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.

  1. {\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)}.
  2. {\Phi_{\alpha}} is weak* closed. Because, for every {\Phi_{\alpha} \ni \phi_{n} \rightarrow_{*} \phi}
    • {\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}}.
  3. 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)}.
  4. 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]}.
  5. 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.
  6. {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

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