Stochastic analysis for obtuse random walks

TL;DR

This paper develops stochastic analysis tools like gradient and divergence for obtuse random walks, a class of discrete-time martingales, enabling advanced analysis and applications such as option hedging.

Contribution

It introduces a new framework for stochastic calculus on obtuse random walks, extending identities and formulas known for Bernoulli walks to this broader class.

Findings

Operators satisfy identities similar to Bernoulli walks

Derived a Clark-Ocone-type representation formula

Established a deviation inequality for obtuse random walks

Abstract

We present a construction of the basic operators of stochastic analysis (gradient and divergence) for a class of discrete-time normal martingales called obtuse random walks. The approach is based on the chaos representation property and discrete multiple stochastic integrals. We show that these operators satisfy similar identities as in the case of the Bernoulli randoms walks. We prove a Clark-Ocone-type predictable representation formula, obtain two covariance identities and derive a deviation inequality. We close the exposition by an application to option hedging in discrete time.