The property of coercivity in stochastic analysis
$begingroup$
Given an SDE
$$ dX_{t}=b(t,X_{t})dt+sigma (t,X_{t}) dW_{t} $$
With $W$ the Wiener process. I have seen some results that under some assumptions on the coefficients $b,sigma$ such as :
i) Lipshitz
ii) Linear growth
iii) $sigma$ bounded
iv) $b$ coercive that is $b(t,X_{t})cdot X_{t} leq -alpha |X_{t}|^{2} $ where $alpha>0$
Then using krylov bogoliubov theorem we can find an invariant measure. My question is regarding the intuition behind the coercivity condition, I can see that the action of $dW_{t}$ is not really 'invariant' since it is a b.m and hence has Guassian distribution, therefore we need the drift term i.e the $dt$ to counteract the non-invarience of the b.m? What is it about coercivity that does this?
stochastic-calculus stochastic-analysis sde stationary-processes coercive
asked Dec 9 '18 at 14:09
MontyMonty
33613
$endgroup$
add a comment |
$begingroup$
Given an SDE
$$ dX_{t}=b(t,X_{t})dt+sigma (t,X_{t}) dW_{t} $$
With $W$ the Wiener process. I have seen some results that under some assumptions on the coefficients $b,sigma$ such as :
i) Lipshitz
ii) Linear growth
iii) $sigma$ bounded
iv) $b$ coercive that is $b(t,X_{t})cdot X_{t} leq -alpha |X_{t}|^{2} $ where $alpha>0$
Then using krylov bogoliubov theorem we can find an invariant measure. My question is regarding the intuition behind the coercivity condition, I can see that the action of $dW_{t}$ is not really 'invariant' since it is a b.m and hence has Guassian distribution, therefore we need the drift term i.e the $dt$ to counteract the non-invarience of the b.m? What is it about coercivity that does this?
stochastic-calculus stochastic-analysis sde stationary-processes coercive
asked Dec 9 '18 at 14:09
MontyMonty
33613
$endgroup$
add a comment |
$begingroup$
Given an SDE
$$ dX_{t}=b(t,X_{t})dt+sigma (t,X_{t}) dW_{t} $$
With $W$ the Wiener process. I have seen some results that under some assumptions on the coefficients $b,sigma$ such as :
i) Lipshitz
ii) Linear growth
iii) $sigma$ bounded
iv) $b$ coercive that is $b(t,X_{t})cdot X_{t} leq -alpha |X_{t}|^{2} $ where $alpha>0$
Then using krylov bogoliubov theorem we can find an invariant measure. My question is regarding the intuition behind the coercivity condition, I can see that the action of $dW_{t}$ is not really 'invariant' since it is a b.m and hence has Guassian distribution, therefore we need the drift term i.e the $dt$ to counteract the non-invarience of the b.m? What is it about coercivity that does this?
stochastic-calculus stochastic-analysis sde stationary-processes coercive
asked Dec 9 '18 at 14:09
MontyMonty
33613
$endgroup$
Given an SDE
$$ dX_{t}=b(t,X_{t})dt+sigma (t,X_{t}) dW_{t} $$
With $W$ the Wiener process. I have seen some results that under some assumptions on the coefficients $b,sigma$ such as :
i) Lipshitz
ii) Linear growth
iii) $sigma$ bounded
iv) $b$ coercive that is $b(t,X_{t})cdot X_{t} leq -alpha |X_{t}|^{2} $ where $alpha>0$
Then using krylov bogoliubov theorem we can find an invariant measure. My question is regarding the intuition behind the coercivity condition, I can see that the action of $dW_{t}$ is not really 'invariant' since it is a b.m and hence has Guassian distribution, therefore we need the drift term i.e the $dt$ to counteract the non-invarience of the b.m? What is it about coercivity that does this?
stochastic-calculus stochastic-analysis sde stationary-processes coercive
stochastic-calculus stochastic-analysis sde stationary-processes coercive
asked Dec 9 '18 at 14:09
MontyMonty
33613
asked Dec 9 '18 at 14:09
MontyMonty
33613
edited Dec 11 '18 at 18:05
Monty
asked Dec 9 '18 at 14:09
MontyMonty
33613
asked Dec 9 '18 at 14:09
MontyMonty
33613
asked Dec 9 '18 at 14:09
MontyMonty
33613
33613
add a comment |
add a comment |
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