Itô isomorphisms for L p -valued Poisson stochastic integrals

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1 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Itô isomorhisms for L -valued Poisson stochastic integrals Sjoerd Dirksen - artly joint work with Jan Maas & Jan van Neerven Universität Bonn King s College, 1 Aril 2014 Itô isomorhisms for L -valued Poisson stochastic integrals

2 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces L -valued Rosenthal inequalities 4 Itô isomorhisms in L -saces 5 One-sided estimates in martingale tye/cotye saces Itô isomorhisms for L -valued Poisson stochastic integrals

3 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Let X be a Banach sace, let F t ) t>0 be a filtration and J, J, ν) be a σ-finite measure sace. Definition Let F : Ω R + J X. F is a simle, adated X -valued rocess if there is a finite artition π = {0 = t 1 <... < t l+1 < } of R +, F ijk L F ti ), x ijk X and disjoint sets A 1,... A m in J satisfying νa j ) < such that F = l m n F ijk χ ti,t i+1 ] A j x ijk. i=1 j=1 k=1 Itô isomorhisms for L -valued Poisson stochastic integrals

4 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Fix a Poisson random measure N on R + J, BR + ) J, dt ν). Set ÑA) := NA) dt νa). Definition Let t 0 and B J. We define the comensated) Poisson stochastic integral of F on 0, t] B by 0,t] B F dñ = l m n F ijk Ñt i t, t i+1 t] A j B))x ijk. i=1 j=1 k=1 Here s t = mins, t). Itô isomorhisms for L -valued Poisson stochastic integrals

5 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Question Can we find a suitable norm,x on the integrand F such that c,x F,X E F dñ X for constants c,x, C,X deending only on and X? C,X F,X, RHS estimate = Bichteler-Jacod inequality or rather Novikov inequality Novikov 75, Marinelli-Röckner 13) Two-sided estimate = Itô isomorhism. Itô isomorhisms for L -valued Poisson stochastic integrals

6 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Motivation: stochastic Cauchy roblem in a Banach sace X of the form dut) = AUt) + f t, Ut))) dt + gt, Ut), z) dñdt, dz) U0) = x. 0,t] J If A generates a nice semigrou e ta ) t 0, we can consider its mild formulation Ut) = e ta x + + t 0 0,t] J e t s)a f s, Us))ds e t s)a gs, Us), z)dñds, dz). Need: L -maximal inequality for stochastic convolution. Itô isomorhisms for L -valued Poisson stochastic integrals

7 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Theorem Novikov 75, Bichteler-Jacod 83) ) Let 2 <. For any simle, adated R-valued rocess F, E max F dñ { E ) } F 2 2 ds dν, E F ds dν. Itô isomorhisms for L -valued Poisson stochastic integrals

8 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Definition Let 1 < <. A Banach sace X is called a UMD -sace if for any finite martingale difference sequence x i ) in L Ω; X ) and any choice of signs ε i ) i 1 in { 1, 1} N E ε i x i i X,X E x i i X. Remark: X is UMD if and only if it is UMD q 1 <, q < ). If this is the case, we call X a UMD-sace. UMD saces: Hilbert saces, L -saces 1 < < ), noncommutative L -saces 1 < < ); not UMD saces: L 1, L, CK). Itô isomorhisms for L -valued Poisson stochastic integrals

9 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Theorem McConnell 89, Hitczenko) If 1 < < and X is a UMD sace, then E F dñ X,X EE c F dñ c where Ñ c is a coy of Ñ which is indeendent of F. F dñ c is a sum of indeendent, mean-zero X -valued random variables if F is simle. X, Itô isomorhisms for L -valued Poisson stochastic integrals

10 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Theorem Rosenthal 70) If 2 and f i ) is a sequence of indeendent, mean-zero random variables in L Ω), then E n i=1 f i { n max i=1 E f i, n E f i 2 } 2. i=1 Original motivation: Banach sace geometry. Nowadays: standard tool in robability theory. Itô isomorhisms for L -valued Poisson stochastic integrals

11 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Lemma Let N be a Poisson distributed random variable with arameter 0 λ 1. Then for every 1 < there exist constants b, c > 0 such that b λ E N λ c λ. Itô isomorhisms for L -valued Poisson stochastic integrals

12 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces ). E F dñ EE c F ijk Ñ c t i, t i+1 ] A j ) i,j,k { max E i,j,k E c F ijk Ñ c t i, t i+1 ] A j ) 2 2, E E c F ijk Ñ c t i, t i+1 ] A j ) } i,j,k Itô isomorhisms for L -valued Poisson stochastic integrals

13 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Proof. { max E F ijk 2 2 t i+1 t i )νa j ), i,j,k E } F ijk t i+1 t i )νa j ) i,j,k { = max E ) F 2 2 ds dν, E } F ds dν. Itô isomorhisms for L -valued Poisson stochastic integrals

14 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Theorem Rosenthal 70) If 2 and f i ) is a sequence of indeendent, mean-zero random variables in L Ω), then E n i=1 f i { n max i=1 E f i, n E f i 2 } 2. i=1 Itô isomorhisms for L -valued Poisson stochastic integrals

15 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Let S be any measure sace. Theorem D. 14) Suose 2 q <. If ξ i ) is a finite sequence of indeendent, mean-zero L q S)-valued random variables, then E i ξ i L q S) {,q max E ξ i 2 2, L q S) i i ) E ξ i 1 L q S), i E ξ i q L q S) q }. Itô isomorhisms for L -valued Poisson stochastic integrals

