This (lengthy) exercise is designed to guide you through a proof that solutions to (40.1) exit bounded sets in finite time, a.s. (1) Suppose
where W is a one-dimensional Brownian motion, and as is an adapted process bounded by K. Let L > K > 0 and t0 > 0. Show that there exists ε > 0, depending only on L, K, and t0 such that
(2) Suppose
where as is as in (1) and M is a continuous martingale with
Use a time change argument to show that there exist L,ε> 0 such that
(3) If now X is a solution to (40.1),
and L given by (40.2) is uniformly elliptic, show by looking at the first coordinate of X that there exist L,ε such that
(4) What you have proved in (3) can be rephrased as saying that if
is a strong Markov process that solves (40.1) for every starting point and
then
where ε does not depend on x. Now use the strong Markov property (cf. the proof of Proposition 21.2) to show
Conclude that
for each starting point x.