How do investors' expectations drive asset prices?

In the remainder of this section we will now briefly discuss the relationship
between the general pricing kernel which is based on the no arbitrage assump-
tion and the pricing kernel in a representative investor economy. We assume
that the representative investor has a state-independent utility function U of
wealth F_τ, which belongs to the set of twice continuous differentiable, strictly
increasing and strictly concave functions defined on (0, ∞). We assume that
F is given by a stochastic differential equation of form (6). It is well known,
that in equilibrium in such an economy, the following equation must hold⁶

∂

₇-U (F_τ)
_ ∂x

o,r —----------

a

for some scalar a > 0. Since E^p' (Φ_0jτ) — 1 we get a — E^p'

Thus, for Φ_tjτ we get the common characterization (see for example Bick [2])

∂

₇-U (F_τ)
____∂x__________
^ep'(J°_x^{u f} ) 1«)

Since Φ_t — Φ₀,_t — E^p' (Φ_o, _τ∖Q'_t) the process Φ can be characterized by a
function h satisfying the Feynman-Kac partial differential equation

by Φ₀,_t — h(t,F_t). We derive the following stochastic differential equation
for the pricing kernel by applying Itô’s formula

rfΦ₀,t — ɪh (t, F) ∑_t F dw; , 0 ≤ t ≤ T,
ox

^φo,o ^{— 1}∙

⁶See for example Pham, Touzi [27] or Decamps, Lazrak [6].

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