Barnett originated the Divisia monetary aggregates, which incontinuous time exactly track any monetary aggregator function underperfect certainty. With user costs measuring the prices of theservices of components, Barnett's aggregates are based on FrancoisDivisia's derivation of the Divisia line integral from thefirst-order conditions for optimizing behavior by economic agentsunder perfect certainty. We derive an extended Divisia index fromthe first-order conditions (Euler equations) that apply under risk.Our extended Divisia index is the first extension of index numbertheory into the domain of decision making under risk and therebyproduces a route for the extension of all index number theory topermit non-risk-neutrality. We generate simulated data from amodeled rational consumer and investigate the tracking accuracy ofthe extended Divisia index to the consumer's exact aggregatorfunction.