Existence and uniqueness on 1-d homogeneous SDE with zero drift.

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

$\displaystyle d X_{t} = \sigma(X_{t}) dW_{t}, \ X_{0} = \mu. \ \ \ \ \ (1)$

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 ${\mu}$ if and only if
(WE) — ${\sigma^{-2}(x)}$ is locally integrable at all ${x}$ satisfying ${\sigma(x) \neq 0}$ .

Any weak solution of SDE (1) is non-exploding.

SDE (1) has a unique weak solution for any initial ${\mu}$ if and only if
(WEU) — ${\sigma^{-2}(x)}$ is locally integrable at ${x}$ if and only if ${\sigma(x) \neq 0}$ .

From the above theorem, one can conclude following simple existence result:

Proposition 2 If ${\sigma}$ is continuous, then SDE (1) has a weak solution for any initial ${\mu}$ .

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

$\displaystyle X_{t} = \mu + B_{A_{t}}, \ \ \ \ \ (2)$

where ${B}$ is ${\mathcal{G}_{s}}$ -B.M., and ${A}$ is an increasing process

$\displaystyle A_{t} := \inf\{s\ge 0; T_{s} >t\},$

and

$\displaystyle T_{s} := \int_{0}^{s+} \sigma^{-2} (\mu + B_{u}) du; \ 0\le s<\infty.$

Then, one can actually verify that

${T}$ is a nondecreasing, extended real-valued process, which is continuous on ${[0, A_{\infty})}$ , where
$\displaystyle A_{\infty} := \inf\{s>0: T_{s} = \infty\},$
${A_{\infty} = \inf\{s\ge 0: \mu + B_{s} \in I(\sigma)\}}$ , where ${I(\sigma) := \{x: \sigma^{-2}(x) \notin L^{1}_{loc}\}.}$
${X}$ 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 ${\sigma(x) = I_{\{x = 0\}}}$ . Then, ${\sigma^{-2}(x)}$ is not integrable at ${x = 0}$ , for which ${\sigma(0) = 1}$ . 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 ${x = 0}$ .

Example 2 Let ${\sigma (x) = |x|^{\gamma}}$ . Since ${\sigma}$ is continuous, one can immediately conclude that there exists at least one non exploding solution of (1). For the uniqueness, we first observe that ${\sigma^{-2}(x)}$ is locally integrable at ${x = 0}$ if and only if ${\gamma<1/2}$ , i.e.

${\int_{-\epsilon}^{\epsilon} \sigma^{-2}(x) dx = 2\int_{0}^{\epsilon} x^{-2\gamma} dx < \infty }$ if and only if ${\gamma<1/2}$ .

Together with (WEU), we conclude that

For ${\gamma < 1/2}$ , (1) has multiple solutions;
while for ${\gamma \ge 1/2}$ , (1) has unique solution.

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

Example 3 Consider ${\sigma(x) = sgn(x)}$ , which takes value ${1}$ for ${x>0}$ , otherwise value ${-1}$ . By Theorem1, SDE (1) has a unique weak solution. However, one can show that there is no strong solution, see Page 302 of [KS91]

Omega

Deterministic Randomness

Existence and uniqueness on 1-d homogeneous SDE with zero drift.

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