I look for examples of SDEs (stochastic differential equations) s.t. the uniqueness of the solution fails, i.e.

$$dX_t = B(t,X_t)dt + \Sigma(t,X_t)dW_t,\quad \forall t\ge 0.$$

More precisely, the coefficients $B,\Sigma$ satisfy :

1. $B: \mathbb R_+\times \mathbb R\to\mathbb R$ is Lipschitz and bounded;
2. $\Sigma:\mathbb R_+\times \mathbb R\to\mathbb R_+$ is of form $\Sigma(t,x)=(1+{\bf 1}_{\{B(t,x)>0\}})\sigma(t,x)$, where $\sigma$ is Lipschitz, bounded and $\inf_{(t,x)}\sigma(t,x)>0$.

The set of discontinuities of $\Sigma$ is clearly included in $E:=\{(t,x)\in\mathbb R_+\times\mathbb R: B(t,x)\}$. Under which conditions the above SDE has a unique solution? Under which conditions the above SDE has more than one solution?

Any answer, comments and references are highly appreciated.

PS : Here we the uniqueness can be considered for either strong solution or weak solution.

Two cases are trivial. If $B$ does not change sign, i.e. $B>0$ or $B\le 0$, there is a unique solution. If $B\equiv B(t)$, and the set $E^o:=\{t\ge 0: B(t)=0\}$ is negligible.

I look for examples of SDEs (stochastic differential equations) s.t. the uniqueness of the solution fails, i.e.

$$dX_t = B(t,X_t)dt + \Sigma(t,X_t)dW_t,\quad \forall t\ge 0.$$

More precisely, the coefficients $B,\Sigma$ satisfy :

1. $B: \mathbb R_+\times \mathbb R\to\mathbb R$ is Lipschitz and bounded;
2. $\Sigma:\mathbb R_+\times \mathbb R\to\mathbb R_+$ is of form $\Sigma(t,x)=(1+{\bf 1}_{\{B(t,x)>0\}})\sigma(t,x)$, where $\sigma$ is Lipschitz, bounded and $\inf_{(t,x)}\sigma(t,x)>0$.

The set of discontinuities of $\Sigma$ is clearly included in $E:=\{(t,x)\in\mathbb R_+\times\mathbb R: B(t,x)=0\}$. Under which conditions the above SDE has a unique solution? Under which conditions the above SDE has more than one solution?

Any answer, comments and references are highly appreciated.

PS : Here we the uniqueness can be considered for either strong solution or weak solution.

Two cases are trivial. If $B$ does not change sign, i.e. $B>0$ or $B\le 0$, there is a unique solution. If $B\equiv B(t)$, and the set $E^o:=\{t\ge 0: B(t)=0\}$ is negligible.

I look for examples of SDEs (stochastic differential equations) s.t. the uniqueness of the solution fails, i.e.

$$dX_t = B(t,X_t)dt + \Sigma(t,X_t)dW_t,\quad \forall t\ge 0.$$

More precisely, the coefficients $B,\Sigma$ satisfy :

1. $B: \mathbb R_+\times \mathbb R\to\mathbb R$ is Lipschitz and bounded;
2. $\Sigma:\mathbb R_+\times \mathbb R\to\mathbb R_+$ is of form $\Sigma(t,x)=(1+{\bf 1}_{\{B(t,x)>0\}})\sigma(t,x)$, where $\sigma$ is Lipschitz, bounded and $\inf_{(t,x)}\sigma(t,x)>0$.

The set of discontinuities of $\Sigma$ is clearly included in $E:=\{(t,x)\in\mathbb R_+\times\mathbb R: B(t,x)\}$. Under which conditions the above SDE has a unique solution? Under which conditions the above SDE has more than one solution?

Any answer, comments and references are highly appreciated.

PS : Two cases are trivial. If $B$ does not change sign, i.e. $B>0$ or $B\le 0$, there is a unique solution. If $B\equiv B(t)$, and the set $E^o:=\{t\ge 0: B(t)=0\}$ is negligible.

I look for examples of SDEs (stochastic differential equations) s.t. the uniqueness of the solution fails, i.e.

$$dX_t = B(t,X_t)dt + \Sigma(t,X_t)dW_t,\quad \forall t\ge 0.$$

More precisely, the coefficients $B,\Sigma$ satisfy :

1. $B: \mathbb R_+\times \mathbb R\to\mathbb R$ is Lipschitz and bounded;
2. $\Sigma:\mathbb R_+\times \mathbb R\to\mathbb R_+$ is of form $\Sigma(t,x)=(1+{\bf 1}_{\{B(t,x)>0\}})\sigma(t,x)$, where $\sigma$ is Lipschitz, bounded and $\inf_{(t,x)}\sigma(t,x)>0$.

The set of discontinuities of $\Sigma$ is clearly included in $E:=\{(t,x)\in\mathbb R_+\times\mathbb R: B(t,x)\}$. Under which conditions the above SDE has a unique solution? Under which conditions the above SDE has more than one solution?

Any answer, comments and references are highly appreciated.

PS : Here we the uniqueness can be considered for either strong solution or weak solution.

Two cases are trivial. If $B$ does not change sign, i.e. $B>0$ or $B\le 0$, there is a unique solution. If $B\equiv B(t)$, and the set $E^o:=\{t\ge 0: B(t)=0\}$ is negligible.