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Entropy and additional utility of a discrete information disclosed progressively in time

Anna Aksamit

Abstract

The additional information carried by an enlarged filtration and its measurement was studied by several authors. Already Meyer (Sur un theoreme de J. Jacod, 1978) and Yor (Entropie d'une partition, et grossissement initial d'une filtration, 1985), investigated stability of martingale spaces with respect to initial enlargement with atomic sigma-field. We extend these considerations to the case where information is disclosed progressively in time. We define the entropy of such information and we prove that its finiteness is enough for stability of some martingale spaces in progressive setting. Finally we calculate additional logarithmic utility of a discrete information disclosed progressively in time.

Entropy and additional utility of a discrete information disclosed progressively in time

Abstract

The additional information carried by an enlarged filtration and its measurement was studied by several authors. Already Meyer (Sur un theoreme de J. Jacod, 1978) and Yor (Entropie d'une partition, et grossissement initial d'une filtration, 1985), investigated stability of martingale spaces with respect to initial enlargement with atomic sigma-field. We extend these considerations to the case where information is disclosed progressively in time. We define the entropy of such information and we prove that its finiteness is enough for stability of some martingale spaces in progressive setting. Finally we calculate additional logarithmic utility of a discrete information disclosed progressively in time.
Paper Structure (10 sections, 6 theorems, 68 equations)

This paper contains 10 sections, 6 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. General progressive enlargement with a partition
  3. Preliminaries
  4. Adding a partition
  5. Entropy of a pair $(\xi, \tau)$
  6. Main result
  7. Proof of Theorem \ref{['entropy2']}
  8. Additional logarithmic utility of a pair $(\xi,\tau)$
  9. Optional projections
  10. Auxiliary lemmas

Key Result

Lemma 2.1

For any ${\mathcal{G}}$-measurable integrable random variable $X$ and $s\leq t$ we have $z^{n,k}_t>0$ and $z^{n,k}_{t-}>0$ for all $t\geq 0$ a.s. on $C_n^k$ for each $n, k$, and

Theorems & Definitions (13)

  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Remark 3.1
  • Theorem 3.2
  • Theorem 4.1
  • proof
  • Definition A.1
  • Definition A.2
  • Lemma B.1
  • ...and 3 more