given in El Karoui et al. [8]). Section 6 summarizes the main results.
2 A Short Survey on Related Papers
Before the seminal paper of Huang [19] continuous time models in finance
were already prevalent. The usual assumption was that the equilibrium asset
prices can be represented by Ito integrals. Huang was the first to give a
sound theoretical justification for this assumption. One main result in his
paper is that if equilibrium asset prices are adapted to a filtration generated
by a Brownian motion, then equilibrium asset prices are Ito integrals. Thus
Huang provided a justification for continuous sample paths of equilibrium
asset prices by linking them to the information flow.
The studies in the 90’s on the foundation of equilibrium asset price
processes addressed the question which characteristics of the state price den-
sity can be supported by sensible assumptions on the utility function of a
representative agent. Connected to this was the question, which utility func-
tions are implied by equilibrium asset prices which are governed by specific
stochastic differential equations. Bick [2] characterizes processes as viable by
the ’’no-trade criteria”, i.e. an asset price process is a possible equilibrium
if there exists a von Neumann-Morgenstern utility function such that it is
optimal for the representative agent to buy the market portfolio in t = O and
hold it until T. Bick requires path-independence of the pricing kernel for
viability which is equivalent to the requirement that the pricing kernel is a
deterministic function of wealth.2 Ensuing papers of He, Leland [14], Hodges,
Carverhill [17], Hodges, Selby [16] and Decamps, Lazrak [6] generalize this
analysis further.
Pham and Touzi [27] tackle the case of stochastic volatility. They provide
utility-theoretic foundations for common assumptions on the risk premia in
stochastic volatility models. Their analysis is similar to the previously men-
tioned, as they start with stochastic differential equations for the asset prices.
The main results of their paper are necessary and sufficient conditions for the
viability of the risk premia. Of special interest may be their analysis of the
classical stochastic volatility model of Hull and White [18] and the concept
of a minimal martingale measure introduced by Follmer and Schweizer [10].
Hull and White were the first to derive an explicit formula for the price of
an European option written on an asset with stochastic volatility. Yet their
2See for example Decamps, Lazrak [6] and section 4 of this paper.