Algebraic Form of Malliavin Calculus: Creation-Annihilation Operators, Conserved Currents and All That

TL;DR

This paper introduces an algebraic reformulation of Malliavin calculus using creation-annihilation operators to simplify its application, demonstrated through financial and stochastic problems.

Contribution

It proposes a novel algebraic formalism replacing analytic methods in Malliavin calculus with operator-based computations.

Findings

Simplified algebraic framework for Malliavin calculus

Application to financial valuation problems

Potential for broader adoption due to reduced complexity

Abstract

The extremely useful method of Malliavin calculus has not yet gained adequate popularity because of the complicated analytic apparatus of this method. The author attempts here to propose a simplified algebraic formalism similar to Malliavin calculus, but based on the notion of creation-annihilation operators instead of Malliavin derivative to replace analytic theorems with algebraic computations. Three test problems: the valuation of portfolio with stochastic payoff function, the expression of the terminal payoff through stochastic integral and the approximate equation for the high-frequency market measure are discussed in Appendices.