On some cadlaguity moment estimates of stochastic jump processes
aa r X i v : . [ m a t h . P R ] J u l ON SOME C `ADL `AGUITY MOMENT ESTIMATES OFSTOCHASTIC JUMP PROCESSES
R. MIKULEVI ˇCIUS AND FANHUI XU
Abstract.
Using X. Fernique’s results on the compactness of distribu-tions of c`adl`ag random functions, we derive some c`adl`aguity momentestimates for stochastic processes with jumps. Introduction
As suggested by Kolmogorov, it was proved in [2] (1956) that if X t , ≤ t ≤ , is a separable real valued process (see [5]) such that(1.1) E [ | X t − X t | p ∧ | X t − X t | p ] ≤ C | t − t | r with r > , p > , and C independent of t , then X has no discontinuities ofthe second kind with probability 1. If(1.2) E [ | X t − X t | p ] ≤ C | t − t | r is assumed instead of of (1.1), then X paths are H¨older continuous (Kol-mogorov, 1934). It can be shown (e.g. [8], [11]), that under (1.2), the H¨oldercontinuity is a consequence of the well-known Sobolev embedding theorem.In that case, X H¨older norm moment estimates can be derived. In this note,we estimate the moments of time supremum and c`adl`ag H¨older coefficientof X in terms of some integrated time differences of X from which, usingassumption (1.1), we can derive the classical claim about the existence of ac`adl`ag modification of X . On the other hand, the estimate obtained couldbe helpful in the construction of the solutions to SPDEs driven by jumpprocesses when the method of characteristics with a time reversal is used(see [4]). Some different type moment estimates were derived in [10] by im-posing assumptions on the cumulative distribution function of the quantitiesintroduced in [3] (see [7], Section 4 of Chapter III, as well).Our note is organized as follows. In Section 2, we introduce some notationand state the main claim. Some auxiliary results are presented in Section 3,and the main theorem is proved in Section 4. Date : January 2, 2019.1991
Mathematics Subject Classification.
Key words and phrases. path regularity of stochastic processes, embedding theorems. Notation and main result
Let E be a Polish space with distance d and D ([0 , , E ) be the standardspace of E -valued c`adl`ag functions on [0 , . For 0 ≤ s ≤ t ≤ u ≤
