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Malliavin calculus for marked binomial processes: portfolio optimisation in the trinomial model and compound Poisson approximation

Hélène Halconruy

Abstract

In this paper we develop a stochastic analysis for marked binomial processes, that can be viewed as the discrete analogues of marked Poisson processes. The starting point is the statement of a chaotic expansion for square-integrable (marked binomial) functionals, prior to the elaboration of a Markov-Malliavin structure within this framework. We take advantage of the new formalism to deal with two main applications. First, we revisit the Chen-Stein method for the (compound) Poisson approximation which we perform in the paradigm of the built Markov-Malliavin structure, before studying in the second one the problem of portfolio optimisation in the trinomial model.

Malliavin calculus for marked binomial processes: portfolio optimisation in the trinomial model and compound Poisson approximation

Abstract

In this paper we develop a stochastic analysis for marked binomial processes, that can be viewed as the discrete analogues of marked Poisson processes. The starting point is the statement of a chaotic expansion for square-integrable (marked binomial) functionals, prior to the elaboration of a Markov-Malliavin structure within this framework. We take advantage of the new formalism to deal with two main applications. First, we revisit the Chen-Stein method for the (compound) Poisson approximation which we perform in the paradigm of the built Markov-Malliavin structure, before studying in the second one the problem of portfolio optimisation in the trinomial model.

Paper Structure

This paper contains 37 sections, 34 theorems, 236 equations.

Table of Contents

  1. Introduction
  2. Stochastic analysis for marked binomial processes, part I
  3. Framework
  4. Integration with respect to a binomial marked process
  5. Stochastic integrals
  6. Multiple integrals
  7. Chaotic decomposition
  8. Doléans exponentials
  9. Stochastic analysis for marked binomial processes, part II
  10. $\mathrm L^1$-theory: the Mecke and Mehler's formulas
  11. The Mecke formula and difference operators on $\mathrm L^1$
  12. $\mathrm L^2$-theory: Malliavin operators
  13. Gradient
  14. Divergence
  15. The Ornstein-Uhlenbeck structure
  16. ...and 22 more sections

Key Result

Proposition 2.4

Any process $u\in\mathcal{U}$ of representative $\mathfrak u$ is integrable with respect to the process $\mathrm R$ by the formula The so-called $\mathcal{R}$-stochastic integral $\mathrm J_1(u\,;\,\mathcal{R})$ of $u$ extends to square-integrable predictable processes via the (conditional) isometry formula where $\tilde{\nu}$ is the measure on $\mathbb X$ defined by $\tilde{\nu}(\{(t,k)\})=\kap

Theorems & Definitions (92)

  • Remark 2.1
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Definition 2.9
  • Lemma 2.10
  • Theorem 2.11
  • ...and 82 more