that V ( t, Y t ) − V ( s, Y s ) = Z ts ∂ r V ( r, Y r )d r + Z ts ∂ k V ( r, Y r ) b k ( r, Y r )d r + Z ts ∂ k V ( r, Y r ) h kij ( r, Y r )d h B i , B j i r + Z ts h ( σ T ∂ x V )( r, Y r ) , d B r i + Z ts Z R d \{ } (cid:16) V ( r, Y r + f ( r, Y r , u )) − V ( r, Y r ) (cid:17) L (d r, d u )+ 12 Z ts ∂ kl V ( r, Y r ) σ ki ( r, Y r ) σ lj ( r, Y r )d h B i , B j i r . (12)By (5) ( H b,h,σ,f )-( H b,h,σ,f ), one can verify that ∂ r V ( r, Y r ) + ∂ k V ( r, Y r ) b k ( r, Y r ) ∈ M G (0 , T ) , k V ( r, Y r ) h kij ( r, Y r ) + 12 ∂ kl V ( r, Y r ) σ ki ( r, Y r ) σ lj ( r, Y r ) ∈ M G (0 , T ) , ( σ T ∂ x V )( r, Y r ) ∈ H G ([0 , T ] , R d ) ,V ( r, Y r + f ( r, Y r , u )) − V ( r, Y r ) ∈ H G ([0 , T ] × ( R d \ { } )) . Thus, by (11) (12) and Lemma 3.1 we know that ∂ r V ( r, Y r ) + h ∂ x V ( r, Y r ) , b ( r, Y r ) i = αG ( g )( r, Y r ) + sup ν ∈V R R d \{ } γg ( r, Y r , u ) ν (d u ) , h ∂ x V ( r, Y r ) , h ij ( r, Y r ) i + h ∂ x V ( r, Y r ) σ i ( r, Y r ) , σ j ( r, Y r ) i = βg ij ( r, Y r ) , ( σ T ∂ x V )( r, Y r ) = g ( r, Y r ) ,V (cid:16) r, Y r + f ( r, Y r , u ) (cid:17) − V ( r, Y r ) = g ( r, Y r , u ) , a.e. × q.s.. Now, we insert r = s, Y s = x ∈ R d into the above equalities and get that ∂ s V ( s, x ) + h ∂ x V ( s, x ) , b ( s, x ) i = αG ( g )( s, x ) + sup ν ∈V R R d \{ } γg ( s, x, u ) ν (d u ) , h ∂ x V ( s, x ) , h ij ( s, x ) i + h ∂ x V ( s, x ) σ i ( s, x ) , σ j ( s, x ) i = βg ij ( s, x ) , ( σ T ∂ x V )( s, x ) = g ( s, x ) ,V (cid:16) s, x + f ( s, x, u ) (cid:17) − V ( t, x ) = g ( s, x, u ) ,s ∈ [0 , T ] , x ∈ R d , u ∈ R d \ { } . Since s, x are arbitrary, we have (10).Next, we treat sufficiency. By the Itˆo formula for G-Itˆo-L´evy processes to V ( t, Y t ),we have (12). And then one can apply (10) to (12) to get (11). That is, F s,t is pathindependent in the sense of (7). The proof is complete. (cid:3) Some special cases.
