let's consider a simple sde of the form
$$dX_t=b(t,X_t)dt+\sigma(t,X_t)dW_t$$
for $t\leq T$ and $b$,$\sigma$ continuous in $(t,x)$, $t \geq 0, x \in \mathbb{R}$.
My question is: does uniqueness in law hold?
Thanks in advance
2026-03-30 08:24:09.1774859049
continuity of coefficients imply uniqueness in law of the solutions of an sde?
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Unfortunatly, no. The generic counterexample is \begin{equation} dX_{t}=|X_{t}|^{\alpha}dB_{t},\quad X_{0}=0 \end{equation} with $\alpha\in(0,1/2)$. Then, $0$ gives one solution and another can be constructed by time-change. Take a look e.g. into the book 'Singular Stochastic Differential Equations' of Cherny et.al.