1, denote∆ ( f ; s, t, u ) = d ( f ( s ) , f ( t )) ∧ d ( f ( t ) , f ( u )) . For σ < τ , let us introduce thestandard modulus of c`adl`aguity∆ ( f ; ( σ, τ )) = sup σ ≤ s ≤ t ≤ u ≤ τ d ( f ( s ) , f ( t )) ∧ d ( f ( t ) , f ( u )) . For µ ∈ (0 , µ -H¨older c`adl`ag function space D µ ([0 , , E ) to bethe set of of all f ∈ D ([0 , , E ) such that[ f ] µ + | f ] µ + [ f | µ < ∞ , where[ f ] µ = sup ≤ s ≤ t ≤ u ≤ d ( f ( s ) , f ( t )) ∧ d ( f ( t ) , f ( u )) | u − s | µ = sup ≤ s ≤ t ≤ u ≤ ∆ ( f ; s, t, u ) | u − s | µ , | f ] µ = sup t ∈ (0 , d ( f (0) , f ( t )) t µ , [ f | µ = sup t ∈ [0 , d ( f (1) , f ( t )) | − t | µ . For µ ∈ (0 , , p > , f ∈ D ([0 , , E ) , let[[ f ]] µ,p = (cid:18)Z Z Z s Theorem 1. Let p > , µ ∈ (0 , . There is C > such that for any f ∈ D µ ([0 , , E )[ f ] µ + | f ] µ + [ f | µ ≤ C ([[ f ]] µ,p + || f ]] µ,p + [[ f || µ,p ) . Moreover, if E = R d , d ( x, y ) = | x − y | , x, y ∈ R d , then sup ≤ t ≤ | f ( t ) | ≤ C (cid:16) | f | L p ([0 , + [[ f ]] µ,p + || f ]] µ,p + [[ f || µ,p (cid:17) . Remark 1. Obviously, for any E -valued measurable function f on [0 , , [[ f ]] pµ,p ≤ Z Z Z s Let (Ω , F , P ) be a probability space and X : [0 , × Ω → E bea measurable function, p > , r > . Assume that E [∆ ( X ; s, t, u ) p ] ≤ C | u − s | r , ≤ s < t < u ≤ , E [ d ( X (1) , X ( t )) p ] ≤ C (1 − t ) r , ≤ t ≤ , E [ d ( X (0) , X ( t )) p ] ≤ C t r , ≤ t ≤ , ADLAGUITY ESTIMATES 3 for some C > . Then for each µ ∈ (0 , , µ < r/p, there is a constant N = N ( µ, r, p ) so that (2.1) E (cid:16) [[ X ]] pµ,p + || X ]] pµ,p + [[ X || pµ,p (cid:17) ≤ N C . If E = R k , d ( x, y ) = | x − y | , x, y ∈ R k , and X · ∈ D µ ([0 . , E ) a.s. with µ ∈ (0 , , µ < r/p, then, in addition, E (cid:20) sup ≤ t ≤ | X t | p (cid:21) ≤ N [ C + E Z | X t | p dt ] . Proof. Let r − µp > 0. Then E (cid:16) [[ X ]] pµ,p (cid:17) ≤ C Z Z s
Let X t , t ∈ [0 , , be a real valued and stochastically continuousprocess. Assume that E [ | X t − X s | p ∧ | X t − X u | p ] ≤ C | u − s | r , E [ | X − X t | p ] ≤ Ct r , E [ | X − X t | p ] ≤ C | − t | r for all ≤ s < t < u ≤ , and some r > , p > . Then X has a c`adl`agmodification.Proof. Let X nt = X π n ( t ) , t ∈ [0 , , where π n ( t ) = k/ n if k/ n ≤ s < ( k + 1) / n , n = 1 , , . . . , k = 0 , . . . , n − π n (1) = 1 − / n . Obviously the sequence X n · ∈ D β ([0 , β ∈ (0 , µ ∈ (0 , , µp < r . It is enough to show that(2.2) sup n E h [[ X n ]] pµ,p + || X n ]] pµ,p + [[ X n || pµ,p i < ∞ . Indeed, every X n induces a probability measure X n ( P ) on D ([0 , { X n ( P ) , n ≥ } isweakly relatively compact (see [7], [9]). Any weak limit of a weakly converg-ing subsequence X n k has c`adl`ag paths with probability 1 and, obviously,the same finite-dimensional