How do investors' expectations drive asset prices?

Acknowledgement

We are grateful for extensive help by Gtinter Franke and Michael Kohlmann.
We strongly benefited also from conversations with Gtinther Bamberg, Jan Beran,
Jürgen Eichberger, Yuanhua Feng, Frank Gerhard, Stamen Gortchev, Joachim
Grammig, Dieter Hess, Johannes Leitner, Christian R. Meurer, Rainer Schobel,
Jack Wahl and Jochen Wilhelm. An earlier version (On the Relationship of In-
formation Processes and Asset Price Processes) has been presented at the Annual
Congress of the European Economic Association in Bolzano, the Annual Congress
of the Verein für Socialpolitik in Berlin, the Annual Congress of the Deutsche
Mathematiker-Vereinigung (DMV) in Dresden and the Annual Congress of the
German Finance Association in Konstanz. We are grateful to the participants for
useful comments.

Financial support by the Center of Finance and Econometrics and the Zentrum
für Europaische Wirtschaftsforschung is gratefully acknowledged. Erik Lüders ap-
preciates a grant by the Deutsche Bundesbank.

A Appendix

The PDE, with indices on the functions indicating partial derivatives,

O = U_t(t,X_t,X₂) - ʌɑ^(t,X₁,X₂) XlX₂U_{x 1}(t,X₁,X₂)

-A⁽²( (t, x₁, X₂) σ^y(t, Xi, x₂) u_x({t, x₁, X₂)

+- (x∣X^{2 u}X₁X₁(t,Xι,x₂ ) + (σ^v (t,x₁,x₂ ))²

^UX₂X₂(t,Xι,X₂ )}

+b(t,X₁,X₂) U_x2(t,x₁,x₂)

u(T, x) = X₁

can be solved for only time-dependent coefficients.

For notational simplicity define the functions: a := — λ¹, β := — λ²σ^y + b,
7 := σ^v = — Д^г. Hence, the PDE can be written as

O = u_t(t,x₁,x₂) + a (t,x₁,x₂) x₁x₂ u_{x 1}(t,x₁,x₂) + β (t,x₁,x₂) u_x2(t,x₁,x₂)

+ -x₁x∣ u_xia-₁(t,x₁,x₂) + -7²(t,X₁,X₂) u_a-_2a,₂(t,x₁,x₂)

with the boundary condition u(T, x) = x₁ .

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