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Consider two SDEs (stochastic differential equations) as follows:

$$dX_t=b^-(t,X_t)dt+a(t,X_t)dW_t;\quad dY_t=b^+(t,Y_t)dt+a(t,Y_t)dW_t,$$

where $b^-,b^+,a$ are Lipschitz such that $b^-<b^+$ pointwise. For any $x\le y$, it is known that, see e.g. Theorem 1.1 of https://www.sciencedirect.com/science/article/pii/0304414994900558

$$X^x_t\le Y^y_t,\quad \forall t\ge 0, $$

where $X^x, Y^y$ denote the solutions to the above SDEs with initial conditions $X^x_0=x, Y^y_0=y$. My question is as follows: Let $X,Y$ be two arbitrary solution to the above SDEs (note that such $X,Y$ are not unique), does

$$\mathbb P\big[X_s\ge Y_s, \forall 0\le s\le t \big|X_t=z=Y_t\big]=1$$

hold for (almost) all $t$ and $z$?

Consider two SDEs (stochastic differential equations) as follows:

$$dX_t=b^-(t,X_t) \, dt+a(t,X_t) \, dW_t;\quad dY_t = b^+(t,Y_t)\,dt+a(t,Y_t)\,dW_t,$$

where $b^-,b^+,a$ are Lipschitz such that $b^-<b^+$ pointwise. For any $x\le y$, it is known that, see e.g. Theorem 1.1 of https://www.sciencedirect.com/science/article/pii/0304414994900558

$$X^x_t\le Y^y_t,\quad \forall t\ge 0, $$

where $X^x, Y^y$ denote the solutions to the above SDEs with initial conditions $X^x_0=x, Y^y_0=y$. My question is as follows: Let $X,Y$ be two arbitrary solution to the above SDEs (note that such $X,Y$ are not unique), does

$$\mathbb P\big[X_s\ge Y_s, \forall 0\le s\le t \mid X_t=z=Y_t\big]=1$$

hold for (almost) all $t$ and $z$?

Consider two SDEs (stochastic differential equations) as follows:

$$dX_t=b^-(t,X_t)dt+a(t,X_t)dW_t;\quad dY_t=b^+(t,Y_t)dt+a(t,Y_t)dW_t,$$

where $b^-,b^+,a$ are Lipschitz such that $b^-<b^+$ pointwise. For any $x\le y$, it is known that, see e.g. Theorem 1.1 of https://www.sciencedirect.com/science/article/pii/0304414994900558

$$X^x_t\le Y^y_t,\quad \forall t\ge 0, $$

where $X^x, Y^y$ denote the solutions to the above SDEs with initial conditions $X^x_0=x, Y^y_0=y$. My question is as follows: Let $X,Y$ satisfy respectively the above SDEs (note that such $X,Y$ may not be unique), does

$$\mathbb P\big[X_s\ge Y_s, \forall 0\le s\le t \big|X_t=z=Y_t\big]=1$$

hold for (almost) all $t$ and $z$?

Consider two SDEs (stochastic differential equations) as follows:

$$dX_t=b^-(t,X_t)dt+a(t,X_t)dW_t;\quad dY_t=b^+(t,Y_t)dt+a(t,Y_t)dW_t,$$

where $b^-,b^+,a$ are Lipschitz such that $b^-<b^+$ pointwise. For any $x\le y$, it is known that, see e.g. Theorem 1.1 of https://www.sciencedirect.com/science/article/pii/0304414994900558

$$X^x_t\le Y^y_t,\quad \forall t\ge 0, $$

where $X^x, Y^y$ denote the solutions to the above SDEs with initial conditions $X^x_0=x, Y^y_0=y$. My question is as follows: Let $X,Y$ be two arbitrary solution to the above SDEs (note that such $X,Y$ are not unique), does

$$\mathbb P\big[X_s\ge Y_s, \forall 0\le s\le t \big|X_t=z=Y_t\big]=1$$

hold for (almost) all $t$ and $z$?

Consider two SDEs (stochastic differential equations) as follows:

$$dX_t=b^-(t,X_t)dt+a(t,X_t)dW_t;\quad dY_t=b^+(t,Y_t)dt+a(t,Y_t)dW_t,$$

where $b^-,b^+,a$ are Lipschitz such that $b^-<b^+$ pointwise. For any $x\le y$, it is known that, see e.g. Theorem 1.1 of https://www.sciencedirect.com/science/article/pii/0304414994900558

$$X^x_t\le Y^y_t,\quad \forall t\ge 0, $$

where $X^x, Y^y$ denote the solutions to the above SDEs with initial conditions $X^x_0=x, Y^y_0=y$. My question is as follows: Pick an arbitrary $z$ such that there exist $X,Y$ satisfying respectively the above SDEs and $X_t=z=Y_t$ (note that such $X,Y$ may not be unique), does

$$\mathbb P[X_s\ge Y_s,\quad \forall 0\le s\le t]=1$$

hold?

Consider two SDEs (stochastic differential equations) as follows:

$$dX_t=b^-(t,X_t)dt+a(t,X_t)dW_t;\quad dY_t=b^+(t,Y_t)dt+a(t,Y_t)dW_t,$$

where $b^-,b^+,a$ are Lipschitz such that $b^-<b^+$ pointwise. For any $x\le y$, it is known that, see e.g. Theorem 1.1 of https://www.sciencedirect.com/science/article/pii/0304414994900558

$$X^x_t\le Y^y_t,\quad \forall t\ge 0, $$

where $X^x, Y^y$ denote the solutions to the above SDEs with initial conditions $X^x_0=x, Y^y_0=y$. My question is as follows: Let $X,Y$ satisfy respectively the above SDEs (note that such $X,Y$ may not be unique), does

$$\mathbb P\big[X_s\ge Y_s, \forall 0\le s\le t \big|X_t=z=Y_t\big]=1$$

hold for (almost) all $t$ and $z$?