Construction of SDEs that admit more than one solution

Question

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.

Answer 1

If the diffusion coefficient is not continuous but uniformly positive, there is a general pathwise uniqueness result for homogenous SDE coefficients:

Theorem (Pathwise Uniqueness). Suppose the SDE coefficients are homogeneous, measurable and bounded; and, the noise coefficient is uniformly positive and of bounded variation on any compact interval. Then pathwise uniqueness holds.

(To be sure, pathwise uniqueness implies strong and hence weak uniqueness.) A reference for this pathwise uniqueness result is:

Nakao, Shintaro, On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations, Osaka J. Math. 9, 513-518 (1972). ZBL0255.60039.

Remark 1. Generalizations of this result to time-dependent coefficients, but with several fairly technical requirements on the coefficients, can be found in:

Nakao, Shintaro, On pathwise uniqueness and comparison of solutions of one-dimensional stochastic differential equations, Osaka J. Math. 20, 197-204 (1983). ZBL0517.60061.

Veretennikov, A. Yu., On the strong solutions of stochastic differential equations, Theory Probab. Appl. 24, 354-366 (1980). ZBL0434.60064.

Remark 2. If one only requires weak uniqueness, then the assumption of bounded variation of the noise coefficient can be dropped (and the result holds for planar SDEs too) but again one requires homogeneity of the coefficients, see:

Krylov, N. V., On Ito's stochastic differential equations, Theory of Probability and Its Applications, 14, No. 2, 330-336 (1969).

Remark 3. Even for the homogeneous SDE $dX_t = \Sigma(X_t) dW_t$, continuity and positivity of the noise coefficient do not guarantee pathwise uniqueness; see

Barlow, M. T., One dimensional stochastic differential equations with no strong solution, J. Lond. Math. Soc., II. Ser. 26, 335-347 (1982). ZBL0456.60062.