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This is a continuation of my question posted in Uniqueness of the solution to some SDE

Consider

$$X_t=X_0 + t + \int_0^t \frac{\sigma(s,X_s)}{1+m(s)}dW_s,\quad \forall t\ge 0,\quad\quad\quad (\ast)$$

where $X_0>0$ is some random variable and $m(t)=\mathbb P[\inf_{0\le s\le t}X_s>0]$ for $t\ge 0$. Iosif Pinelis has proved the existence/uniqueness of the solution to $(\ast)$ when $\sigma\equiv 1$. Can we show the uniqueness for $\sigma=\sigma(t,x)$? If not, do we have an example? Here we may assume $\sigma$ is as nice as possible, e.g. $\sigma$ is Lipschitz and $1\le \sigma(t,x)\le 2$ uniformly in $(t,x)$.

Any answer, comments or references are highly appreciated.

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  • $\begingroup$ Have you tried his arguments using the fact $\sigma(t,X_s)\approx \sigma(t,X_{t_0})$ for $t\in [t_0, t_0+h]$ (for $h$ small enough) ? $\endgroup$
    – user128095
    Commented Feb 23, 2022 at 6:57
  • $\begingroup$ @Neymar Yes, while I don't know how to estimate $D_1$ and $D_2$ in the reasoning $\endgroup$
    – GJC20
    Commented Feb 23, 2022 at 9:03

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