We discuss the existence and uniqueness of solutions for 1-d SDE in the form of

This note is based on Section 5.5 of the book [Karatzas and Shreve 1991].

**Theorem 1** * We have the following results: *

*
*
- 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 . *

Existence result under (WE) relies on a construction of weak solution as follows. Set

where is -B.M., and is an increasing process

and

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

- For , (1) has multiple solutions;
- while for , (1) has unique solution.

* *

Another interesting example yet to be mentioned in this context is the famous Tanaka’s example.

**Example 3** * Consider , which takes value for , otherwise value . By Theorem1, SDE (1) has a unique weak solution. However, one can show that there is no strong solution, see Page 302 of [KS91] *

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