Martingales and Stochastic Integrals [chapter 1 and 2, 1 ed.] 9780511897221

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1 Weak compactness and uniform integrability

The construction of canonical processes leads us naturally into infinitedimensional function spaces. On the other hand, Brownian motion and Poisson processes deal with families of random variables with rather special distributions. For more general processes, convergence theorems will require a priori boundedness and compactness conditions. We therefore dip briefly into functional analysis to isolate, in particular, a characterisation of relatively weakly compact sets in l}(£l, ^,P). The reader may wish to omit the details here and move straight to Chapter 2, where we begin the study of martingales in discrete time. Even in that context the need for certain weak compactness properties will become apparent.

1.1.

Duality and weak compactness in reflexive Banach spaces The Banach spaces Uf = Lp(Q^,P) were introduced in Definition 0.1.3. We assume that the reader is familiar with elementary Banach space theory, but we recall the following ideas (details may be found in [27], [77]). We deal only with real Banach spaces. 1.1.1.

Definition: Let £ be a real Banach space. The vector space E of all bounded linear functional on E, with the norm ||x'|| = sup{x'(x): xeE, \\x\\ ^1}, is a Banach space, the dual of E. The weak topology on E9 a{E,E'\ is defined by the system of neighbourhoods of 0 of the form Ve = {xeE: |xj(x)|^6, i^n}, where neN, e > 0 and x\eE' for all i^n. Thus a sequence (xn) in E converges weakly to xeE if x'(xn)—>x'(x) for all x'eF. The second dual E" of E is defined as (£')'. Note that E is isometrically embedded in E" by the map x—>x, where x(x') = x\x) for all x'eE'. E is said to be reflexive if this embedding is onto E". 22 Downloaded from https://www.cambridge.org/core. UFRJ-SiBI-Sistema de Bibliotecas e Informacao, on 30 Nov 2020 at 19:29:00, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511897221.003

Duality and weak compactness in reflexive Banach spaces

23

1.1.2.

Theorem: Let l^p l , and q = oo if p= 1. Then any bounded linear functional 0 on LF corresponds uniquely to an element geB under the map (/)=$QfgdP, and | |0| | = | \g\ \r So we can identify the dual of IP with 13. (Note that /gel} by the Holder inequality, which shows that \\f.g\\x ^||/|| p ||^|| € for all fell, geB, with equality ifff=Ag for some Uniqueness: if Jo/0idP=j* Q /0 2 dP for all fell, then gi=g2 a.s. (take/= 1^, for all AetF). So gfGZ? is unique if it exists, and then Holder's inequality shows that for/elf, |0(/)IHjn/0dP|oo. But (/> is a continuous map Zf->R, so ;i(£k) = (/>(lBk)--0(1J = A(A). Hence X is a-additive. If P(A) = 09 then 111^1^ = 0, so /l(X) = (/)(l>1) = 0, and X < P. Hence the Radon-Nikodym theorem provides gel}{P) such that A(A) = $AgdP = $QlAgdP for AeJ^. By linearity and continuity of 0, (/) = jo/#dP for all/GL°°(P), since each/GL00 is a uniform limit of simple functions. If we can show that geB and \\g\\q ^ 110||, the continuous functions /—•Jo/^dP and 0 coincide on the dense subspace L00 of IF, hence are equal. Thus it remains to show that | \g\ \q ^ | \\ |, where l/p + l/q=l. First take p = l: if ^ = {^>||(/)||} has positive measure, then 11(/)| \P(A)0 such that (o(x')=f((o)>(x+e and x"{x') =f{x") 0, P(A")-+0 as n—•oo. Now use Proposition 1.2.4 to deduce that j / 4»|/ n |dP(i) will require a form of the Vitali-Hahn-Saks theorem ([67], [27]). Proof that (I)=>(II): Embed L1 in its second dual and let 3f be the image of Jf. Consider the weak*-closure JTW of St in (L00)'. X is ^-bounded and Jf—*Jf is an isometry, so Jf w is weak*-bounded and hence (by Alaoglu's theorem) weak*-compact. If eJTw and /IE = (1£) then the uniform integrability of JT ensures (Proposition 1.2.4) that for given e > 0 we can find 0 such that sup / e j r J £ |/|dP{y) - 4>(u) u—x ^ y —x y—u Downloaded from https://www.cambridge.org/core. UFRJ-SiBI-Sistema de Bibliotecas e Informacao, on 30 Nov 2020 at 19:29:29, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511897221.004

Jensen's inequality

39

Fig. 2

It follows that if j?(x) denotes the right-hand derivative of at x, (y) — (x) ^ P(x)(y — x) for all x and y in R: if y ^ x, choose w in ]x, >>[, then

y-x

u—x

and as wjx, the right-hand side converges to P(x). Ifx^y, consider

4>iy) - < y-x

choose u>x and

4>(x) x-y

u—x

so that as ( y - x ) ^ O , 0(y)-(/)(x)^(y-x)j8(x) again. Now apply this inequality to y=f(co) and x = E(/|^)(co): = gf(co), where/ in L1 and co in ft are arbitrary. We obtain a.s. Hence of^°g + {f—g)$og. Now if