distributions as X . Therefore X has a c`adl`agmodification according to Lemma 2.24 in [9]. R. MIKULEVIˇCIUS AND FANHUI XU In order to show (2.2), we estimate, using assumptions imposed, E (cid:20)Z | X n − X nt | p t µp +1 dt (cid:21) ≤ n − X k =1 Z ( k +1) / n k/ n E (cid:2)(cid:12)(cid:12) X − X k/ n (cid:12)(cid:12) p (cid:3) ( k/ n ) µp +1 dt ≤ C − n n − X k =1 (cid:18) k n (cid:19) r − µp − = C (cid:0) − n (cid:1) r − µp n − X k =1 k r − µp − ≤ C, n ≥ . Similarly, E (cid:20)Z | X n − X nt | p | − t | µp +1 dt (cid:21) ≤ C, n ≥ . In the same vein, E Z Z Z s Following [6], for 0 ≤ σ < τ ≤ 1, we introduce another modulus ofc`adl`aguity N ( f ; ( σ, τ )) = inf σ<θ ≤ τ sup s ∈ [ σ,θ ) ,u ∈ [ θ,τ ] [ d ( f ( σ ) , f ( s )) ∨ d ( f ( u ) , f ( τ ))] . Denote for η > ,N ( f ; η ) = sup σ<τ ≤ σ + η N ( f ; ( σ, τ )) = sup <τ − σ ≤ η N ( f ; ( σ, τ )) . Clearly, N ( f ; η ) is increasing in η . ADLAGUITY ESTIMATES 5 Remark 2. According to Lemma 1.0 in [6] ,(a) For any σ < τ , N ( f ; ( σ, τ )) ≤ ∆ ( f ; ( σ, τ )) ≤ N ( f ; ( σ, τ )) . In particular, ∆ ( f ; (0 , ≤ s ≤ t ≤ u ≤ d ( f ( s ) , f ( t )) ∧ d ( f ( t ) , f ( u )) ≤ N ( f ; (0 , . (b) For ( s, t ) ⊆ ( σ, τ ) ,N ( f ; ( s, t )) ≤ N ( f ; ( σ, τ )) , ∆ ( f ; ( s, t )) ≤ ∆ ( f ; ( σ, τ )) . For µ ∈ (0 , f ] ˜ µ := sup η> N ( f ; η ) η µ , f ∈ D µ ([0 , . Remark 3. Obviously, [ f ] µ = sup ≤ s ≤ t ≤ u ≤ ∆ ( f ; s, t, u ) | u − s | µ = sup a> sup ≤ s ≤ t ≤ u ≤ , | u − s |≤ a ∆ ( f ; s, t, u ) | u − s | µ = sup a> sup | u − s |≤ a,s ≤ t ≤ u ∆ ( f ; s, t, u ) a µ , and (3.1) [ f ] µ ≤ sup s ≤ t ≤ u, | u − s |≤ / ∆ ( f ; s, t, u ) | u − s | µ + 2 µ ∆ ( f ; (0 , also, [ f ] ˜ µ = sup η> N ( f ; η ) η µ = sup a> sup η ≤ a N ( f ; η ) η µ = sup a> sup η ≤ a N ( f ; η ) a µ . We show that [ f ] µ and [ f ] ˜ µ are equivalent. Lemma 1. Let µ ∈ (0 , . For any f ∈ D µ ([0 , , E ) , 12 [ f ] ˜ µ ≤ [ f ] µ ≤ f ] ˜ µ . Proof. Since for each σ ≤ r ≤ τ , τ ≤ σ + η,d ( f ( σ ) , f ( r )) ∧ d ( f ( r ) , f ( τ )) η µ ≤ ∆ ( f ; ( σ, τ )) η µ ≤ ∆ ( f ; ( σ, τ ))( τ − σ ) µ = sup σ ≤ s ≤ t ≤ u ≤ τ ∆ ( f ; s, t, u ) | τ − σ | µ ≤ sup σ ≤ s ≤ t ≤ u ≤ τ ∆ ( f ; s, t, u ) | u − s | µ ≤ [ f ] µ it follows thatsup σ ≤ r ≤ τ,τ ≤ σ + η d ( f ( σ ) , f ( r )) ∧ d ( f ( r ) , f ( τ )) η µ ≤ sup σ ≤ τ,τ ≤ σ + η ∆ ( f ; ( σ, τ )) η µ ≤ [ f ] µ , R. MIKULEVIˇCIUS AND FANHUI XU and, using Remark 3,(3.2) [ f ] µ = sup η> sup σ ≤ τ,τ ≤ σ + η ∆ ( f ; ( σ, τ )) η µ Hence for any η > , by Remark 2(a),12 N ( f ; η ) = 12 