How do investors' expectations drive asset prices?

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, P_t, P) where (P_t)_{t e}j_or∣ 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, I_t is a positive P-martingale and hence admits the following
representation³

The martingale representation theorem provides that there exist two processes
σ¹_t and σf^,v. By this theorem one only knows that these processes are adapted

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 I_t. Later we will give some economic
arguments for this special representation.

In the remainder of this paper we assume that the volatility σ¹_t of the
information process is governed by the following stochastic differential equa-
tion:

^σt = ^σo ⁺ [ ^b(s,I_sPs^{) ds + σ} σ^v(s,Is^,^_a^{) dw}' ^{, 0} < ^t < ^{t,    (}2)

Jo                 Jo

where b is the drift and σ^v describes the volatility of σ¹_t (these two functions
are assumed to be deterministic). Since I_t represents investors’ expecta-
tions in t about the value of the asset in T, the process I_t and the forward

³See for example Karatzas and Shreve [22].

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