Mixing
Example of ANY stochastic process (SDE), with reversible distribution
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Can anyone provide an example (as simple as they like) of a process $X_t$ on $mathbb{R}$ solution to $dX=sigma (X,t)dt+b(X,t)dW$. Where $W$ is a Brownian Motion, and $sigma$ and $b$ can be any coefficients.
And there exists a reversible distribution $pi$ for $X$ defined in the usual way through the semi-group.
Any example will do, I don't know one.
Edit :
The answerer is free to choose the SDE, the probability space, and the distribution $pi$. But please can the solution to the SDE $X_t$ take values in $mathbb{R}$.
Please no degenerate answers, like a process which is deterministic.
stochastic-processes stochastic-calculus stochastic-analysis stationary-processes
asked Jan 22 at 17:25
MontyMonty
34613
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add a comment |
$begingroup$
Can anyone provide an example (as simple as they like) of a process $X_t$ on $mathbb{R}$ solution to $dX=sigma (X,t)dt+b(X,t)dW$. Where $W$ is a Brownian Motion, and $sigma$ and $b$ can be any coefficients.
And there exists a reversible distribution $pi$ for $X$ defined in the usual way through the semi-group.
Any example will do, I don't know one.
Edit :
The answerer is free to choose the SDE, the probability space, and the distribution $pi$. But please can the solution to the SDE $X_t$ take values in $mathbb{R}$.
Please no degenerate answers, like a process which is deterministic.
stochastic-processes stochastic-calculus stochastic-analysis stationary-processes
asked Jan 22 at 17:25
MontyMonty
34613
$endgroup$
$begingroup$
Is $pi$ given or can the answerer pick it too?
$endgroup$
– Ian
Jan 22 at 17:34
$begingroup$
@Ian answerer can pick it. Im just looking for a solution to an sde, for which there exists a reversible distribution. Any will do, something as 'close' (in some sense of the word :) ) to Brownian motion as possible would be best. I don't want some degenerate case where everything is deterministic.
$endgroup$
– Monty
Jan 22 at 18:32
add a comment |
$begingroup$
Can anyone provide an example (as simple as they like) of a process $X_t$ on $mathbb{R}$ solution to $dX=sigma (X,t)dt+b(X,t)dW$. Where $W$ is a Brownian Motion, and $sigma$ and $b$ can be any coefficients.
And there exists a reversible distribution $pi$ for $X$ defined in the usual way through the semi-group.
Any example will do, I don't know one.
Edit :
The answerer is free to choose the SDE, the probability space, and the distribution $pi$. But please can the solution to the SDE $X_t$ take values in $mathbb{R}$.
Please no degenerate answers, like a process which is deterministic.
stochastic-processes stochastic-calculus stochastic-analysis stationary-processes
asked Jan 22 at 17:25
MontyMonty
34613
$endgroup$
Can anyone provide an example (as simple as they like) of a process $X_t$ on $mathbb{R}$ solution to $dX=sigma (X,t)dt+b(X,t)dW$. Where $W$ is a Brownian Motion, and $sigma$ and $b$ can be any coefficients.
And there exists a reversible distribution $pi$ for $X$ defined in the usual way through the semi-group.
Any example will do, I don't know one.
Edit :
The answerer is free to choose the SDE, the probability space, and the distribution $pi$. But please can the solution to the SDE $X_t$ take values in $mathbb{R}$.
Please no degenerate answers, like a process which is deterministic.
stochastic-processes stochastic-calculus stochastic-analysis stationary-processes
stochastic-processes stochastic-calculus stochastic-analysis stationary-processes
asked Jan 22 at 17:25
MontyMonty
34613
asked Jan 22 at 17:25
MontyMonty
34613
edited Jan 22 at 18:35
Monty
asked Jan 22 at 17:25
MontyMonty
34613
asked Jan 22 at 17:25
MontyMonty
34613
asked Jan 22 at 17:25
MontyMonty
34613
34613
$begingroup$
Is $pi$ given or can the answerer pick it too?
$endgroup$
– Ian
Jan 22 at 17:34
$begingroup$
@Ian answerer can pick it. Im just looking for a solution to an sde, for which there exists a reversible distribution. Any will do, something as 'close' (in some sense of the word :) ) to Brownian motion as possible would be best. I don't want some degenerate case where everything is deterministic.
$endgroup$
– Monty
Jan 22 at 18:32
add a comment |
$begingroup$
Is $pi$ given or can the answerer pick it too?
$endgroup$
– Ian
Jan 22 at 17:34
$begingroup$
@Ian answerer can pick it. Im just looking for a solution to an sde, for which there exists a reversible distribution. Any will do, something as 'close' (in some sense of the word :) ) to Brownian motion as possible would be best. I don't want some degenerate case where everything is deterministic.
$endgroup$
– Monty
Jan 22 at 18:32
$begingroup$
Is $pi$ given or can the answerer pick it too?
$endgroup$
– Ian
Jan 22 at 17:34
$begingroup$
Is $pi$ given or can the answerer pick it too?
$endgroup$
– Ian
Jan 22 at 17:34
$begingroup$
@Ian answerer can pick it. Im just looking for a solution to an sde, for which there exists a reversible distribution. Any will do, something as 'close' (in some sense of the word :) ) to Brownian motion as possible would be best. I don't want some degenerate case where everything is deterministic.
$endgroup$
– Monty
Jan 22 at 18:32
$begingroup$
@Ian answerer can pick it. Im just looking for a solution to an sde, for which there exists a reversible distribution. Any will do, something as 'close' (in some sense of the word :) ) to Brownian motion as possible would be best. I don't want some degenerate case where everything is deterministic.
$endgroup$
– Monty
Jan 22 at 18:32
add a comment |
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$begingroup$
Is $pi$ given or can the answerer pick it too?
$endgroup$
– Ian
Jan 22 at 17:34
$begingroup$
@Ian answerer can pick it. Im just looking for a solution to an sde, for which there exists a reversible distribution. Any will do, something as 'close' (in some sense of the word :) ) to Brownian motion as possible would be best. I don't want some degenerate case where everything is deterministic.
$endgroup$
– Monty
Jan 22 at 18:32