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 Fτ, 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 (Fτ)
_ ∂x
(7)
o,r —----------
a
for some scalar a > 0. Since Ep' (Φ0jτ) — 1 we get a — Ep'
Thus, for Φtjτ we get the common characterization (see for example Bick [2])
∂
φt,τ —
7-U (Fτ)
____∂x__________
ep'(J°xu f ) 1«)
Since Φt — Φ0,t — Ep' (Φo, τ∖Q't) the process Φ can be characterized by a
function h satisfying the Feynman-Kac partial differential equation
0 — |
∂h + ∂h ^1 ∂2h ∑2 2 7°-U (x) |
h (T,x) — |
∂x____ a |
by Φ0,t — h(t,Ft). 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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