= Tanaka equation
{c}
{title2=$dX_t=s(X_t)\,dB_t$}
{wiki}
With $s(x)=1$ for $x\geq0$ and $s(x)=-1$ for $x<0$, the equation starting at zero has <uniqueness in law> but not <pathwise uniqueness>. For existence, take a <Brownian motion> $W$ and set $B=\int s(W)dW$; the <Lévy characterization of Brownian motion> and <associativity of stochastic integration> give $W=\int s(W)dB$. Every solution has <quadratic variation> $t$ and hence the <Brownian motion> law. Both $W$ and $-W$ are solutions driven by the same $B$, since the time spent at zero is zero and $s(-W)=-s(W)$ away from zero. Using $s(0)=0$ instead would allow the constant solution and invalidate this uniqueness-in-law example.
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