conditional expectations of investors about a future cash flow. Thus, this
process describes all the information available to the investors. We focus on
the question how the asset price process is depending on the characteristics
of such an information process I while we take the information process as
given.
We consider a market with the given time horizon T > 0 and the two
dimensional standard Brownian motion W = {(W/, W)) : t ∈ [0,T]} on a
given probability space (Ω, P, Pt, P) where (Pt)t ejor∣ is the filtration gener-
ated by W augmented by all the P -null sets, with P = Pτ. The information
process I is defined on this probability space. As already mentioned, this
process is assumed to represent investors’ expectations about the future cash
flow paid to the shareholder of a company. We assume that this strictly
positive dividend is paid only in T. Since investors are assumed to act to-
tally rational, It is a positive P-martingale and hence admits the following
representation3
It = Io +
I I. (У dw’ + <
Jo
dWsv)
(1)
The martingale representation theorem provides that there exist two processes
σ1t and σf,v. By this theorem one only knows that these processes are adapted
= 1 and P (Jθr (σ1ay}2 ds < ∞) = 1.
(∕0r (p1s )2 ds < ∞)
to Pt and that P
In the following we make the assumption σf, = = 0 for all t ∈ [0, T] and we
require a special characterization of σ{. Of course, with these assumptions
we assume a special representation of It. Later we will give some economic
arguments for this special representation.
In the remainder of this paper we assume that the volatility σ1t of the
information process is governed by the following stochastic differential equa-
tion:
σt = σo + [ b(s,IsPs) ds + σ σv(s,Is,^a) dw' , 0 < t < t, (2)
Jo Jo
where b is the drift and σv describes the volatility of σ1t (these two functions
are assumed to be deterministic). Since It represents investors’ expecta-
tions in t about the value of the asset in T, the process It and the forward
3See for example Karatzas and Shreve [22].