On some expectation and derivative operators related to integral representations of random variables with respect to a PII process

TL;DR

This paper derives explicit decompositions and variance formulas for square integrable random variables based on processes with independent increments, aiding in quadratic risk minimization in finance.

Contribution

It introduces explicit Kunita-Watanabe and Föllmer-Schweizer decompositions using expectation and derivative operators tied to PII process characteristics.

Findings

Explicit decompositions for a broad class of random variables.

Closed-form expression for hedging error in quadratic risk minimization.

Application to global and local risk minimization problems.

Abstract

Given a process with independent increments $X$ (not necessarily a martingale) and a large class of square integrable r.v. $H=f(X_T)$, $f$ being the Fourier transform of a finite measure $\mu$, we provide explicit Kunita-Watanabe and F\"ollmer-Schweizer decompositions. The representation is expressed by means of two significant maps: the expectation and derivative operators related to the characteristics of $X$. We also provide an explicit expression for the variance optimal error when hedging the claim $H$ with underlying process $X$. Those questions are motivated by finding the solution of the celebrated problem of global and local quadratic risk minimization in mathematical finance.