On uniqueness of solutions to stochastic equations: A counter-example (Q1872286)

By the following counter-example it is shown that the solution \(X_t\) of the stochastic differential equation (i) \( dX_t= b(X_t)dM_t\), \(X_0=0\) (\(b(x)\) a locally integrable function, \(M_t\) a continuous local martingale) with the property that (ii) \(X_t\) has no occupation time in the zeros of \(b\), is not unique in the sense that \(\text{law}(X,M)\) is not unique. Counter-example: Let \[ b(x):= \text{sgn}(x):= \begin{cases} +1&\text{for \(x \geq 0\)},\\ -1&\text{for \(x < 0\)},\end{cases} \] and \(M_t:= \int_0^t b(X_s)dXs\), where \[ X_t:= \begin{cases} W_t &\text{for \(W_1 \geq 0\)}, \\ W_t &\text{for \(W_1 < 0\) and \(0 \leq t \leq 1\)} \\ W_1-\sqrt{2} (W_t-W_1)&\text{for \(W_1 < 0\) and \(t>1\)}, \end{cases}, \] \((W_t, {\mathcal F})\) a standard Wiener process, then \(X_t\) as well as \(\tilde{X}_t=-X_t\) solve the SDE (i) with (ii), but \(\text{law}(X,M) \neq \text{law}(\tilde{X},M)\). Here \((M,{\mathcal F})\) does not possess the representation property of continuous local martingales. It is conjectured that the solution of (i) and (ii) is unique iff the martingale satisfies this property.