Vector measures of bounded γ-variation and stochastic integrals

Transcription

1 Vector measures of bounded γ-variation and stochastic integrals Jan van Neerven and Lutz Weis bstract. We introduce the class of vector measures of bounded γ-variation and study its relationship with vector-valued stochastic integrals with respect to Brownian motions. Mathematics ubject Classification 000). 46G10, 60H05. Keywords. Vector measures, bounded randomised variation, stochastic integration. 1. Introduction It is well known that stochastic integrals can be interpreted as vector measures, the identification being given by the identity F) = φdb. Here, the driving process B is a semi)martingale for instance, a Brownian motion), and φ is a stochastic process satisfying suitable measurability and integrability conditions. This observation has been used by various authors as the starting point of a theory of stochastic integration for vector-valued processes. Let X be a Banach space. In [5] we characterized the class of functions φ : 0, 1) X which are stochastically integrable with respect to a Brownian motion W t ) t [0,1] as being the class of functions for which the operator T φ : L 0, 1) X, T φ f := 1 0 ft)φt) dt, The first named author is supported by VICI subsidy of the Netherlands Organisation for cientific Research NWO).

2 Jan van Neerven and Lutz Weis belongs to the operator ideal γl 0, 1), X) of all γ-radonifying operators. Indeed, we established the Itô isomorphism 1 f dw = T f γl 0,1),X). 0 The linear subspace of all operators in γl 0, 1), X) of the form T = T f for some function f : 0, 1) X is dense, but unless X has cotype it is strictly smaller than γl 0, 1), X). This means that in general there are operators T γl 0, 1), X) which are not representable by an X-valued function. ince the space of test functions D0, 1) embeds in L 0, 1), by restriction one could still think of such operators as X-valued distributions. It may be more intuitive, however, to think of T as an X-valued vector measure. We shall prove see Theorem.3 and the subsequent remark) that if X does not contain a closed subspace isomorphic to c 0, then the space γl 0, 1), X) is isometrically isomorphic in a natural way to the space of X-valued vector measures on 0, 1) which are of bounded γ-variation. This gives a measure theoretic description of the class of admissible integrands for stochastic integrals with respect to Brownian motions. The condition c 0 X can be removed if we replace the space of γ-radonifying operators by the larger space of all γ-summing operators which contains the space of all γ-radonifying operators isometrically as a closed subspace). Vector measures of bounded γ-variation behave quite differently from vector measures of bounded variation. For instance, the question whether an X-valued vector measure of bounded γ-variation can be represented by an X-valued function is not linked to the Radon-Nikodým property, but rather to the type and cotype properties of X see Corollaries.5 and.6). In section 3 we consider yet another class of vector measures whose variation is given by certain random sums, and we show that a function φ : 0, 1) X is stochastically integrable with respect to a Brownian motion W t ) t [0,1] on a probability space Ω, P) if and only if the formula F) := φdw defines an L Ω; X)-valued vector measure F in this class.. Vector measures of bounded γ-variation Let, Σ) be a measurable space, X a Banach space, and γ n ) n 1 a sequence of independent standard Gaussian random variables defined on a probability space Ω, F, P). Definition.1. We say that a countably additive vector measure F has bounded γ-variation with respect to a probability measure µ on, Σ) if F Vγµ;X) <, where F n ) F Vγµ;X) := sup γ n )1, µn ) the supremum being taken over all finite collections of disjoint sets 1,..., N Σ such that µ n ) > 0 for all n = 1,...,N.

