An example of existence proof of optimality using weak* topology

The proof of Lemma 4.3 in [RUDLOFF and KARATZAS 2010] is very succinct. In this below, the proof will be rephrased for an exercise of utilization of weak* topology.

The problem is the following. Let ${(\Omega, \mathcal{F}, R)}$ be a reference probability space. ${\Phi}$ is the space of ${\mathcal F}$ -measurable random variables ${\phi}$ satisfying ${\phi \in [0,1]}$ almost surely in ${R}$ . Let ${\mathcal P}$ and ${\mathcal Q}$ are two given sets of probability measures on ${(\Omega, \mathcal F)}$ whose elements are all absolutely continuous with respect to ${R}$ . Define ${\Phi_{\alpha}}$ for some constant ${\alpha\in (0,1)}$ by

$\displaystyle \Phi_{\alpha} = \{ \phi\in \Phi: \sup_{P\in \mathcal{P}} \mathbb E^{P}[\phi] \le \alpha\}.$

[RK10] studies max-min problem

$\displaystyle V^{-} = \sup_{\phi \in \Phi_{\alpha}} \inf_{Q\in \mathcal Q} \mathbb E^{Q}[\phi].$

In this below, we will present existence of the optimality in ${\Phi_{\alpha}}$ .

Proposition 1 There exists ${\phi^{*} \in \Phi_{\alpha}}$ such that

$\displaystyle V^{-} = \inf_{Q\in \mathcal Q} \mathbb E^{Q}[\phi^{*}].$

We need some preliminary terminologies for the preparation of the proof. For short, we use ${\mathbb E[\cdot]}$ to denote ${\mathbb E^{R}[\cdot]}$ . We consider the following two layers of function spaces.

The lower level function space is
$\displaystyle L^{1} := L^{1}(\Omega, \mathcal F, R) = \{X\in \mathcal F/\mathcal B: \mathbb E[|X|] <\infty\}$

endowed with norm ${\|X\|_{1} = \mathbb E[|X|].}$
The upper level function space is its dual space of ${L^{1}}$ , that is,
$\displaystyle L^{\infty}:=L^{\infty}(\Omega, \mathcal F, R) = \{X\in \mathcal F/\mathcal B: \hbox{esssup}_{\Omega} |X| <\infty\}$

endowed with weak* topology.

In general, the dual space of ${L^{1}}$ is bigger than ${L^{\infty}}$ . However, since ${R}$ is a ${\sigma}$ -finite measure as a probability measure, we can identify that the dual of ${L^{1}}$ is simply ${L^{\infty}}$ . Recall that, we say ${\phi_{n} \rightarrow_{*} \phi}$ (weak* convergence) in ${L^{\infty}}$ if

$\displaystyle \lim_{n\rightarrow \infty} \mathbb E[\phi_{n} X] = \mathbb E[\phi X], \quad \forall X\in L^{1}. \ \ \ \ \ (1)$

We also denote ${Z[{P}] = d P/ dR}$ for any probability measure ${P<<R}$ , ${Z[{\mathcal P}] = \{Z[{P}] : P\in \mathcal P\}}$ , and ${Z[{\mathcal Q}] = \{Z[{Q}] : Q\in \mathcal Q\}}$ . Note that ${Z[\mathcal P] \cup Z[\mathcal Q] \subset L^{1}}$ , and ${\Phi_{\alpha} \subset \Phi \subset L^{\infty}}$ .

Now, we are ready for the proof.

Proof: First we prove step by step with a few small related facts.

${\Phi_{\alpha}}$ is a subset of the closed (normed) unit ball ${B(1) = \{ \phi \in L^{\infty}: \|\phi\|_{\infty}\le 1\}}$ , since for all ${\phi \in \Phi_{\alpha}\subset \Phi \subset B(1)}$ .
is weak* closed. Because, for every
- ${\sup_{P\in \mathcal P} \mathbb E^{P} [\phi] = \sup_{Z\in Z[\mathcal P]} \mathbb E [Z\phi] = \sup_{Z\in Z[\mathcal P]} \lim_{n} \mathbb E [Z\phi_{n}] \le \alpha;}$
- ${\phi \in \Phi}$ , otherwise there exists ${A\in \mathcal F}$ and ${R(A) >0}$ . But this contradicts to (1) by taking ${X = I_{A}}$ .
Banach-Alaoglu’s theorem tells us that ${B(1)}$ is weak* compact, and hence ${\Phi_{\alpha}}$ is also weak* compact as a closed subset of ${B(1)}$ .
Fix arbitrary ${Q\in \mathcal Q}$ . Then, the function ${\Phi_{\alpha} \ni \phi \mapsto \mathbb E^{Q} [ \phi ] \in \mathbb R}$ is a continuous function under weak* topology, due to (1) by taking ${X = Z[Q]}$ .
The function ${\Phi_{\alpha} \ni \phi \mapsto \inf_{Q\in \mathcal Q}\mathbb E^{Q} [ \phi ] \in \mathbb R}$ is a upper semicontinuous function under weak* topology.
${V^{-}}$ attains maximum for some ${\phi^{*}}$ since ${\phi \mapsto \inf_{Q\in \mathcal Q}\mathbb E^{Q} [ \phi ]}$ is upper semicontinuous and ${\Phi_{\alpha}}$ is compact under weak* topology.

$\Box$

Omega

Deterministic Randomness

An example of existence proof of optimality using weak* topology

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