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This question can be seen as a variant of the post Bounded density for diffusions with diffusion coefficients bounded away from $0$ by Iosif Pinelis. Namely, consider the diffusion

$$X_t=\int_0^t a(s,X_s){\bf 1}_{\{|X_s|<1\}}dW_s,\quad \forall t\ge 0,$$

where $(W_t)_{t\ge 0}$ is a standard Brownian motion and $a$ is smooth s.t. $\inf_{(t,x)}a(t,x)\ge c>0$. Denote by $\mu_t$ the distribution of $X_t$. Can we write (under suitable conditions)

$$\mu_t(dx) = q^+_t\delta_{1}(dx)+q^-_t\delta_{-1}(dx)+ p_t(x)dx?$$

Here $q^{\pm}_t=\mathbb P[X_t=\pm 1]$ and $p_t$ (up to a normalization) is the conditional density function of $X_t$ knowing $\{\tau>t\}$, where $\tau:=\inf\{t\ge 0: |X_t|\ge 1\}$. So an alternative formulation is whether $X_t$ admits a density on the event $\{\tau>t\}$.

Any solution, references or comments are appreciated.

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The answer is yes. Indeed, let $$Y_t=\int_0^t a(s,Y_s)\,dW_s\quad \forall t\ge 0.$$ Then $X_t=Y_t$ on the event $\{\tau>t\}$. So, for any Borel set $A\subseteq(-1,1)$, we have $$P(X_t\in A)=P(Y_t\in A,\tau>t)\le P(Y_t\in A).$$ So, the distribution of $X_t$ is absolutely continuous with respect to the distribution of $Y_t$. By the previous answer, for $t>0$, the distribution of $Y_t$ has a density (with respect to the Lebesgue measure). Thus, for $t>0$, the distribution $\mu_t$ of $X_t$ has a density $p_t$ on the interval $(-1,1)$ as well, so that indeed $$\mu_t(dx) = q^+_t\delta_{1}(dx)+q^-_t\delta_{-1}(dx)+ p_t(x)dx$$ for some nonnegative $q^\pm_t$.

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  • $\begingroup$ Very tricky inequality! Thanks a lot Iosif for your answer $\endgroup$
    – GJC20
    Nov 1, 2021 at 16:51
  • $\begingroup$ Btw, is it known that this density $p_t$ is related to any PDE (as shown in the paper that you found previously)? $\endgroup$
    – GJC20
    Nov 1, 2021 at 16:58
  • $\begingroup$ @GJC20 : I vaguely remember seeing some work on this long ago, but this is all I remember in this regard, unfortunately. $\endgroup$ Nov 1, 2021 at 17:01

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