3 Vector measures of bounded γ-variation 3 It is routine to check e.g. by an argument similar to [4, Proposition 5.]) that the space V γ µ; X) of all countably additive vector measures F : Σ X which have bounded γ-variation with respect to µ is a Banach space with respect to the norm Vγµ;X). Furthermore, every vector measure which is of bounded γ-variation is of bounded -semivariation. In order to give a necessary and sufficient condition for a vector measure to have bounded γ-variation we need to introduce the following terminology. bounded operator T : H X, where H is a Hilbert space, is said to be γ- summing if there exists a constant C such that for all finite orthonormal systems {h 1,..., h N } in H one has γ n Th n C. The least constant C for which this holds is called the γ-summing norm of T, notation T γ H,X). With respect to this norm, the space γ H, X) of all γ- summing operators from H to X is a Banach space which contains all finite rank operators from H to X. In what follows we shall make free use of the elementary properties of γ-summing operators. For a systematic exposition of these we refer to [, Chapter 1] and the lecture notes [4]. Theorem.. Let be an algebra of subsets of which generates the σ-algebra Σ, and let F : X be a finitely additive mapping. If, for some 1 p <, T : L p µ) X is a bounded operator such that F) = T1,, then F has a unique extension to a countably additive vector measure on Σ which is absolutely continuous with respect to µ. If T : L µ) X is γ-summing, then the extension of F has bounded γ-variation with respect to µ and we have F Vγµ;X) T γ L µ),x). Proof. We define the extension F : Σ X by F) := T1, Σ. To see that F is countably additive, consider a disjoint union = n 1 n with n, Σ. Then lim N 1 = 1 N n in L p µ) and therefore lim F n ) = lim T N 1 n = T1 = F). N N The absolute continuity of F is clear. To prove uniqueness, suppose F : Σ X is another countably additive vector measure extending F. For each x X, F, x and F, x are finite measures on Σ which agree on, and therefore by Dynkin s lemma they agree on all of Σ. This being true for all x X, it follows that F = F by the Hahn-Banach theorem. uppose next that T : L µ) X is γ-summing, and consider a finite collection of disjoint sets 1,..., N in Σ such that µ n ) > 0 for all n = 1,...,N.

4 4 Jan van Neerven and Lutz Weis The functions f n = 1 n / µ n ) are orthonormal in L µ) and therefore F n ) γ n = γ n Tf n T γ L µn ) µ),x). It follows that F has bounded γ-variation with respect to µ and that F Vγµ;X) T γ L µ),x). Theorem.3. For a countably additive vector measure F : Σ X the following assertions are equivalent: 1) F has bounded γ-variation with respect to µ; ) There exists a γ-summing operator T : L µ) X such that In this situation we have F) = T1, Σ. F Vγµ;X) = T γ L µ),x). Proof. 1) ): uppose that F has bounded γ-variation with respect to µ. For a simple function f = N c n1 n, where the sets n Σ are disjoint and of positive µ-measure, define Tf := c n F n ). By the Cauchy-chwarz inequality, for all x X we have Tf, x F n ), x = E γ m c m µm ) γ n µn ) m=1 E γ n c n µn ) N c n µ n ) ) 1 F n ), x E γ n ) 1 µn ) )1 = f L µ) F Vγµ;X) x. F Vγµ;X) x It follows that T is bounded and T LL µ),x) F Vγµ;X). To prove that T is γ-summing we shall first make the simplifying assumption that the σ-algebra Σ is countably generated. Under this assumption there exists an increasing sequence of finite σ-algebras Σ n ) n 1 such that Σ = n 1 Σ n. Let P n be the orthogonal projection in L µ) onto L Σ n, µ) and put T n := T P n. These operators are of finite rank and we have lim n T n T in the strong operator topology of L L µ), X). Fix an index n 1 for the moment. ince Σ n is finitely generated there exists a partition = N j, where the disjoint sets 1,..., N generate Σ n. ssuming that µ j ) > 0 for all j = 1,...,M and µ j ) = 0 for j = M +1,...,N,

