Statistics for Finance

Appendix A

Projections in Hilbert spaces

A.1 Introduction

In many situations we want information about variables that are not directly measured, assuming that we have information about some variables which are correlated with the unmeasured variable. If this (cross)correlation is known or estimated, then it can be used for estimating the value of the unmeasured variables.

Consider for instance the interest rates. Short term interest rates are quoted on a daily basis in the money markets for maturities up to, say, one year; but longer term interest rates are traded only indirectly through the bond markets. Theoretically, options dependent on interest rates are priced according to a stochastic process describing the evolution in continuous time ...