n this section we
are repeatedly using the formulas (
Chain rule
) and
(
Ito_formula
) without further reference. The
filtration is generated by
.
The reference is
[Kohn]
.
A onedimensional process
is given by SDE
for some smooth functions
and
and standard Brownian motion
.
Let
be an integrable function
.
Proposition
(Backward Kolmogorov for
running payoff) The
function
is a solution of the
problem


(Backward Kolmogorov with running payoff)

Proposition
(Backward Kolmogorov for
discounted payoff) The function
is a solution of the
problem
Proof
The proof was already given for the situation when
is away from the boundary. On the boundary the statement is obvious.