5 Vector measures of bounded γ-variation 5 the functions g j = 1 j / µ j ), j = 1,...,M, form an orthonormal basis for L Σ n, µ) and T n γ L µ),x) = T n γ L Σ n,µ),x) M M F j ) = γ j Tg j = E γ j F V µn ), γµ;x) the first identity being a consequence of [4, Corollary 5.5] and the second of [4, Lemma 5.7]. It follows that the sequence T n ) n 1 is bounded in γ L µ), X). By the Fatou lemma, if {f 1,..., f k } is any orthonormal family in L µ), then k k γ j Tf j liminf E γ j T n f j Tn n γ L µ),x) F V. γµ;x) This proves that T is γ-summing and T γ L µ),x) F Vγµ;X). It remains to remove the assumption that Σ is countably generated. The preceding argument shows that if we define T in the above way, then its restriction to L Σ, µ) is γ-summing for every countably generated σ-algebra Σ Σ, with a uniform bound T γ L Σ,µ),X) F Vγµ;X). ince every finite orthonormal family {f 1,...,f k } in L µ) is contained in L Σ, µ) for some countably generated σ-algebra Σ Σ, we see that k γ j Tf j T γ L Σ,µ),X) F V. γµ;x) It follows that T is γ-summing and T γ L µ),x) F Vγµ;X). ) 1): This implication is contained in Theorem.. By a theorem of Hoffmann-Jørgensen and Kwapień [3, Theorem 9.9], if X is a Banach space not containing an isomorphic copy of c 0, then for any Hilbert space H one has γ H, X) = γh, X), where by definition γh, X) denotes the closure in γ H, X) of the finite rank operators from H to X. ince any operator in this closure is compact we obtain: Corollary.4. If X does not contain an isomorphic copy of c 0 and F : Σ X has bounded γ-variation with respect to µ, then F has relatively compact range. Using the terminology of [5], a theorem of Rosiński and uchanecki [6] asserts that if X has type we have a continuous inclusion L µ; X) γl µ), X) and that if X has cotype we have a continuous inclusion γ L µ), X) L µ; X). In both cases the embedding is contractive, and the relation between the operator T and the representing function φ is given by Tf = fφdµ, f L µ).

6 6 Jan van Neerven and Lutz Weis If diml µ) =, then in the converse direction the existence of a continuous embedding L µ; X) γ L µ), X) respectively γl µ), X) L µ; X)) actually implies the type property respectively the cotype property) of X. Corollary.5. Let X have type. For all φ L µ; X) the formula F) := φdµ, Σ, defines a countably additive vector measure F : Σ X which has bounded γ- variation with respect to µ. Moreover, F Vγµ;X) φ L µ;x). If diml µ) =, this property characterises the type property of X. Proof. By the theorem of Rosiński and uchanecki, φ represents an operator T γl µ), X) such that T1 = φdµ = F) for all Σ. The result now follows from Theorem.. The converse direction follows from Theorem.3 and the preceding remarks. Corollary.6. Let X have cotype. If F : Σ X has bounded γ-variation with respect to µ, there exists a function φ L µ; X) such that F) = φdµ, Σ. Moreover, φ L µ;x) F Vγµ;X). If diml µ) =, this property characterises the cotype property of X. Proof. By Theorem.3 there exists an operator T γ L µ), X) such that F) = T1 for all Σ. ince X has cotype, X does not contain an isomorphic copy of c 0 and therefore the theorem of Hoffmann-Jørgensen and Kwapień implies that T γl µ), X). Now the theorem of Rosiński and uchanecki shows that T is represented by a function φ L µ; X). The converse direction follows from Theorem. and the remarks preceding Corollary Vector measures of bounded randomised variation Let, Σ) be a measurable space and r n ) n 1 a Rademacher sequence, i.e., a sequence of independent random variables with Pr n = ±1) = 1. Definition 3.1. countably additive vector measure F : Σ X is of bounded randomised variation if F V r µ;x) <, where F V r µ;x) = sup r n F n ) ) 1, the supremum being taken over all finite collections of disjoint sets 1,..., N Σ.

