Let , where
is an open domain in
. Consider equation
We are going to identify sufficient condition, under which continuous stochastic solution is indeed a classical solution. The results are from ([JT06]).
First, we define stochastic solution of (1). Consider -valued stochastic process
solving
where . Note that, (3) has unique strong solution if
Assumption 1 (Local regularity)
,
,
, or
if
.
We should note that by Lemma 6.1.1 of [Friedman 1976] ([Fri76]), under Assumption 4, , (or
) exists iff
(or
). To avoid the explosion of
from (3), we assume
Definition 1 (Stochastic solution) A stochastic solution to (1) is a function
defined on
satisfying,
where
is the first exit time from
, i.e.
We need one more assumption to avoid explosion of stochastic solution from (4).
Next, we present sufficient condition, under which a classical solution is a stochastic solution.
Theorem 2 Suppose
is a classical solution, and Assumptions 1 and 3 hold. In addition, if either of two followings are ture,
is bounded;
, Assumption 2 hold, and
satisfies
then
is a stochastic solution.
Proof: Assumption 1 is needed to have strong solution of (3). Ito’s formula shows that
is a local martingale. Also, one can show
, hence
is a uniformly integrable martingale in , and so is a martingale.
Next, we consider converse of Theorem 2: when do we have stochatics solution being a classical solution.
Lemma 3 Consider a bounded cylinder
being
for some
. Suppose
is nondegenerate with constant of ellipticity
. Also, assume
. Let
be a continuous function on the parabolic boundary
. Then, there is a unique classical solution to (1) in
with boundary data
.
If boundary data satisfies smooth and consistency conditions of (10.4.2-3) of [Kry96], then it is the result of Theorem 10.4.1 of [Kry96]. However,
is just assumed to be continuous without any regularity. In this case, Shauder’s interior estimate will be useful. A proof of similar result of Lemma 3 is provided in [JT06]. (The proof will be completed later)
The next is Theorem 2.7 of [JT06].
Theorem 4 Suppose Assumptions 1, 2, 3, and 4 hold. If
is a continuous stochastic solution, then
is a classical solution.
Proof: Consider arbitrary with
with
. By strong Markov property, for any
, we have
where is the first hitting time of
to
. This implies,
is stochastic solution in
with continuous boundary data
. Thus,
. Therefore,
.