Backward Kolmogorov equation for simple markov process
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The following exercise is from a course on SDE's and I am a bit stumped. Consider the process.
$dX_t=lambdaleft(xi-X_t right)dt+gammasqrt{|X_t|}dB_t$
$lambda,xi,gamma>0$
Find $mathbb{P}^{X_t=x}left( X_t>2right)$.
Now I would start by finding transition probabilities $p(trightarrow T,xrightarrow X_T)$ in the backwards kolmogorov equation $frac{partial p}{partial t }+Lp=0$, where $L$ is the generator for the diffusion. How would I then find the relevant area in the state space? And how to find the solution to the kolmogorov equation for these parameters?
probability-theory stochastic-processes stochastic-calculus markov-process sde
asked Dec 3 '18 at 22:08
thaumoctopusthaumoctopus
9519
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add a comment |
$begingroup$
The following exercise is from a course on SDE's and I am a bit stumped. Consider the process.
$dX_t=lambdaleft(xi-X_t right)dt+gammasqrt{|X_t|}dB_t$
$lambda,xi,gamma>0$
Find $mathbb{P}^{X_t=x}left( X_t>2right)$.
Now I would start by finding transition probabilities $p(trightarrow T,xrightarrow X_T)$ in the backwards kolmogorov equation $frac{partial p}{partial t }+Lp=0$, where $L$ is the generator for the diffusion. How would I then find the relevant area in the state space? And how to find the solution to the kolmogorov equation for these parameters?
probability-theory stochastic-processes stochastic-calculus markov-process sde
asked Dec 3 '18 at 22:08
thaumoctopusthaumoctopus
9519
$endgroup$
$begingroup$
Can this be explicitly calculated? In any case, $X$ can be transformed into a squared Bessel process, which is extensively studied.
$endgroup$
– AddSup
Dec 4 '18 at 8:47
1
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I was looking at another post, which had a link to the following paper: citeseerx.ist.psu.edu/viewdoc/…. In page 4, you can find the expression for the transition density of a CIR process. It should be possible to verify that it satisfies the Kolmogorov equations.
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– AddSup
Dec 7 '18 at 7:10
$begingroup$
That is a great article, love it when it is written so clearly, thank you
$endgroup$
– thaumoctopus
Dec 7 '18 at 12:07
add a comment |
$begingroup$
The following exercise is from a course on SDE's and I am a bit stumped. Consider the process.
$dX_t=lambdaleft(xi-X_t right)dt+gammasqrt{|X_t|}dB_t$
$lambda,xi,gamma>0$
Find $mathbb{P}^{X_t=x}left( X_t>2right)$.
Now I would start by finding transition probabilities $p(trightarrow T,xrightarrow X_T)$ in the backwards kolmogorov equation $frac{partial p}{partial t }+Lp=0$, where $L$ is the generator for the diffusion. How would I then find the relevant area in the state space? And how to find the solution to the kolmogorov equation for these parameters?
probability-theory stochastic-processes stochastic-calculus markov-process sde
asked Dec 3 '18 at 22:08
thaumoctopusthaumoctopus
9519
$endgroup$
The following exercise is from a course on SDE's and I am a bit stumped. Consider the process.
$dX_t=lambdaleft(xi-X_t right)dt+gammasqrt{|X_t|}dB_t$
$lambda,xi,gamma>0$
Find $mathbb{P}^{X_t=x}left( X_t>2right)$.
Now I would start by finding transition probabilities $p(trightarrow T,xrightarrow X_T)$ in the backwards kolmogorov equation $frac{partial p}{partial t }+Lp=0$, where $L$ is the generator for the diffusion. How would I then find the relevant area in the state space? And how to find the solution to the kolmogorov equation for these parameters?
probability-theory stochastic-processes stochastic-calculus markov-process sde
probability-theory stochastic-processes stochastic-calculus markov-process sde
asked Dec 3 '18 at 22:08
thaumoctopusthaumoctopus
9519
asked Dec 3 '18 at 22:08
thaumoctopusthaumoctopus
9519
asked Dec 3 '18 at 22:08
thaumoctopusthaumoctopus
9519
asked Dec 3 '18 at 22:08
thaumoctopusthaumoctopus
9519
asked Dec 3 '18 at 22:08
thaumoctopusthaumoctopus
9519
9519
$begingroup$
Can this be explicitly calculated? In any case, $X$ can be transformed into a squared Bessel process, which is extensively studied.
$endgroup$
– AddSup
Dec 4 '18 at 8:47
1
$begingroup$
I was looking at another post, which had a link to the following paper: citeseerx.ist.psu.edu/viewdoc/…. In page 4, you can find the expression for the transition density of a CIR process. It should be possible to verify that it satisfies the Kolmogorov equations.
$endgroup$
– AddSup
Dec 7 '18 at 7:10
$begingroup$
That is a great article, love it when it is written so clearly, thank you
$endgroup$
– thaumoctopus
Dec 7 '18 at 12:07
add a comment |
$begingroup$
Can this be explicitly calculated? In any case, $X$ can be transformed into a squared Bessel process, which is extensively studied.
$endgroup$
– AddSup
Dec 4 '18 at 8:47
1
$begingroup$
I was looking at another post, which had a link to the following paper: citeseerx.ist.psu.edu/viewdoc/…. In page 4, you can find the expression for the transition density of a CIR process. It should be possible to verify that it satisfies the Kolmogorov equations.
$endgroup$
– AddSup
Dec 7 '18 at 7:10
$begingroup$
That is a great article, love it when it is written so clearly, thank you
$endgroup$
– thaumoctopus
Dec 7 '18 at 12:07
$begingroup$
Can this be explicitly calculated? In any case, $X$ can be transformed into a squared Bessel process, which is extensively studied.
$endgroup$
– AddSup
Dec 4 '18 at 8:47
$begingroup$
Can this be explicitly calculated? In any case, $X$ can be transformed into a squared Bessel process, which is extensively studied.
$endgroup$
– AddSup
Dec 4 '18 at 8:47
1
1
$begingroup$
I was looking at another post, which had a link to the following paper: citeseerx.ist.psu.edu/viewdoc/…. In page 4, you can find the expression for the transition density of a CIR process. It should be possible to verify that it satisfies the Kolmogorov equations.
$endgroup$
– AddSup
Dec 7 '18 at 7:10
$begingroup$
I was looking at another post, which had a link to the following paper: citeseerx.ist.psu.edu/viewdoc/…. In page 4, you can find the expression for the transition density of a CIR process. It should be possible to verify that it satisfies the Kolmogorov equations.
$endgroup$
– AddSup
Dec 7 '18 at 7:10
$begingroup$
That is a great article, love it when it is written so clearly, thank you
$endgroup$
– thaumoctopus
Dec 7 '18 at 12:07
$begingroup$
That is a great article, love it when it is written so clearly, thank you
$endgroup$
– thaumoctopus
Dec 7 '18 at 12:07
add a comment |
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$begingroup$
Can this be explicitly calculated? In any case, $X$ can be transformed into a squared Bessel process, which is extensively studied.
$endgroup$
– AddSup
Dec 4 '18 at 8:47
1
$begingroup$
I was looking at another post, which had a link to the following paper: citeseerx.ist.psu.edu/viewdoc/…. In page 4, you can find the expression for the transition density of a CIR process. It should be possible to verify that it satisfies the Kolmogorov equations.
$endgroup$
– AddSup
Dec 7 '18 at 7:10
$begingroup$
That is a great article, love it when it is written so clearly, thank you
$endgroup$
– thaumoctopus
Dec 7 '18 at 12:07