7 Vector measures of bounded γ-variation 7 Clearly, if F is of bounded variation, then F is of bounded randomised variation. The converse fails; see Example 1. If X has finite cotype, standard comparison results for Banach space-valued random sums [, 3] imply that an equivalent norm is obtained when the Rademacher variables are replaced by Gaussian variables. It is routine to check that the space V r µ; X) of all countably additive vector measures F : Σ X of bounded randomised variation is a Banach space with respect to the norm V r µ;x). In Theorem 3. below we establish a connection between measures of bounded randomised variation and the theory of stochastic integration. For this purpose we need the following terminology. Brownian motion on Ω, F, P) indexed by another probability space, Σ, µ) is a mapping W : Σ L Ω) such that: i) For all Σ the random variable W) is centred Gaussian with variance EW)) = µ); ii) For all disjoint, B Σ the random variables W) and WB) are independent. strongly µ-measurable function φ : X is stochastically integrable with respect to W if for all x X we have φ, x L µ) i.e, f belongs to L µ) scalarly) and for all Σ there exists a strongly measurable random variable Y : Ω X such that for all x X we have Y, x = φ, x dw almost surely. Note that each Y is centred Gaussian and therefore belongs to L Ω; X) by Fernique s theorem; the above equality then holds in the sense of L Ω). We define the stochastic integral of φ over by φdw := Y. For more details and various equivalent definitions we refer to [5]. Theorem 3.. Let W : Σ L Ω) be a Brownian motion. For a strongly µ- measurable function φ : X the following assertions are equivalent: 1) φ is stochastically integrable with respect to W; ) φ belongs to L µ) scalarly and there exists a countably additive vector measure F : Σ X, of bounded γ-variation with respect to µ, such that for all x X we have F), x = φ, x dµ, Σ; 3) φ belongs to L µ) scalarly and there exists a countably additive vector measure G : Σ L Ω; X) of bounded randomised variation such that for all x X we have G), x = φ, x dw, Σ.

8 8 Jan van Neerven and Lutz Weis In this situation we have F Vγµ;X) = G V r µ;l Ω;X)) = φdw )1 Proof. 1) ): This equivalence is immediate from Theorem.3 and the fact, proven in [5], that φ is stochastically integrable with respect to W if and only there exists an operator T γl µ), X) such that Tf = fφdµ, f L µ). In this case we also have T γl µ),x) = φdw )1 In view of Theorem.3, this proves the identity F Vγµ;X) = φdw ) 1. 1) 3): Define G : Σ L Ω; X) by G) := φdw, Σ. By the γ-dominated convergence theorem [5], G is countably additive. To prove that G is of bounded randomised variation we consider disjoint sets 1,..., N Σ. If r n ) n 1 is a Rademacher sequence on a probability space Ω, F, P), then by randomisation we have Ẽ r n G n ) = ẼE N r n φdw L Ω;X) n. = φdw n. φdw, with equality if N n =. In the second identity we used that the X-valued random variables n φdw are independent and symmetric. The final inequality follows by, e.g., covariance domination [5] or an application of the contraction principle. It follows that G is a countably additive vector measure of bounded randomised variation and G V r µ;x) = φdw. )1 3) 1): This is immediate from the definition of stochastic integrability. Example 1. If W is a standard Brownian motion on Ω, F, P) indexed by the Borel interval [0, 1], B, m), then W is a countably additive vector measure with values in L Ω) which is of bounded randomised variation, but of unbounded variation. The first claim follows from Theorem 3. since W) = 1 dw for all

9 Vector measures of bounded γ-variation 9 Borel sets. To see that W is of unbounded variation, note that for any partition 0 = t 0 < t 1 < < t N 1 < t N = 1 we have Wt n 1, t n )) L Ω) = tn t n 1. The supremum over all possible partitions of [0, 1] is unbounded. References [1] J. Diestel and J.J. Uhl, Vector measures. Mathematical urveys, Vol. 15, mer. Math. oc., Providence 1977). [] J. Diestel, H. Jarchow and. Tonge, bsolutely summing operators. Cambridge tudies in dv. Math., Vol. 34, Cambridge, [3] M. Ledoux and M. Talagrand, Probability in Banach spaces. Ergebnisse d. Math. u. ihre Grenzgebiete, Vol. 3, pringer-verlag, [4] J.M..M. van Neerven, tochastic evolution equations. Lecture notes of the 11th International Internet eminar, TU Delft, downloadable at [5] J.M..M. van Neerven and L. Weis, tochastic integration of functions with values in a Banach space. tudia Math ), [6] J. Rosiński and Z. uchanecki, On the space of vector-valued functions integrable with respect to the white noise. Colloq. Math ), Jan van Neerven Delft University of Technology Delft Institute of pplied Mathematics P.O. Box 5031, 600 G Delft The Netherlands J.M..M.vanNeerven@TUDelft.nl Lutz Weis University of Karlsruhe Mathematisches Institut I D-7618 Karlsruhe Germany Lutz.Weis@math.uni-karlsruhe.de