In the subsection, we analysis some special cases.If b ( t, x ) = 0 , h ij ( t, x ) = 0 , σ ( t, x ) = I d , f ( t, x, u ) = u , Eq.(1) becomesd Y t = d B t + Z R d \{ } uL (d t, d u ) = d X t . We take α = 1 , β = , γ = 1. Thus, by Theorem 3.2, it holds that F s,t is path independentin the sense of (7) if and only if ( V, g , g , g ) satisfies the partial integral-differentialequation ∂ t V ( t, x ) = G ( ∂ x V )( t, x ) + sup ν ∈V R R d \{ } ( V ( t, x + u ) − V ( t, x )) ν (d u ) ,∂ x V ( t, x ) = g ( t, x ) ,∂ x V ( t, x ) = g ( t, x ) ,V ( t, x + u ) − V ( t, x ) = g ( t, x, u ) ,t ∈ [0 , T ] , x ∈ R d , u ∈ R d \ { } . Besides, by Theorem 2.7, it holds that for φ ∈ C b,lip ( R d ), V ( t, x ) = ¯ E [ φ ( x + X t )] is theunique viscosity solution of the following partial integro-differential equation: ∂ t V ( t, x ) − G ( ∂ x V )( t, x ) − sup ν ∈V Z R d \{ } (cid:16) V ( t, x + u ) − V ( t, x ) (cid:17) ν (d u ) = 0with the initial condition V (0 , x ) = φ ( x ). So, g ( t, x ) = ∂ x ¯ E [ φ ( x + X t )] ,g ( t, x ) = ∂ x ¯ E [ φ ( x + X t )] , ( t, x, u ) = ¯ E [ φ ( x + u + X t )] − ¯ E [ φ ( x + X t )] . That is, we can find g , g , g .If d = 1 , α = 1 , β = 0 , γ = 0, it follows from Theorem 3.2 that F s,t is path independentin the sense of (7) if and only if ( V, g , g , g ) satisfies the partial integral-differentialequation ∂ t V ( t, x ) + ∂ x V ( t, x ) b ( t, x ) = G ( g )( t, x ) ,∂ x V ( t, x ) h ( t, x ) + ∂ x V ( t, x ) σ ( t, x ) = 0 , ( σ∂ x V )( t, x ) = g ( t, x ) ,V (cid:16) t, x + f ( t, x, u ) (cid:17) − V ( t, x ) = g ( t, x, u ) ,t ∈ [0 , T ] , x ∈ R , u ∈ R \ { } . And then if σ ( t, x ) = 0, the unique solution of the above second equation is V ( t, x ) = V ( t,
0) + ∂ x V ( t, Z x e − R z h ( t,v ) σ t,v ) d v d z. Besides, since Q is bounded away from 0, G is invertible. Thus, g ( t, x ) = G − (cid:16) ∂ t V ( t, x ) + b ( t, x ) ∂ x V ( t, e − R x h ( t,v ) σ t,v ) d v (cid:17) ,g ( t, x ) = σ ( t, x ) ∂ x V ( t, e − R x h ( t,v ) σ t,v ) d v ,g ( t, x, u ) = ∂ x V ( t, Z x + f ( t,x,u ) x e − R z h ( t,v ) σ t,v ) d v d z. That is, we also give out g , g , g in the case. Remark 3.3.
In the above special cases, we can describe concretely g , g , g . This isinteresting and also is one of our motivations. Comparison with some known results.
In the subsection, we compare our resultwith some known results.First, if we take f ( t, x, u ) = 0 in Eq.(1) and g ( t, x, u ) = 0 in (6), Theorem 3.2 becomes[12, Theorem 2]. Therefore, our result is more general.Second, we take M ( R d \ { } ) = { ν } , Q = { I d } in Subsection 2.5. Thus, B is a classicalBrownian motion with h B i , B j i t = I i = j t and L (d t, d u ) is a classical Poisson randommeasure. And then Eq.(1) goes intod Y t = b ( t, Y t )d t + d X i =1 h ii ( t, Y t )d t + σ ( t, Y t )d B t + Z R d \{ } f ( t, Y t , u ) L (d t, d u ) . (13)Note that ν ( R d \ { } ) < ∞ . So, by (5) ( H b,h,σ,f )-( H b,h,σ,f ), [4, Theorem 13] admits us toobtain that Z t Z R d \{ } f ( s, Y s , u ) ˜ L (d s, d u ) := Z t Z R d \{ } f ( s, Y s , u ) L (d s, d u ) − Z t Z R d \{ } f ( s, Y s , u ) ν (d u )d s is a B t -martingale, where B t := σ { ω s , s t } , t T . Therefore, we can rewriteEq.