This note is based on Section 5.5 of the book [Karatzas and Shreve 1991].
- SDE (1) has a weak solution for any initial if and only if
(WE) — is locally integrable at all satisfying .
Any weak solution of SDE (1) is non-exploding.
- SDE (1) has a unique weak solution for any initial if and only if
(WEU) — is locally integrable at if and only if .
From the above theorem, one can conclude following simple existence result:
Proposition 2 If is continuous, then SDE (1) has a weak solution for any initial .
where is -B.M., and is an increasing process
Then, one can actually verify that
- is a nondecreasing, extended real-valued process, which is continuous on , where
- , where
- satisfies (1) on some filtered probability space.
Next, we will present some examples, where SDEs have non-existence, existence but non-uniqueness, unique existence for each.
Example 1 Let . Then, is not integrable at , for which . Thus, SDE (1) violates (WE) in Theorem1, and does not have weak solution for some initials. In fact, one can directly prove non-existence when the initial is given by .
Example 2 Let . Since is continuous, one can immediately conclude that there exists at least one non exploding solution of (1). For the uniqueness, we first observe that is locally integrable at if and only if , i.e.
if and only if .
Together with (WEU), we conclude that
Another interesting example yet to be mentioned in this context is the famous Tanaka’s example.