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 Lp(µ), with 1 ≤ p < ∞; then f ∈ Lp(µ) ⇔

• Ω Φ(f)dµ < +∞,

where Φ(x) := |x|p.

• 1. Φ(x) is an increasing convex function
• 2. Φ(0) = 0

3. lim

x→+∞ Φ(x) = +∞

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 ˜ LΦ(µ) := {u r.v. :

• Ω Φ(|u|)dµ < ∞}
• ˜

LΦ(µ) is a convex space

• it is not a vector space in general!

We introduce the Orlicz space LΦ(µ) := {u r. v. : ∃α > 0 s.t. Eµ[Φ(αu)] < ∞} .

• LΦ(µ) is a vector space
• it is indeed a Banach space by endowing it

with the norm ||u||(Φ,µ) := inf

• k > 0 : Eµ
• Φ

u

k

• ≤ 1
• .

2

The space Lcosh −1(p · µ)

• Let Φ1(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 LΦ1(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

Ep·µ

• etu

< ∞, for every t ∈ (−ε, ε). Proposition . The Cram´ er class at p coincides with the space LΦ1(p · µ)

• Pistone and Sempi (1995) -

3

Change of measures Let u ∈ LΦ1(r · µ); consider the one-dimensional exponential model p(t) := etu−ψ(t)r t ∈ (−ε, ε), ε > 0, (1) Ψ(t) := log Er

• etu

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. LΦ1(p · µ) ≃ LΦ1(q · µ)

• see Pistone and Sempi (1995), Pistone and

Rogantin (1999) and Cena (2003)-

4

The space Ln,Φ1(p · µ) In analogy with Lebesgue spaces Lp, we define Ln,Φ1(p · µ) := {u r. v. : un ∈ LΦ1(p · µ)}

• Ln,Φ1(p · µ) is indeed an Orlicz space with

respect to the Young function Φn

1(x) := cosh(xn) − 1.

• ||u||(Φn

1,p·µ) = ||un|| 1 n

(Φ1,p·µ).

• LΦn+1

1

(p · µ) ⊂ LΦn

1(p · µ)

∀n ≥ 1. Proposition . The product uv L2,Φ1(p·µ)×L2,Φ1(p·µ) ∋ (u, v) → uv ∈ LΦ1(p·µ) is a continuous bilinear form.

5

Application to martingales theory

• (Ω, F, µ, (Ft)t∈[0,T]) filtered probability space

satisfying usual conditions

• M ∈ MC

loc loc. martingale with c.t.

Classical inequalities within LΦ(µ) (see Dellacherie - Meyer, (1975) ) Let Φ(t) =

t

0 φ(s)ds and

α(Φ) := sup

t>0

tφ(t) Φ(t) < ∞.

• Let Z ∈ LΦ(µ). If At is an increasing pro-

cess s.t. Yt := E[A∞ − At|Ft] ≤ E[Z|Ft],

6

then ||A∞||(Φ,µ) ≤ α(Φ)||Y∞||(Φ,µ).

• Let Xt be a mart s.t. sup

t

||Xt||(Φ,µ) < ∞; then Xt converges in LΦ(µ). Furthermore, if Ψ denotes the conjugate Young function

• f Φ and α(Ψ) < ∞, then

|| sup

t

Xt||(Φ,µ) ≤ c||X∞||(Φ,µ). New inequalities within LΦ1(µ) Proposition . Let τ be a bounded stopping time and suppose Mτ ∈ LΦ1(µ); ||Mτ||2

(Φ1,µ) ≤

√ 2||Mτ||(Φ1,µ). (2) Lemma . Let τ be a bounded stopping time. If Mτ ∈ LΦ1(µ), there exists a strictly positive constant c such that ||Mτ||(Φ1,qα·µ) ≤ c||Mτ||2

(Φ1,µ),

(3) where qα := E(αMτ) and α = α(||Mτ||(Φ1,µ)).

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It is possible to show that measures qα · µ and µ can be connected by an exponential model. Proposition . Under the same hypothesis, there exists a strictly positive constant C such that ||Mτ||(Φ1,µ) ≤ C||Mτ||2

(Φ1,µ).

(4) Corollary . Mτ ∈ LΦ1(µ) ⇔ Mτ ∈ LΦ1(µ).

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Continuity properties By using the previous results, the following statement follows. Theorem . Let Mt, Nt ∈ LΦ1(µ) and τ be a bounded stopping time. Then there exists a constant k > 0 such that ||M, Nτ||(Φ1,µ) ≤ k||Mτ||(Φ1,µ)||Nτ||(Φ1,µ), i.e. the crochet is a continuous bilinear form within LΦ1(µ).

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Let X be a Banach space; we denote by Lp(X, F), 1 ≤ p < ∞, the Banach space endowed with the following norm: for f : Ω → X, ||f||(p,X) :=

• ||f||p

XdF

p

. Proposition . Let M be a cont. mart such that Mt = F(t); then the map η → (η · M)(•) :=

0 ηs(ω)dMs(ω)

is a continuous linear operator from L2(L2,Φ1(µ), F) to LΦ1(µ). Similar results can be obtained in more general cases: vector measures mathematical frame- work is needed - see Diestel and Uhl (1977).

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References Pistone,

• G. and Sempi, C. (1995) An in-

finite dimensional geometric structure on the space of all the probability measures equivalent to a given one. Ann. Statist. 23, 1543 − 1561. Pistone, G. and Rogantin, M.P. (1999) The exponential statistical manifold: mean pa- rameters, orthogonality and space transfor-

• mations. Bernoulli 5(4), 721 − 760

Rao, M.M. and Ren, Z.D. (1991)Theory of Orlicz Spaces. New York: Dekker. Cena, A. (2003)Geometric structures on the non-parametric statistical manifold. PhD

• thesis. Milan: Polytechnic of Milan.

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