## On uniqueness of solutions to stochastic equations: A counter-example.(English)Zbl 1029.60045

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.

### MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60G44 Martingales with continuous parameter 60J57 Multiplicative functionals and Markov processes 60J65 Brownian motion
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