Continuity properties of local martingales within Orlicz spaces - PDF document

Continuity properties of local martingales within Orlicz spaces Daniele Imparato Politecnico of Turin daniele.imparato@polito.it September 16, 2005

Introduction to Orlicz spaces Let (Ω , F , µ ) be a probability space; consider the Lebesgue space L p ( µ ), with 1 ≤ p < ∞ ; then � f ∈ L p ( µ ) ⇔ Ω Φ( f ) dµ < + ∞ , where Φ( x ) := | x | p . 1. Φ( x ) is an increasing convex function 2. Φ(0) = 0 3. x → + ∞ Φ( x ) = + ∞ lim Definition . Any function Ψ satisfying 1 ., 2 ., 3 . is called a Young function . 1

Let us consider the Orlicz class associated to the Young function Φ � L Φ ( µ ) := { u r.v. : ˜ Ω Φ( | u | ) dµ < ∞} L Φ ( µ ) is a convex space • ˜ • it is not a vector space in general! We introduce the Orlicz space L Φ ( µ ) := { u r. v. : ∃ α > 0 s.t. E µ [Φ( αu )] < ∞} . • L Φ ( µ ) is a vector space • it is indeed a Banach space by endowing it with the norm � u � � �� � || u || (Φ ,µ ) := inf k > 0 : E µ Φ ≤ 1 . k 2

The space L cosh − 1 ( p · µ ) • Let Φ 1 ( x ) := cosh( x ) − 1 • Let M (Ω , F , µ ) be the set of the µ -almost surely strictly positive densities. ∀ p ∈ M (Ω , F , µ ), consider L Φ 1 ( p · µ ). Definition . Let p ∈ M (Ω , F , µ ) be given; we define the Cram´ er Class at p as the set of r.v. u on (Ω , F , µ ) such that there exists ε > 0 for which � e tu � < ∞ , E p · µ for every t ∈ ( − ε, ε ) . Proposition . The Cram´ er class at p coincides with the space L Φ 1 ( p · µ ) -Pistone and Sempi (1995) - 3

Change of measures Let u ∈ L Φ 1 ( r · µ ); consider the one-dimensional exponential model p ( t ) := e tu − ψ ( t ) r t ∈ ( − ε, ε ) , ε > 0 , (1) � e tu � Ψ( t ) := log E r is the cumulant function. Proposition . Let p, q ∈ M (Ω , F , µ ) connected by the above exponential model; then the cor- respondent Orlicz spaces are isomorphic, i.e. L Φ 1 ( p · µ ) ≃ L Φ 1 ( q · µ ) -see Pistone and Sempi (1995), Pistone and Rogantin (1999) and Cena (2003)- 4

The space L n, Φ 1 ( p · µ ) In analogy with Lebesgue spaces L p , we define u n ∈ L Φ 1 ( p · µ ) } L n, Φ 1 ( p · µ ) := { u r. v. : • L n, Φ 1 ( p · µ ) is indeed an Orlicz space with respect to the Young function Φ n 1 ( x ) := cosh( x n ) − 1 . 1 1 ,p · µ ) = || u n || n • || u || (Φ n (Φ 1 ,p · µ ) . • L Φ n +1 ( p · µ ) ⊂ L Φ n 1 ( p · µ ) ∀ n ≥ 1 . 1 Proposition . The product uv L 2 , Φ 1 ( p · µ ) × L 2 , Φ 1 ( p · µ ) ∋ ( u, v ) �→ uv ∈ L Φ 1 ( p · µ ) is a continuous bilinear form. 5

Application to martingales theory • (Ω , F , µ, ( F t ) t ∈ [0 ,T ] ) filtered probability space satisfying usual conditions • M ∈ M C loc loc. martingale with c.t. Classical inequalities within L Φ ( µ ) (see Dellacherie - Meyer, (1975) ) � t Let Φ( t ) = 0 φ ( s ) ds and tφ ( t ) α (Φ) := sup Φ( t ) < ∞ . t> 0 • Let Z ∈ L Φ ( µ ). If A t is an increasing pro- cess s.t. Y t := E [ A ∞ − A t |F t ] ≤ E [ Z |F t ] , 6

then || A ∞ || (Φ ,µ ) ≤ α (Φ) || Y ∞ || (Φ ,µ ) . • Let X t be a mart s.t. sup || X t || (Φ ,µ ) < ∞ ; t then X t converges in L Φ ( µ ). Furthermore, if Ψ denotes the conjugate Young function of Φ and α (Ψ) < ∞ , then || sup X t || (Φ ,µ ) ≤ c || X ∞ || (Φ ,µ ) . t New inequalities within L Φ 1 ( µ ) Proposition . Let τ be a bounded stopping time and suppose � M � τ ∈ L Φ 1 ( µ ) ; √ || M τ || 2 (Φ 1 ,µ ) ≤ 2 ||� M � τ || (Φ 1 ,µ ) . (2) Lemma . Let τ be a bounded stopping time. If M τ ∈ L Φ 1 ( µ ) , there exists a strictly positive constant c such that ||� M � τ || (Φ 1 ,q α · µ ) ≤ c || M τ || 2 (3) (Φ 1 ,µ ) , where q α := E ( αM τ ) and α = α ( || M τ || (Φ 1 ,µ ) ) . 7

It is possible to show that measures q α · µ and µ can be connected by an exponential model. Proposition . Under the same hypothesis, there exists a strictly positive constant C such that ||� M � τ || (Φ 1 ,µ ) ≤ C || M τ || 2 (4) (Φ 1 ,µ ) . Corollary . M τ ∈ L Φ 1 ( µ ) ⇔ � M � τ ∈ L Φ 1 ( µ ) . 8

Continuity properties By using the previous results, the following statement follows. Theorem . Let M t , N t ∈ L Φ 1 ( µ ) and τ be a bounded stopping time. Then there exists a constant k > 0 such that ||� M, N � τ || (Φ 1 ,µ ) ≤ k || M τ || (Φ 1 ,µ ) || N τ || (Φ 1 ,µ ) , i.e. the crochet is a continuous bilinear form within L Φ 1 ( µ ) . 9

Let X be a Banach space; we denote by L p ( X, F ), 1 ≤ p < ∞ , the Banach space endowed with the following norm: for f : Ω → X , � p �� || f || p || f || ( p,X ) := X dF . Proposition . Let M be a cont. mart such that � M � t = F ( t ) ; then the map � • η �→ ( η · M )( • ) := 0 η s ( ω ) dM s ( ω ) is a continuous linear operator from L 2 ( L 2 , Φ 1 ( µ ) , F ) to L Φ 1 ( µ ) . Similar results can be obtained in more general cases: vector measures mathematical frame- work is needed - see Diestel and Uhl (1977). 10

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