Confusion about the uniqueness of an SDE

Question

Suppose we are given a SDE of the form $$ dX_t=\sigma(t,X_t)dW_t + b(t,X_t)dt,\quad X_0=\xi. $$

For simplicity, let $\xi$ be a constant.

My understanding is that

• Strong solution: for a given probability space, filtration, and a Brownian motion (BM), if there exists $X$ that solves the above SDE, then $X$ is a strong solution.
• Weak solution: if we can find a probability space, filtration, a BM, and $X$, then we say that the a tuple of these is a weak solution.

For the uniqueness, we say that

• Unique in law: if the law of $X$ is unique. (E.g. a BM $B$ and $-B$ both solve the Tanaka SDE, and their laws are the same)

• Pathwise uniqueness: if there are two solutions $X$ and $X'$, then $\mathbb{P}(X_t=X_t',\forall t)$.

My confusion is related to Ito's theorem and Yamada-Watanabe Theorem. Ito's theorem says that if $\sigma$ and $b$ are Lipshitz and of linear growth, then the solution exists and is unique. Yamada-Watanabe says that a weak solution + pathwise uniqueness is equivalent to the existence of a unique strong solution.

My confusions are as follows.

• In Ito's theorem, what does it mean by ``unique''? Is it pathwise uniqueness?
• In Ito's theorem, what does it mean by ``solution''? Is is strong solution?
• If Ito's theorem is about a strong and pathwise unique solution, then since strong solution is weak, Yamada-Watanabe seems to be moot.

I consulted Karatzas-Shereve, Rogers-Williams, and Klenke, but they use slightly different terminology and I am confused.

Answer 1

Indeed globally-Lipschitz regularity gives you strong-solution and strong-uniqueness (see Karatzas-Shreve 5.1 "Strong solutions"). As explained in Karatzas-Shreve the difference between pathwise and strong-uniqueness is:

• Pathwise uniqueness is said to hold among solutions that spend zero time at 0 if whenever $(X,W), (X,\tilde{W} )$ are two weak solutions with a common Brownian motion $W$ (relative to possibly different filtrations $\mathcal{F}_{t},\tilde{\mathcal{F}}_{t}$) on a common probability space and with common initial value, then $P(Xt = \tilde{X}_t \text{ for all }t \geq 0) = 1$.
• Strong uniqueness almost the same above with the extra assumption "with respect to a common Brownian motion $W$ and with the same filtration" $\mathcal{F}_{t}$.

One theorem of Yamada-Watanabe proves that one can lift a weak-solution with pathwise-uniqueness to a strong solution.

They also prove that a strong-solution is possible with a weaker requirement on the coefficients (described in Karatzas-Shreve as you mentioned) (note that below they actually mean strong-uniqueness) eg. in LECTURE NOTES ON THE YAMADA–WATANABE CONDITION FOR THE PATHWISE UNIQUENESS OF SOLUTIONS OF CERTAIN STOCHASTIC DIFFERENTIAL EQUATIONS.

For example, Revuz-Yor never uses the phrase "strong-uniqueness" but instead just talks about "pathwise-uniqueness" (which they define having the same filtration i.e. the strong uniqueness from Karatzas-Shrever).