(13) to get thatd Y t = b ( t, Y t )d t + d X i =1 h ii ( t, Y t )d t + Z R d \{ } f ( t, Y t , u ) ν (d u )d t + σ ( t, Y t )d B t Z R d \{ } f ( t, Y t , u ) ˜ L (d t, d u ) . This is a classical stochastic differential equation with jumps. Thus, by [7, Theorem 1.2]it holds that under ( H b,h,σ,f )-( H b,h,σ,f ), the above equation has a unique solution.In the following, note that G ( g ) = tr( g ) = d P i =1 g ii . And then F s,t can be representedas F s,t = Z ts (cid:16) α β (cid:17) d X i =1 g ii ( r, Y r )d r + Z ts h g ( r, Y r ) , d B r i + Z ts Z R d \{ } g ( r, Y r , u ) ˜ L (d r, d u )+ Z ts Z R d \{ } (1 + γ ) g ( r, Y r , u ) ν (d u )d r. This is just right [11, (3)] without the distribution of Y r for r ∈ [ s, t ]. So, in the caseDefinition 2.11 and Theorem 3.2 are Definition 2.1 and Theorem 3.2 in [11] without thedistribution of Y r for r ∈ [ s, t ], respectively. Therefore, our result overlaps [11, Theorem3.2] in some sense. Acknowledgements:
The authors are very grateful to Professor Xicheng Zhang for valuable discussions. Thefirst author also thanks Professor Renming Song for providing her an excellent environ-ment to work in the University of Illinois at Urbana-Champaign.
References [1] F. Gao: Pathwise properties and hoemeomorphic flows for stochastic differential equations drivenby G-Brownian motion,
Stochastic Processes and their Applications , 119(2009)3356-3382.[2] M. Hu and S. Peng: G-L´evy processes under sublinear expectations. arXiv:0911.3533v1.[3] K. Paczka: It calculus and jump diffusions for G-L´evy processes. arXiv:1211.2973v3.[4] K. Paczka: G-martingale representation in the G-L´evy setting. arXiv:1404.2121v1.[5] S. Peng: Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation,
Stochastic Processes and their Applications , 118(2008)2223-2253.[6] S. Peng:
Nonlinear Expectations and Stochastic Calculus under Uncertainty with Robust CLT andG-Brownian Motion . Probability Theory and Stochastic Modelling 95, Springer, 2019.[7] H. Qiao: Euler-Maruyama approximation for SDEs with jumps and non-Lipschitz coefficients,
Osaka Journal of Mathematics , 51(2014)47-66.[8] H. Qiao: The cocycle property of stochastic differential equations driven by G-Brownian motion,
Chinese Annals of Mathematics, Series B , 36(2015)147-160.[9] H. Qiao and J.-L. Wu: Characterising the path-independence of the Girsanov transformation fornon-Lipschnitz SDEs with jumps,
Statistics and Probability Letters , 119(2016)326-333.[10] H. Qiao and J.-L. Wu: On the path-independence of the Girsanov transformation for stochasticevolution equations with jumps in Hilbert spaces,
Discrete and Continuous Dynamical Systems-B ,24(2019)1449-1467.[11] H. Qiao and J.-L. Wu: Path independence of the additive functionals for McKean-Vlasov stochasticdifferential equations with jumps, http://arxiv.org/abs/1911.03830.[12] P. Ren and F. Yang: Path independence of additive functionals for stochastic differential equationsunder G-framework,
Front. Math. China
14 (2019), no. 1, 135-148.[13] A. Truman, F.-Y. Wang, J.-L. Wu, W. Yang: A link of stochastic differential equations to nonlinearparabolic equations,
SCIENCE CHINA Mathematics , 55 (2012) 1971-1976.[14] B. Wang and H. Gao: Exponential stability of solutions to stochastic differential equations drivenby G-L´evy process, appear in
Applied Mathematics & Optimization.
15] J.-L. Wu and W. Yang: On stochastic differential equations and a generalised Burgers equation, pp425-435 in
Stochastic Analysis and Its Applications to Finance: Essays in Honor of Prof. Jia-AnYan (eds. T S Zhang, X Y Zhou) , Interdisciplinary Mathematical Sciences, Vol. 13, World Scientific,Singapore, 2012., Interdisciplinary Mathematical Sciences, Vol. 13, World Scientific,Singapore, 2012.