sup σ<τ ≤ σ + η N ( f ; ( σ, τ )) ≤ sup σ<τ ≤ σ + η ∆ ( f ; ( σ, τ )) ≤ N ( f ; η ) , and by (3.2),12 sup η> N ( f ; η ) η µ ≤ sup η> sup σ ≤ τ,τ ≤ σ + η ∆ ( f ; ( σ, τ )) η µ ≤ η> N ( f ; η ) η µ . (cid:3) The following key estimate was pointed out in [6], Lemma 1.2.4, as anextraction from Theorem 12.5 in [1] (cf. inequality 12.76 in [1]). For thesake of completeness we provide its proof. Lemma 2. (Lemma 1.2.4 in [6] ) For any f ∈ D (0 . , E ) and every triplet ≤ σ < t < τ ≤ , (3.3) N ( f ; ( σ, τ )) ≤ N ( f ; ( σ, t )) ∨ N ( f ; ( t, τ )) + ∆ ( f ; ( σ, t, τ )) . Proof. Let 0 ≤ σ < t < τ ≤ 1. By the definition of N , for each ε ∈ (0 , σ < θ ≤ t < θ ≤ τ such that d ( f ( σ ) , f ( s )) ∨ d ( f ( u ) , f ( t )) ≤ N ( f ; ( σ, t )) + ε, s ∈ [ σ, θ ) , u ∈ [ θ , t ] ,d ( f ( t ) , f ( s )) ∨ d ( f ( u ) , f ( τ )) ≤ N ( f ; ( t, τ )) + ε, s ∈ [ t, θ ) , u ∈ [ θ , τ ] . We split the proof in two cases: d ( f ( σ ) , f ( t )) ≤ d ( f ( t ) , f ( τ )) and d ( f ( σ ) , f ( t )) >d ( f ( t ) , f ( τ )) respectively. Case 1 : d ( f ( σ ) , f ( t )) ≤ d ( f ( t ) , f ( τ )), i.e. ∆ ( f ; σ, t, τ ) = d ( f ( σ ) , f ( t )).Obviously,(3.4) N ( f ; ( σ, τ )) ≤ sup s ∈ [ σ,θ ) d ( f ( σ ) , f ( s )) ∨ sup u ∈ [ θ ,τ ] d ( f ( u ) , f ( τ )) , and(3.5) sup u ∈ [ θ ,τ ] d ( f ( u ) , f ( τ )) ≤ N ( f ; ( t, τ )) + ε. Now, d ( f ( σ ) , f ( s )) ≤ N ( f ; ( σ, t )) + ε, if s ∈ [ σ, θ ) , and d ( f ( σ ) , f ( s )) ≤ d ( f ( s ) , f ( t )) + d ( f ( σ ) , f ( t )) ≤ N ( f ; ( σ, t )) + ε + ∆ ( f ; σ, t, τ ) , if s ∈ [ θ , t ] ,d ( f ( σ ) , f ( s )) ≤ d ( f ( s ) , f ( t )) + d ( f ( σ ) , f ( t )) ≤ N ( f ; ( t, τ )) + ε + ∆ ( f ; σ, t, τ ) , if s ∈ ( t, θ ) . ADLAGUITY ESTIMATES 7 Hence(3.6) sup s ∈ [ σ,θ ) d ( f ( σ ) , f ( s )) ≤ N ( f ; ( σ, t )) ∨ N ( f ; ( t, τ )) + ∆ ( f ; σ, t, τ ) + ε. Using (3.4)-(3.6), we have(3.7) N ( f ; ( σ, τ )) ≤ N ( f ; ( σ, t )) ∨ N ( f ; ( t, τ )) + ∆ ( f ; σ, t, τ ) + ε. Case 2 : d ( f ( σ ) , f ( t )) > d ( f ( t ) , f ( τ )), i.e. ∆ ( f ; σ, t, τ ) = d ( f ( t ) , f ( τ )).Obviously,(3.8) N ( f ; ( σ, τ )) ≤ sup s ∈ [ σ,θ ) d ( f ( σ ) , f ( s )) ∨ sup u ∈ [ θ ,τ ] d ( f ( u ) , f ( τ )) , and(3.9) sup s ∈ [ σ,θ ) d ( f ( σ ) , f ( s )) ≤ N ( f ; ( σ, t )) + ε. Now, d ( f ( u ) , f ( τ )) ≤ N ( f ; ( t, τ )) + ε, if u ∈ [ θ , τ ] ,d ( f ( u ) , f ( τ )) ≤ d ( f ( u ) , f ( t )) + d ( f ( t ) , f ( τ )) ≤ N ( f ; ( σ, t )) + ε + ∆ ( f ; σ, t, τ ) , if u ∈ [ θ , t ] ,d ( f ( u ) , f ( τ )) ≤ N ( f ; ( t, τ )) + ε + ∆ ( f ; σ, t, τ ) , if u ∈ [ t, θ )Therefore, again, N ( f ; ( σ, τ )) ≤ N ( f ; ( σ, t )) ∨ N ( f ; ( t, τ )) + ∆ ( f ; σ, t, τ ) + ε. Since ε ∈ (0 , 1) is arbitrary, N ( f ; ( σ, τ )) ≤ N ( f ; ( σ, t )) ∨ N ( f ; ( t, τ )) + ∆ ( f ; σ, t, τ ) . (cid:3) For µ ∈ (0 , , define for f ∈ D ([0 , , E ) , [ f ] ˆ µ = sup Lemma 3. Let µ ∈ (0 , . For any f ∈ D µ ([0 , , E ) , [ f ] ˆ µ ≤ [ f ] µ ≤ f ] ˜ µ ≤ − − µ [ f ] ˆ µ . Proof. Let K = [ f ] ˜ µ . According to Remark 3, for any a > ,N ( f ; a/ ≤ Ka µ − µ . R. MIKULEVIˇCIUS AND FANHUI XU By (3.3), for every σ < τ ≤ | τ − σ | ≤ a, we have, by Lemma 2, N ( f ; ( σ, τ )) ≤ N (cid:18) f ; (cid:18) σ, σ + τ (cid:19)(cid:19) ∨ N (cid:18) f ; (cid:18) σ + τ , τ (cid:19)(cid:19) + ∆ (cid:18) f ; σ, σ + τ , τ (cid:19) ≤ N ( f ; a/ 2) + ∆ (cid:18) f ; σ, σ + τ , τ (cid:19) ≤ Ka µ − µ + ∆ (cid:18) f ; σ, σ + τ , τ (cid:19) . Hence N ( f ; a ) ≤ Ka µ − µ + sup <τ − σ ≤ a ∆ (cid:18) f ; (cid:18) σ, σ + τ , τ (cid:19)(cid:19) and N ( f ; a ) a µ ≤ − µ K + sup <τ − σ ≤ a ∆ (cid:0) f ; (cid:0) σ, σ + τ , τ (cid:1)(cid:1) a µ Taking sup in a > K ≤ − µ K + sup a> sup <τ − σ ≤ a ∆ (cid:0) f ; σ, σ + τ , τ (cid:1) a µ or [ f ] ˜ µ = K ≤ − − µ sup a> sup <τ − σ ≤ a ∆ (cid:0) f ; σ, σ + τ , τ (cid:1) a µ = 11 − − µ [ f ] ˆ µ . The statement follows by Lemma 1. (cid:3) Proof of Theorem 1 First we show that there is C = C ( µ, p ) so that with any δ ∈ (0 , , (4.1) d ( f (0) , f ( t )) ≤ Ct µ (cid:16) [ f ] µ δ µ + δ − /p || f ]] µ,p (cid:17) , t ≤ / . Assume 0 < t ≤ / 4. Taking ε < t , we have for τ ′ ∈ ( t − ε, t ) , τ ′′ ∈ ( t, t + ε ), d ( f (0) , f ( t )) ≤ ∆ (cid:0) f ; τ ′ , t, τ ′′ (cid:1) + d (cid:0) f (0) , f (cid:0) τ ′ (cid:1)(cid:1) + d (cid:0) f (0) , f (cid:0) τ ′′ (cid:1)(cid:1) . Integrating with respect to τ ′ , τ ′′ over ¯ Q = [ t − ε, t ] × [ t, t + ε ] ,ε d ( f (0) , f ( t ))(4.2) ≤ ε Z tt − ε d (cid:0) f (0) , f (cid:0) τ ′ (cid:1)(cid:1) dτ ′ + ε Z t + εt d (cid:0) f (0) , f (cid:0) τ ′′ (cid:1)(cid:1) dτ ′′ + Z Z ¯ Q ∆ (cid:0) f ; τ ′ , t, τ ′′ (cid:1) dτ ′ dτ ′′ = A + A + B. ADLAGUITY ESTIMATES 9 Now B ≤ [ f ] µ Z tt − ε Z t + εt (cid:0) τ ′′ − τ ′ (cid:1) µ dτ ′′ dτ ′ ≤ C [ f ] µ ε µ . Taking ε = δt with any δ ∈ (0 , , we have(4.3) ε − B ≤ C [ f ] µ δ µ t µ . By H¨older inequality, κ = µ + 1 /p, /p + 1 /q = 1 ,A = ε Z tt − ε d ( f (0) , f ( τ ′ )) τ ′ τ ′ dτ ′ ≤ ε (cid:18)Z tt − ε d ( f (0) , f ( τ ′ )) p τ ′ κp dτ ′ (cid:19) /p (cid:18)Z tt − ε τ ′ κq dτ ′ (cid:19) /q = Cε h t κq − ( t − ε ) κq i /q || f ]] µ,p ≤ Ct κ ε /q || f ]] µ,p . Taking ε = δt , with any δ ∈ (0 , , (4.4) ε − A ≤ Ct κ ε /q − A = Ct κ ε − /p A = Ct µ δ − /p || f ]] µ,p and, the same way,(4.5) ε − A ≤ Ct µ δ − /p || f ]] µ,p . The inequality (4.1) follows from (4.2)-(4.5).Similarly, with obvious changes, we prove that(4.6) d ( f (1) , f ( t )) ≤ C (1 − t ) µ (cid:16) [ f ] µ δ µ + δ − /p [[ f || µ,p (cid:17) , t ≥ / , for some C = C ( µ, p ) with any δ ∈ (0 , . Finally, (4.1), (4.6) imply that there is C = C ( µ, p ) so that with any δ ∈ (0 , , d ( f (1) , f (0))(4.7) ≤ d (cid:18) f (cid:18) (cid:19) , f (0) (cid:19) + d (cid:18) f (1) , f (cid:18) (cid:19)(cid:19) ≤ C (cid:18) (cid:19) µ (cid:16) [ f ] µ δ µ + δ − /p [[ f || µ,p (cid:17) + C (cid:18) (cid:19) µ δ − /p || f ]] µ,p . Now, d ( f (1) , f ( t )) ≤ d ( f (1) , f (0)) + d ( f (0) , f ( t )) if t ∈ (0 , / ,d ( f (0) , f ( t )) ≤ d ( f (1) , f (0)) + d ( f (1) , f ( t )) if t ∈ (3 / , . Hence, by (4.1), (4.6) and (4.7), there is C = C ( µ, p ) so that for all δ ∈ (0 , , t ∈ (0 , , (4.8) [ f | µ + | f ] µ ≤ C (cid:16) [ f ] µ δ µ + δ − /p [[ f || µ,p + δ − /p || f ]] µ,p (cid:17) . Now we estimate ∆ ( f ; s, t, u ) with 0 ≤ s < t < u ≤ t = s + u .(i) Assume | s | > | u − s | and | − u | > | u − s | . Let ε < ( u − s ) , s ′ ∈ ( s − ε, s ) , s ′′ ∈ ( s, s + ε ) , t ′ ∈ ( t − ε, t ) , t ′′ ∈ ( t, t + ε ) , u ′ ∈ ( u − ε, u ) , u ′′ ∈ ( u, u + ε ) , and A = ∆ (cid:0) f ; s ′ , s, s ′′ (cid:1) + ∆ (cid:0) f ; t ′ , t, t ′′ (cid:1) + ∆ (cid:0) f ; u ′ , u, u ′′ (cid:1) ,B = ∆ (cid:0) f ; s ′ , t ′ , u ′′ (cid:1) + ∆ (cid:0) f ; s ′ , t ′ , u ′ (cid:1) + ∆ (cid:0) f ; s ′ , t ′′ , u ′′ (cid:1) + ∆ (cid:0) f ; s ′ , t ′′ , u ′ (cid:1) +∆ (cid:0) f ; s ′′ , t ′ , u ′′ (cid:1) + ∆ (cid:0) f ; s ′′ , t ′ , u ′ (cid:1) + ∆ (cid:0) f ; s ′′ , t ′′ , u ′′ (cid:1) + ∆ (cid:0) f ; s ′′ , t ′′ , u ′ (cid:1) . Let Q = ( s − ε, s ) × ( s, s + ε ) × ( t − ε, t ) × ( t, t + ε ) × ( u − ε, u ) × ( u, u + ε ).Then | Q | = ε , and(4.9) ∆ ( f ; s, t, u ) ≤ A + B. Let˜ A = ε − Z Q A = Z ss − ε Z s + εs ∆ (cid:0) f ; s ′ , s, s ′′ (cid:1) ds ′′ ds ′ + Z tt − ε Z t + εt ∆ (cid:0) f ; t ′ , t, t ′′ (cid:1) dt ′′ dt ′ + Z uu − ε Z u + εu ∆ (cid:0) f ; u ′ , u, u ′′ (cid:1) du ′′ du ′ , and ˜ B = ε − Z Q B = Z ss − ε Z tt − ε Z u + εu ∆ (cid:0) f ; s ′ , t ′ , u ′′ (cid:1) du ′′ dt ′ ds ′ + Z ss − ε Z tt − ε Z uu − ε ∆ (cid:0) f ; s ′ , t ′ , u ′ (cid:1) du ′ dt ′ ds ′ + Z ss − ε Z t + εt Z u + εu ∆ (cid:0) f ; s ′ , t ′′ , u ′′ (cid:1) du ′′ dt ′′ ds ′ + Z ss − ε Z t + εt Z uu − ε ∆ (cid:0) f ; s ′ , t ′′ , u ′ (cid:1) du ′ dt ′′ ds ′ + Z s + εs Z tt − ε Z u + εu ∆ (cid:0) f ; s ′′ , t ′ , u ′′ (cid:1) du ′′ dt ′ ds ′′ + Z s + εs Z tt − ε Z uu − ε ∆ (cid:0) f ; s ′′ , t ′ , u ′ (cid:1) du ′ dt ′ ds ′′ + Z s + εs Z t + εt Z u + εu ∆ (cid:0) f ; s ′′ , t ′′ , u ′′ (cid:1) du ′′ dt ′ ds ′′ + Z s + εs Z t + εt Z uu − ε ∆ (cid:0) f ; s ′′ , t ′′ , u ′ (cid:1) du ′ dt ′′ ds ′′ = ˜ B + . . . + ˜ B . Integrating (4.9) over Q, (4.10) ∆ ( f ; s, t, u ) ≤ ε − ˜ A + ε − ˜ B. Now, Z ss − ε Z s + εs ∆ (cid:0) f ; s ′ , s, s ′′ (cid:1) ds ′ ds ′′ ≤ [ f ] µ Z ss − ε Z s + εs ( s ′′ − s ′ ) µ ds ′ ds ′′ ≤ C [ f ] µ ε µ . ADLAGUITY ESTIMATES 11 Similarly we estimate the other two terms in ˜ A and see that(4.11) ε − ˜ A ≤ C [ f ] µ ε µ . Using H¨older inequality, 1 /q + 1 /p = 1 , p > , with κ = µ + 3 /p,ε − ˜ B = ε − Z s + εs Z tt − ε Z u + εu d ( f ( s ′ ) , f ( t ′ )) ∧ d ( f ( u ′′ ) , f ( t ′ )) | s ′′ − u ′′ | κ (cid:12)(cid:12) s ′′ − u ′′ (cid:12)(cid:12) κ ds ′ dt ′ du ′′ ≤ ε − (cid:18)Z ss − ε Z tt − ε Z u + εu ( u ′′ − s ′ ) κq ds ′ dt ′ du ′′ (cid:19) /q × (cid:18)Z ss − ε Z tt − ε Z u + εu ∆ ( f ; s ′ , t ′ , u ′′ ) p | u ′′ − s ′ | κ ds ′ dt ′ du ′′ (cid:19) /p ≤ ε − (cid:18)Z s + εs Z tt − ε Z u + εu (cid:12)(cid:12) s ′′ − u ′′ (cid:12)(cid:12) qκ ds ′′ dt ′ du ′′ (cid:19) /q [[ f ]] µ,p . Since Z ss − ε Z tt − ε Z u + εu ( u ′′ − s ′ ) κq ds ′ dt ′ du ′′ ≤ Cε ( u − s ) κq +2 ε ( u − s ) = Cε ( u − s ) κq , we have ε − ˜ B ≤ Cε − q ( u − s ) κ [[ f ]] µ,p . Similarly estimating the other terms in ˜ B , we get ε − ˜ B ≤ Cε − /q ( u − s ) κ [[ f ]] µ,p = Cε − /p ( u − s ) κ [[ f ]] µ,p . = Cδ − /p ( u − s ) κ − /p [[ f ]] µ,p = Cδ − /p ( u − s ) µ [[ f ]] µ,p . Hence by (4.10) and (4.11), for some C = C ( µ, p ) , (4.12) ∆ ( f ; s, t, u ) ≤ C [ f ] µ ε µ + Cε − /p ( u − s ) κ [[ f ]] µ,p if | s | > | u − s | and | − u | > | u − s | . Taking ε = δ ( u − s ) with any δ ∈ (0 , , we have(4.13) ∆ ( f ; s, t, u ) ≤ C ( u − s ) µ (cid:16) [ f ] µ δ µ + δ − p [[ f ]] µ,p (cid:17) for some C = C ( µ, p ) if | s | > | u − s | and | − u | > | u − s | .