Let $X$ be the solution to some stochastic differential equation

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

Here $b,a: \mathbb R^d \to\mathbb R^d$ are bounded and Lipschitz and $W$ denotes a real Brownian motion. Fix some $\theta>0$ (small enough). For every $\varepsilon>0$, does there exist $\delta\equiv \delta(\varepsilon)>0$ (under suitable conditions on $b,a$) such that for $x\in \mathbb R^d$ it holds

$$\mathbb P[\sup_{t-\theta\le s\le t}|X_s-x|\le \varepsilon \mid X_t=x]\ge \delta\text{?}$$

The same question is asked for more general Itô (multidimensional) process $X$, i.e. $dX_t=\alpha_t \, dt + \beta_t \, dW_t$ (with suitable $\alpha,\beta$).

PS: It follows that

$$\mathbb P[\sup_{t-\theta\le s\le t}|X_s-x|\le \varepsilon \mid X_t=x]\ge \mathbb P[\sup_{t-\theta\le s\le t}|X_s-X_{t-\theta}|\le \varepsilon/2 \mid X_t=x]$$

and BDG's inequality yields

$$\mathbb E\Big[\mathbb P[\sup_{t-\theta\le s\le t}|X_s-X_{t-\theta}|\le \varepsilon/2 \mid X_t]\Big]=\mathbb P[\sup_{t-\theta\le s\le t}|X_s-X_{t-\theta}|\le \varepsilon/2] \ge C/\varepsilon^2. $$

Can we say something about the desired inequality using these the above two inequalities?

Let $X$ be the solution to some stochastic differential equation

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

Here $b,a: \mathbb R^d \to\mathbb R^d$ are bounded and Lipschitz and $W$ denotes a real Brownian motion. Fix some $\theta>0$ (small enough). For every $\varepsilon>0$, does there exist $\delta\equiv \delta(\varepsilon)>0$ (under suitable conditions on $b,a$) such that for $x\in \mathbb R^d$ it holds

$$\mathbb P[\sup_{t-\theta\le s\le t}|X_s-x|\le \varepsilon \mid X_t=x]\ge \delta\text{?}$$

The same question is asked for more general Itô (multidimensional) process $X$, i.e. $dX_t=\alpha_t \, dt + \beta_t \, dW_t$ (with suitable $\alpha,\beta$).

PS: It follows that

$$\mathbb P[\sup_{t-\theta\le s\le t}|X_s-x|\le \varepsilon \mid X_t=x]\ge \mathbb P[\sup_{t-\theta\le s\le t}|X_s-X_{t-\theta}|\le \varepsilon/2 \mid X_t=x]$$

and

$$\mathbb E\Big[\mathbb P[\sup_{t-\theta\le s\le t}|X_s-X_{t-\theta}|\le \varepsilon/2 \mid X_t]\Big]=\mathbb P[\sup_{t-\theta\le s\le t}|X_s-X_{t-\theta}|\le \varepsilon/2] >0. $$

Can we say something about the desired inequality using these the above two inequalities?

Let $X$ be the solution to some stochastic differential equation

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

Here $b,a: \mathbb R^d \to\mathbb R^d$ are bounded and Lipschitz and $W$ denotes a real Brownian motion. Fix some $\theta>0$ (small enough). For every $\varepsilon>0$, does there exist $\delta\equiv \delta(\varepsilon)>0$ (under suitable conditions on $b,a$) such that for $x\in \mathbb R^d$ it holds

$$\mathbb P[\sup_{t-\theta\le s\le t}|X_s-x|\le \varepsilon \mid X_t=x]\ge \delta\text{?}$$

The same question is asked for more general Itô (multidimensional) process $X$, i.e. $dX_t=\alpha_t \, dt + \beta_t \, dW_t$ (with suitable $\alpha,\beta$).

Let $X$ be the solution to some stochastic differential equation

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

Here $b,a: \mathbb R^d \to\mathbb R^d$ are bounded and Lipschitz and $W$ denotes a real Brownian motion. Fix some $\theta>0$ (small enough). For every $\varepsilon>0$, does there exist $\delta\equiv \delta(\varepsilon)>0$ (under suitable conditions on $b,a$) such that for $x\in \mathbb R^d$ it holds

$$\mathbb P[\sup_{t-\theta\le s\le t}|X_s-x|\le \varepsilon \mid X_t=x]\ge \delta\text{?}$$

The same question is asked for more general Itô (multidimensional) process $X$, i.e. $dX_t=\alpha_t \, dt + \beta_t \, dW_t$ (with suitable $\alpha,\beta$).

PS: It follows that

$$\mathbb P[\sup_{t-\theta\le s\le t}|X_s-x|\le \varepsilon \mid X_t=x]\ge \mathbb P[\sup_{t-\theta\le s\le t}|X_s-X_{t-\theta}|\le \varepsilon/2 \mid X_t=x]$$

and BDG's inequality yields

$$\mathbb E\Big[\mathbb P[\sup_{t-\theta\le s\le t}|X_s-X_{t-\theta}|\le \varepsilon/2 \mid X_t]\Big]=\mathbb P[\sup_{t-\theta\le s\le t}|X_s-X_{t-\theta}|\le \varepsilon/2] \ge C/\varepsilon^2. $$

Can we say something about the desired inequality using these the above two inequalities?

Let $X$ be the solution to some stochastic differential equation

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

Here $b,a: \mathbb R^d \to\mathbb R^d$ are bounded and Lipschitz and $W$ denotes a real Brownian motion. Fix some $\theta>0$ (small enough). For every $\epsilon>0$, does there exist $\delta\equiv \delta(\epsilon)>0$ (under suitable conditions on $b,a$) such that for $x\in \mathbb R^d$ it holds

$$\mathbb P[\sup_{t-\theta\le s\le t}|X_s-x|\le \epsilon \mid X_t=x]\ge \delta?$$

The same question is asked for more general Itô (multidimensional) process $X$, i.e. $dX_t=\alpha_tdt + \beta_t dW_t$ (with suitable $\alpha,\beta$).

Let $X$ be the solution to some stochastic differential equation

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

Here $b,a: \mathbb R^d \to\mathbb R^d$ are bounded and Lipschitz and $W$ denotes a real Brownian motion. Fix some $\theta>0$ (small enough). For every $\varepsilon>0$, does there exist $\delta\equiv \delta(\varepsilon)>0$ (under suitable conditions on $b,a$) such that for $x\in \mathbb R^d$ it holds

$$\mathbb P[\sup_{t-\theta\le s\le t}|X_s-x|\le \varepsilon \mid X_t=x]\ge \delta\text{?}$$

The same question is asked for more general Itô (multidimensional) process $X$, i.e. $dX_t=\alpha_t \, dt + \beta_t \, dW_t$ (with suitable $\alpha,\beta$).