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

TL;DR

This paper develops a stochastic analysis framework for marked binomial processes, enabling applications like Poisson approximation and portfolio optimization in trinomial models, through a new chaotic expansion and Markov-Malliavin structure.

Contribution

It introduces a novel chaotic expansion and Markov-Malliavin calculus for marked binomial processes, facilitating advanced probabilistic and financial applications.

Findings

Chaotic expansion for square-integrable marked binomial functionals

Application of Markov-Malliavin calculus to Poisson approximation

Portfolio optimization in trinomial models using new stochastic tools

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.