Let $X$ be the solution to some stochastic differential equation (unidimensional or multidimensional) :

$$dX_t = b(t,X_t)dt + a(t,X_t)dW_t\quad \forall t\ge 0,$$

where $b, a$ are both Lipschitz. Let $X$ denote the evolution of some system, and we aim to observe $X_{t}$. If we miss $X_t$ while we are able to observe $X_{t+\Delta t}$ with $\Delta t>0$. How to estimate

$$\mathbb P[\|X_t-x\|\le \varepsilon | X_{t+\Delta t}=x]?$$

Can we find some nice function $g(x,\Delta t, \varepsilon)$ to dominate the above conditional probability?

PS : Nice function means $g(x,0+, \varepsilon)=1$ for all $\varepsilon>0$. A natural attempt is to do the time-inversion, while for my case $b\equiv b(x)$, $a\equiv a(x)$ may degenerate, i.e. $\exists x_0$ such that $b(x_0)=0=a(x_0)$.

Let $X$ be the solution to some stochastic differential equation (unidimensional or multidimensional) :

$$dX_t = b(t,X_t)\,dt + a(t,X_t)\,dW_t\quad \forall t\ge 0,$$

where $b, a$ are both Lipschitz. Let $X$ denote the evolution of some system, and we aim to observe $X_{t}$. If we miss $X_t$ while we are able to observe $X_{t+\Delta t}$ with $\Delta t>0$. How to estimate

$$\mathbb P[\|X_t-x\|\le \varepsilon \mid X_{t+\Delta t}=x]?$$

Can we find some nice function $g(x,\Delta t, \varepsilon)$ to dominate the above conditional probability?

PS : Nice function means $g(x,0+, \varepsilon)=1$ for all $\varepsilon>0$. A natural attempt is to do the time-inversion, while for my case $b\equiv b(x)$, $a\equiv a(x)$ may degenerate, i.e. $\exists x_0$ such that $b(x_0)=0=a(x_0)$.