Mixing
What is $E[X_1]$ where $dX_t=(1-X_t^2)^{-1}dB_t,$ $X_0=1$?
$begingroup$
I have a stochastic equation $$dX_t=(1-X_t^2)^{-1}dB_t,$$ $X_0=1$ and $B$ is a Brownian motion.
I know there exists a strong solution $X$.
How can I compute $$E[X_1]?$$
I thought about trying showing that $int_0^t(1-X_s^2)^{-1}dB_s$ is a martingale, which means I have to show that $E[int_0^t(1-X_s^2)^{-2}ds]<infty$. But I do not know how to show this.
Is this the correct way, and if so, how can I finish it? Does there exist any other way to compute $E[X_1]?$
brownian-motion stochastic-analysis stochastic-pde
asked Jan 25 at 0:47
RavonripRavonrip
898
$endgroup$
add a comment |
$begingroup$
I have a stochastic equation $$dX_t=(1-X_t^2)^{-1}dB_t,$$ $X_0=1$ and $B$ is a Brownian motion.
I know there exists a strong solution $X$.
How can I compute $$E[X_1]?$$
I thought about trying showing that $int_0^t(1-X_s^2)^{-1}dB_s$ is a martingale, which means I have to show that $E[int_0^t(1-X_s^2)^{-2}ds]<infty$. But I do not know how to show this.
Is this the correct way, and if so, how can I finish it? Does there exist any other way to compute $E[X_1]?$
brownian-motion stochastic-analysis stochastic-pde
asked Jan 25 at 0:47
RavonripRavonrip
898
$endgroup$
$begingroup$
May I ask where did you get this SDE? It seems that $X_0=1$ makes this SDE singular at the initial moment.
$endgroup$
– hypernova
Jan 25 at 2:19
add a comment |
$begingroup$
I have a stochastic equation $$dX_t=(1-X_t^2)^{-1}dB_t,$$ $X_0=1$ and $B$ is a Brownian motion.
I know there exists a strong solution $X$.
How can I compute $$E[X_1]?$$
I thought about trying showing that $int_0^t(1-X_s^2)^{-1}dB_s$ is a martingale, which means I have to show that $E[int_0^t(1-X_s^2)^{-2}ds]<infty$. But I do not know how to show this.
Is this the correct way, and if so, how can I finish it? Does there exist any other way to compute $E[X_1]?$
brownian-motion stochastic-analysis stochastic-pde
asked Jan 25 at 0:47
RavonripRavonrip
898
$endgroup$
I have a stochastic equation $$dX_t=(1-X_t^2)^{-1}dB_t,$$ $X_0=1$ and $B$ is a Brownian motion.
I know there exists a strong solution $X$.
How can I compute $$E[X_1]?$$
I thought about trying showing that $int_0^t(1-X_s^2)^{-1}dB_s$ is a martingale, which means I have to show that $E[int_0^t(1-X_s^2)^{-2}ds]<infty$. But I do not know how to show this.
Is this the correct way, and if so, how can I finish it? Does there exist any other way to compute $E[X_1]?$
brownian-motion stochastic-analysis stochastic-pde
brownian-motion stochastic-analysis stochastic-pde
asked Jan 25 at 0:47
RavonripRavonrip
898
asked Jan 25 at 0:47
RavonripRavonrip
898
asked Jan 25 at 0:47
RavonripRavonrip
898
asked Jan 25 at 0:47
RavonripRavonrip
898
asked Jan 25 at 0:47
RavonripRavonrip
898
898
$begingroup$
May I ask where did you get this SDE? It seems that $X_0=1$ makes this SDE singular at the initial moment.
$endgroup$
– hypernova
Jan 25 at 2:19
add a comment |
$begingroup$
May I ask where did you get this SDE? It seems that $X_0=1$ makes this SDE singular at the initial moment.
$endgroup$
– hypernova
Jan 25 at 2:19
$begingroup$
May I ask where did you get this SDE? It seems that $X_0=1$ makes this SDE singular at the initial moment.
$endgroup$
– hypernova
Jan 25 at 2:19
$begingroup$
May I ask where did you get this SDE? It seems that $X_0=1$ makes this SDE singular at the initial moment.
$endgroup$
– hypernova
Jan 25 at 2:19
add a comment |
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$begingroup$
May I ask where did you get this SDE? It seems that $X_0=1$ makes this SDE singular at the initial moment.
$endgroup$
– hypernova
Jan 25 at 2:19