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.


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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