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Fix $\epsilon>0$ and let $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ be a stochastic base, and let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a continous function with $f(0)=0$. Is there a family of Markov diffusion processse $X_t^{x,\epsilon}$ satisfying an SDE of the form: $$ d X_t^{x,\epsilon} = \mu(t,X_t^{x,\epsilon},\epsilon)dt + \Sigma(t,X_t^{x,\epsilon},\epsilon)dW_t, X_0^{x,\epsilon}=f(x) $$ and such that $ \mathbb{P}\left( \sup_{t \in [0,1],x \in \mathbb{R}^n}|X_t^{x,\epsilon} - f(x)| < \epsilon \right) $ holds with high probability?

Note: Of course, here, we require that $\Sigma(t,x,\epsilon)>0$ to avoid trivial solutions.

Fix $\epsilon>0$ and let $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ be a stochastic base, and let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a continous function with $f(0)=0$. Is there a family of Markov diffusion processse $X_t^{x,\epsilon}$ satisfying an SDE of the form: $$ \begin{cases} d X_t^{x,\epsilon} = \mu(t,X_t^{x,\epsilon},\epsilon)dt + \Sigma(t,X_t^{x,\epsilon},\epsilon)dW_t \\ X_0^{x,\epsilon}=f(x) \end{cases} $$ and such that $ \mathbb{P}\left( \sup_{t \in [0,1],x \in \mathbb{R}^n}|X_t^{x,\epsilon} - f(x)| < \epsilon \right) $ holds with high probability?

Note: Of course, here, we require that $\Sigma(t,x,\epsilon)>0$ to avoid trivial solutions.

Fix $\epsilon>0$ and let $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ be a stochastic base, and let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a continous function with $f(0)=0$. Is there a family of Markov diffusion processse $X_t^{x,\epsilon}$ satisfying an SDE of the form: $$ d X_t^{x,\epsilon} = \mu(t,X_t^{x,\epsilon},\epsilon)dt + \Sigma(t,X_t^{x,\epsilon},\epsilon)dW_t, X_0^{x,\epsilon} $$ and such that $ \mathbb{P}\left( \sup_{t \in [0,1],x \in \mathbb{R}^n}|X_t^{x,\epsilon} - f(x)| < \epsilon \right) $ holds with high probability?

Note: Of course, here, we require that $\Sigma(t,x,\epsilon)>0$ to avoid trivial solutions.

Fix $\epsilon>0$ and let $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ be a stochastic base, and let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a continous function with $f(0)=0$. Is there a family of Markov diffusion processse $X_t^{x,\epsilon}$ satisfying an SDE of the form: $$ d X_t^{x,\epsilon} = \mu(t,X_t^{x,\epsilon},\epsilon)dt + \Sigma(t,X_t^{x,\epsilon},\epsilon)dW_t, X_0^{x,\epsilon}=f(x) $$ and such that $ \mathbb{P}\left( \sup_{t \in [0,1],x \in \mathbb{R}^n}|X_t^{x,\epsilon} - f(x)| < \epsilon \right) $ holds with high probability?

Note: Of course, here, we require that $\Sigma(t,x,\epsilon)>0$ to avoid trivial solutions.

Fix $\epsilon>0$ and let $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ be a stochastic base, and let $f:\mathbb{R}^n\to \mathbb{R}$ be a continous function with $f(0)=0$. Is there a family of Markov diffusion processse $X_t^{x,\epsilon}$ satisfying an SDE of the form: $$ d X_t^{x,\epsilon} = \mu(t,X_t^{x,\epsilon},\epsilon)dt + \Sigma(t,X_t^{x,\epsilon},\epsilon)dW_t, X_0^{x,\epsilon} $$ and such that $ \mathbb{P}\left( \sup_{t \in [0,1]}|X_t^{x,\epsilon} - f(x)| < \epsilon \right) $ holds with high probability?

Note: Of course, here, we require that $\Sigma(t,x,\epsilon)>0$ to avoid trivial solutions.

Fix $\epsilon>0$ and let $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ be a stochastic base, and let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a continous function with $f(0)=0$. Is there a family of Markov diffusion processse $X_t^{x,\epsilon}$ satisfying an SDE of the form: $$ d X_t^{x,\epsilon} = \mu(t,X_t^{x,\epsilon},\epsilon)dt + \Sigma(t,X_t^{x,\epsilon},\epsilon)dW_t, X_0^{x,\epsilon} $$ and such that $ \mathbb{P}\left( \sup_{t \in [0,1],x \in \mathbb{R}^n}|X_t^{x,\epsilon} - f(x)| < \epsilon \right) $ holds with high probability?

Note: Of course, here, we require that $\Sigma(t,x,\epsilon)>0$ to avoid trivial solutions.