16 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Recall the following fact: suose X, Y are Banach saces which are continuously embedded in some Hausdorff toological vector sace. Then the intersection X Y and the sum X + Y are Banach saces under the norms and z X Y = max{ z X, z Y } z X +Y = inf{ x X + y Y : z = x + y, x X, y Y }. If X Y is dense in X and Y, then X Y ) = X + Y, X + Y ) = X Y. Itô isomorhisms for L -valued Poisson stochastic integrals

17 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Theorem Reformulation) Suose 2 q <. Set ξ i ) Sq = E ξ i 2 2 ; L q S) ) ξ i ) D,q = E ξ i 1 L q S). i If ξ i ) is a finite sequence of indeendent, mean-zero L q S)-valued random variables, then E i ξ i L q S) where s,q = S q D q,q D,q. i,q ξ i ) s,q, Itô isomorhisms for L -valued Poisson stochastic integrals

18 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Theorem D. 14, general case) Let 1 <, q <. If ξ i ) is a finite sequence of indeendent, mean-zero L q S)-valued random variables, then E i ξ i L q S),q ξ i ) s,q. Itô isomorhisms for L -valued Poisson stochastic integrals

19 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Theorem Continued) Here s,q is given by S q D q,q D,q if 2 q < ; S q D q,q + D,q ) if 2 q < ; S q D q,q ) + D,q if 1 < < 2 q < ; S q + D q,q ) D,q if 1 < q < 2 < ; S q + D q,q D,q ) if 1 < q 2; S q + D q,q + D,q if 1 < q 2. For = q this result is a secial case of the main result in Junge & Xu 03), but with better constants. Ingredients: s,q ) = s,q, hyercontractivity, Khintchine, Kahane inequalities, tye of L q -saces. Itô isomorhisms for L -valued Poisson stochastic integrals

20 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Let 1 <, q <. We set F Sq = F D,q = F Dq,q = E E E F 2 ds dν 2 L q S) F q L q S) ds dν ) q ; F q L q S) ds dν q. ; Itô isomorhisms for L -valued Poisson stochastic integrals

21 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Theorem D. 14) Let 1 <, q <. For any simle, adated L q S)-valued rocess F, E su F dñ,q F I,q. t>0 0,t] J L q S) Itô isomorhisms for L -valued Poisson stochastic integrals

22 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Theorem Continued) Here I,q is given by S q D q,q D,q if 2 q < ; S q D q,q + D,q ) if 2 q < ; S q D q,q ) + D,q if 1 < < 2 q < ; S q + D q,q ) D,q if 1 < q < 2 < ; S q + D q,q D,q ) if 1 < q 2; S q + D q,q + D,q if 1 < q 2. Different roof via stochastic analysis by Marinelli 13. Result extends to noncommutative L q -saces. Itô isomorhisms for L -valued Poisson stochastic integrals

23 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces In comarison, if 1 <, q < and W is a Gaussian random measure on R + J, then E su t>0 0,t] J F dw L q S),q F Sq. Hence, if 2, q <, the class of L -integrable rocesses is bigger in the Gaussian case. If 1 <, q < 2 it is bigger in the Poisson case. Itô isomorhisms for L -valued Poisson stochastic integrals

24 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Definition If X is a Banach sace and 1 s 2, then X has martingale tye s if every finite martingale difference sequence d i ) satisfies E i d i s X s s,x E d i s X i s. On the other hand, if 2 r <, then X has martingale cotye r if every finite martingale difference sequence d i ) in satisfies E d i r X i r r,x E i d i r X r. Itô isomorhisms for L -valued Poisson stochastic integrals

25 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Let 1, s <. We consider F D = E s,x F D,X = E F s X ds dν ) s ; F X ds dν. Itô isomorhisms for L -valued Poisson stochastic integrals

26 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Theorem D., Maas, van Neerven 13) Let X be a Banach sace with martingale tye 1 < s 2 If s <, then E If 1 < < s then E F dñ X F dñ X,s,X F D s,x D,X.,s,X F D s,x +D,X. This result extends/imroves earlier results obtained using either Burkholder-Davis-Gundy inequalities or Itô s formula e.g. Kunita 04, Hausenblas 05, 11, Brzezniak & Hausenblas 09, Marinelli, Prévôt & Röckner 09, Marinelli & Röckner 10). Itô isomorhisms for L -valued Poisson stochastic integrals

27 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Theorem D., Maas, van Neerven 13) Let X be a Banach sace with cotye 2 s <. If s < then F D s,x D,s,X E F dñ.,x If 1 < < s then F D s,x +D,X,s,X E X F dñ X. Follows by duality of martingale tye and cotye Pisier, 75). These estimates match u with the ones in the revious result if and only if X is isomorhic to) a Hilbert sace! Itô isomorhisms for L -valued Poisson stochastic integrals

28 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces Corollary Let H be a Hilbert sace and let S be a C 0 -semigrou of contractions on H. If 2 < then E su t>0 If 1 < 2 then E su t>0 0,t] J 0,t] J St s)f s) dñ H St s)f s) dñ H F D s,h D,H. F D s,h +D,H. By the revious two results this is the best we can exect. Itô isomorhisms for L -valued Poisson stochastic integrals

29 L -valued Rosenthal inequalities Itô isomorhisms in L -saces One-sided estimates in martingale tye/cotye saces References: Itô-isomorhisms for L -valued Poisson stochastic integrals, S. Dirksen, Ann. Prob., to aear Poisson stochastic integration in Banach saces, S. Dirksen, J. Maas, J. van Neerven, Elec. J. Probab., 2013 On maximal inequalities for urely discontinuous martingales in infinite dimensions, C. Marinelli, M. Röckner, rerint, 2013 On maximal inequalities for urely discontinuous L q -valued martingales, C. Marinelli, rerint, 2013 Itô isomorhisms for L -valued Poisson stochastic integrals