# On the marginal distributions of an absorbed diffusion

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.

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 $$q^\pm_t$$.
• Btw, is it known that this density $p_t$ is related to any PDE (as shown in the paper that you found previously)? Nov 1, 2021 at 16:58