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:

  1. 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.

  2. 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

  1. {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\},

  2. {A_{\infty} = \inf\{s\ge 0: \mu + B_{s} \in I(\sigma)\}}, where {I(\sigma) := \{x: \sigma^{-2}(x) \notin L^{1}_{loc}\}.}
  3. {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

  1. For {\gamma < 1/2}, (1) has multiple solutions;
  2. 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]

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