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perfectconan
perfectconan
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Assume the time homogeneousinhomogeneous SDE $dX(t)=\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$ has a solution $X(t)$. If we replace $\sigma$ with its absolute value, does the new SDE $dY(t)=\mu(t,Y(t))dt+|\sigma(t,Y(t))|dW(t)$ have a solution $Y(t)$ which has the same distribution as $X(t)$? Here, the initial condition for both SDEs are assumed to have the same distribution.

My guess is yes. And I have a non rigorous proof coming from the numerical solution of this SDE $X(t_{k+1})=X(t_k)+\mu(t_k,X(t_k))(t_{k+1}-t_k)+\sigma(t_k,X(t_k))(W(t_{k+1})-W(t_{k}))$. For any path of $X(t_k)$ such that $X(t_k)=x$, on the corresponding path of $Y(t_k)$ such that $Y(t_k)=x$, the drift parts are the same, and the volality parts should have the same distribution. But these are all purely heuristic. Can someone offer some ideas?

Thank you!

Assume the time homogeneous SDE $dX(t)=\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$ has a solution $X(t)$. If we replace $\sigma$ with its absolute value, does the new SDE $dY(t)=\mu(t,Y(t))dt+|\sigma(t,Y(t))|dW(t)$ have a solution $Y(t)$ which has the same distribution as $X(t)$? Here, the initial condition for both SDEs are assumed to have the same distribution.

My guess is yes. And I have a non rigorous proof coming from the numerical solution of this SDE $X(t_{k+1})=X(t_k)+\mu(t_k,X(t_k))(t_{k+1}-t_k)+\sigma(t_k,X(t_k))(W(t_{k+1})-W(t_{k}))$. For any path of $X(t_k)$ such that $X(t_k)=x$, on the corresponding path of $Y(t_k)$ such that $Y(t_k)=x$, the drift parts are the same, and the volality parts should have the same distribution. But these are all purely heuristic. Can someone offer some ideas?

Thank you!

Assume the time inhomogeneous SDE $dX(t)=\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$ has a solution $X(t)$. If we replace $\sigma$ with its absolute value, does the new SDE $dY(t)=\mu(t,Y(t))dt+|\sigma(t,Y(t))|dW(t)$ have a solution $Y(t)$ which has the same distribution as $X(t)$? Here, the initial condition for both SDEs are assumed to have the same distribution.

My guess is yes. And I have a non rigorous proof coming from the numerical solution of this SDE $X(t_{k+1})=X(t_k)+\mu(t_k,X(t_k))(t_{k+1}-t_k)+\sigma(t_k,X(t_k))(W(t_{k+1})-W(t_{k}))$. For any path of $X(t_k)$ such that $X(t_k)=x$, on the corresponding path of $Y(t_k)$ such that $Y(t_k)=x$, the drift parts are the same, and the volality parts should have the same distribution. But these are all purely heuristic. Can someone offer some ideas?

Thank you!

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perfectconan
perfectconan
  • 23
  • 4

Assume the time homogeneous SDE $dX(t)=\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$ has a solution $X(t)$. If we replace $\sigma$ with its absolute value, does the new SDE $dY(t)=\mu(t,Y(t))dt+|\sigma(t,Y(t))|dW(t)$ have a solution $Y(t)$ which has the same distribution as $X(t)$? Here, the initial condition for both SDEs are assumed to have the same distribution.

My guess is yes. And I have a non rigorous proof coming from the numerical solution of this SDE $X(t_{k+1})=X(t_k)+\mu(t_k,X(t_k))(t_{k+1}-t_k)+\sigma(t_k,X(t_k))(W(t_{k+1})-W(t_{k}))$. OnFor any path of $X(t_k)$ such that $X(t_k)=x$, on the corresponding path whereof $X(t_k)=Y(t_k)$$Y(t_k)$ such that $Y(t_k)=x$, the drift parts are the same, and the volality parts should have the same distribution. But these are all purely heuristic. Can someone offer some ideas?

Thank you!

Assume the time homogeneous SDE $dX(t)=\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$ has a solution $X(t)$. If we replace $\sigma$ with its absolute value, does the new SDE $dY(t)=\mu(t,Y(t))dt+|\sigma(t,Y(t))|dW(t)$ have a solution $Y(t)$ which has the same distribution as $X(t)$? Here, the initial condition for both SDEs are assumed to have the same distribution.

My guess is yes. And I have a non rigorous proof coming from the numerical solution of this SDE $X(t_{k+1})=X(t_k)+\mu(t_k,X(t_k))(t_{k+1}-t_k)+\sigma(t_k,X(t_k))(W(t_{k+1})-W(t_{k}))$. On the path where $X(t_k)=Y(t_k)$, the drift parts are the same, and the volality parts should have the same distribution. But these are all purely heuristic. Can someone offer some ideas?

Thank you!

Assume the time homogeneous SDE $dX(t)=\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$ has a solution $X(t)$. If we replace $\sigma$ with its absolute value, does the new SDE $dY(t)=\mu(t,Y(t))dt+|\sigma(t,Y(t))|dW(t)$ have a solution $Y(t)$ which has the same distribution as $X(t)$? Here, the initial condition for both SDEs are assumed to have the same distribution.

My guess is yes. And I have a non rigorous proof coming from the numerical solution of this SDE $X(t_{k+1})=X(t_k)+\mu(t_k,X(t_k))(t_{k+1}-t_k)+\sigma(t_k,X(t_k))(W(t_{k+1})-W(t_{k}))$. For any path of $X(t_k)$ such that $X(t_k)=x$, on the corresponding path of $Y(t_k)$ such that $Y(t_k)=x$, the drift parts are the same, and the volality parts should have the same distribution. But these are all purely heuristic. Can someone offer some ideas?

Thank you!

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perfectconan
perfectconan
  • 23
  • 4

change the sign of volatility

Assume the time homogeneous SDE $dX(t)=\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$ has a solution $X(t)$. If we replace $\sigma$ with its absolute value, does the new SDE $dY(t)=\mu(t,Y(t))dt+|\sigma(t,Y(t))|dW(t)$ have a solution $Y(t)$ which has the same distribution as $X(t)$? Here, the initial condition for both SDEs are assumed to have the same distribution.

My guess is yes. And I have a non rigorous proof coming from the numerical solution of this SDE $X(t_{k+1})=X(t_k)+\mu(t_k,X(t_k))(t_{k+1}-t_k)+\sigma(t_k,X(t_k))(W(t_{k+1})-W(t_{k}))$. On the path where $X(t_k)=Y(t_k)$, the drift parts are the same, and the volality parts should have the same distribution. But these are all purely heuristic. Can someone offer some ideas?

Thank you!