result crucially depends on the assumption that the volatility risk premium
is independent of the underlying asset. Since in an incomplete market the
equivalent martingale measure and, thus, the risk premia are not uniquely
determined by arbitrage arguments, some restriction has to be imposed on
the risk premia. The analysis of Pham and Touzi establishes that this kind
of volatility risk premium is consistent with constant relative risk aversion.
For the specification of the equivalent martingale measure in incomplete mar-
kets Follmer and Schweizer [10] propose the concept of a minimal martingale
measure. Loosely stated the minimal martingale measure is defined such
that only traded risk is priced, hence, risk that is uncorrelated with traded
assets has a price of zero. As intuition suggests this kind of equilibrium is
supported by logarithmic preferences.
The analysis of Franke, Stapleton and Subrahmanyam [12] differs in var-
ious ways from the former papers. First, they do not assume the existence
of a representative investor, instead they simply assume that markets do
not admit arbitrage possibilities and hence, a pricing kernel exists. Second,
they do not take the asset price process as given. Their approach is more
constructive as the basis of the model is a process for conditional expecta-
tions of the exogenously given asset price at some terminal date. From the
assumption of rational investors they deduce the martingale property of the
process of conditional expectations. With the assumption that the process of
conditional expectations is governed by a geometric Brownian motion with-
out drift their analysis establishes a strong relationship between the process
of conditional expectations and the asset price process. In particular, they
show that the asset price process follows a geometric Brownian motion if
conditional expectations follow a geometric Brownian motion without drift
and the pricing kernel has constant elasticity. They also derive properties
of the price process for a pricing kernel with either declining or increasing
elasticity. In these cases asset returns are autocorrelated and the variance of
the asset price is higher than with constant elasticity of the pricing kernel.
3 Characterization and Viability of Informa-
tion Processes
In this section we introduce an information process I similar to the one
in Franke, Stapleton, Subrahmanyam [12]. It is defined as the process of