(ii) Assume | s | ≤ | u − s | or | − u | ≤ | u − s | . If s ≤ | u − s | (recall t = s + u ), then s ≤ / t = s + u − s ≤ 34 ( u − s ) ≤ . By (4.1), there is C = c ( µ, p ) so that for any δ ∈ (0 , , s ≤ | u − s | , ∆ ( f ; s, t, u ) ≤ d ( f ( s ) , f (0)) ∧ d ( f ( u ) , f (0)) + d ( f ( t ) , f (0)) ≤ C | u − s | µ (cid:16) [ f ] µ δ µ + δ − /p || f ]] µ,p (cid:17) . If 1 − u ≤ | u − s | , then u ≥ and t ≥ / − t = 1 − u + s − u + u − s ≤ 34 ( u − s ) ≤ . By (4.6), there is C = C ( µ, p ) so that for any δ ∈ (0 , , − u ≤ | u − s | , we have∆ ( f ; s, t, u ) ≤ d ( f ( s ) , f (1)) ∧ d ( f ( u ) , f (1)) + d ( f ( t ) , f (1)) ≤ C | u − s | µ (cid:16) [ f ] µ δ µ + δ − /p [[ f || µ,p (cid:17) . Hence(4.14) ∆ ( f ; s, t, u ) ≤ C ( u − s ) µ (cid:16) [ f ] µ δ µ + δ − /p [[ f || µ,p + δ − /p || f ]] µ,p (cid:17) if | s | ≤ | u − s | or | − u | ≤ | u − s | . According to (4.13) and (4.14), there is C = C ( µ, p ) so that∆ ( f ; s, t, u ) ≤ C ( u − s ) µ (cid:16) [ f ] µ δ µ + + δ − p [[ f ]] µ,p + δ − /p [[ f || µ,p + δ − /p || f ]] µ,p (cid:17) , for any 0 ≤ s < t < u ≤ , t = ( s + u ) / . Hence for all δ ∈ (0 , , (4.15) [ f ] ˆ µ ≤ C (cid:16) [ f ] µ δ µ + [[ f ]] µ,p δ − /p (cid:17) for some C = C ( µ, p ). Then by Lemma 3 and (4.8), (4.15), there is C = C ( µ, p ) so that for all δ ∈ (0 , 1) we have[ f ] µ + | f ] µ + [ f | µ ≤ C (cid:16) [ f ] µ δ µ + δ − p [[ f ]] µ,p + δ − p || f ]] µ,p + δ − p [[ f || µ,p (cid:17) . Choosing δ so that δ µ C ≤ / C = C ( µ, p ) , [ f ] µ + | f ] µ + [ f | µ ≤ C ([[ f ]] µ,p + || f ]] µ,p + [[ f || µ,p ) , f ∈ D µ ([0 , , E ) . If E = R k , d ( x, y ) = | x − y | , x, y ∈ R k , then we can estimate the supre-mum of f . For each t, | f ( t ) | ≤ | f ( τ ) − f (0) | + | f ( t ) − f (0) | + | f ( τ ) |≤ | f ] µ + | f ( τ ) | , τ ∈ [0 , . Hence | f ( t ) | ≤ | f ] µ + Z | f ( τ ) | dτ , and sup ≤ t ≤ | f ( t ) | ≤ | f ] µ + (cid:18)Z | f ( τ ) | p dτ (cid:19) /p . ADLAGUITY ESTIMATES 13 The claim of Theorem 1 follows. 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A., Properties of sample functions of stochastic processes and embeddingtheorems, Teor. Veroyatn. Primen., 18(3), 1973, pp. 468-480.[9] Kallenberg, O., Foundations of Modern Probability, Springer, 1997.[10] Peszat, S. and Zabczyk, J., Time regularity of solutions to linear equations with Levynoise in infinite dimensions, Stochastic Processes and App., v. 123, 2013, pp. 719-751.[11] Schilling, R.S., Sobolev embedding for stochastic processes, Expositiones Mathemat-icae, 2000, v.18, pp. 239-242. E-mail address : [email protected] Department of Mathematics, University of Southern California, Los An-geles E-mail address : [email protected]@usc.edu sup | σ − τ |≤ a ∆ (cid:0) f ; σ, σ + τ , τ (cid:1) a µ . We will need the following equivalence claim.