How do investors' expectations drive asset prices?

We have to consider the two cases x₁ = 0 and x₁ = 0. If x₁ = 0 we have
the trivial solution. Thus, in the following we choose x_i = 0.

Assume that the coefficients of the PDE do not depend on x_i. Then и can
be separated as follows: u(t,xι,x₂) = ^(x1,x2)≠(t,x₂). With the boundary
condition u(T, x_i,x₂) = xι it follows ^(x_i,x₂)≠(T,x₂) = xι and because of
x₁ = 0 : φ(T,x2) = 0 for all x2. Hence, the following relationship holds:
H^{x, x}2⁾ = , <r._j∙., r

After some computation the partial differential equation can be written as
(without noting the variables of the functions a, β and 7)

∩      , /ɪ ʌ, ,/. ∖(      /A₂^(T,x2⁾ I 2≠≈_a^(T,x2⁾ 7²≠z₂U^T,x2Λ

⁰ = ≠, (_i,^x) ₊ ≠(_i,^x) 7∙ -     ,     ⁺      /■ ^- ɪ ≠(T,x_a) )

, , /. √n    ₂≠UT^,x)ʌ , У ∣ _t. ʌ

⁺^x₂(^t,x2⁾ (ʃ ⁷ ^(T x2) / + 2 ^^x2æ2⁽.’^x2)j
without loss of generality choose the boundary condition ≠(T, x₂) = 1.

Now assume that all coefficients are only time-dependent.

In the case of constant volatility, that is u^v ≡ 0 and b ≡ 0 and hence
β ≡ 0 and 7 ≡ 0, we have the following solution of the PDE

∙^(t,x₂) = exp ^x₂ ʃ a(s)ds^ .

With this knowledge, for the general PDE with time-dependent coefficients
we try the ansatz

∙^(t,x₂) = exp —-X2 a a(s)ds^ A(t).

with A(T) = 1. This leads to an ordinary first-order differential equation for
A which has the solution

A(t) = exp ^2 ʃ 2β(r) ^ʃ α(s)ds^ + 7²(r) ^ʃ a(s)ds^ dr^ .

Hence, the solution of the PDE for u is

u(t,xι,x₂) = xι exp ^x₂     a(s)ds^

exp ^2    2^(r) ^ʃ a(s)ds^ + 7²(r) ^ʃ a(s)ds^ dr

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