We know that the existence of solutions and pathwise uniqueness lead to the unique strong solution. And also that the strong solution is peculiar because of the fixed filtration.
I always see the discussion about "the unique strong solution". But I was wondering whether there could be strong solutions that are not unique? For example,can there be two strong solutions for an SDE given the same initial distribution?
Thanks a lot!
Example 1: Note that stochastic differential equations (SDEs) are a generalization of ordinary differential equations (ODEs); in particular, any ODE which does not have a unique solution does not have a unique (strong) solution. Consider for instance the differential equation
$$dX_t =2 \text{sgn}(X_t) \, \sqrt{|X_t|} \, dt, \qquad X_0 = 0;$$
it does not have a unique (strong) solution since both $X_t^{(1)} := t^2$ and $X_t^{(2)} := - t^2$ are a solution to the equation.
Example 2: The stochastic differential equation
$$dX_t = 1_{\mathbb{R} \backslash \{0\}}(X_t) \, dB_t, \qquad X_0 = 0$$
has more than one (strong) solution; just note that both $X_t^{(1)} := 0$ and $X_t^{(2)} := B_t$ solve the SDE.