A question on the defintion of Markov process
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Included in the defintion of the Markov process is the following
$text{ for } x in mathbb{R}^d,s,t ge 0, Gamma in mathcal{B}(mathbb{R}^d)$
$$,
P^x[X_{t+s} in Gamma mid X_s=y]=P^y[X_t in Gamma],P^xX_s^{-1}text{- a.s.} y
$$
Now what is bothering me is that if $X_s$ has a continuous density this conditional probability is not well defined as the set ${X_s=y}$ has measure zero for all $y$. But for any nice stochastic process to be markov, it has to satisfy this condition(and some others which I havent written here). How do I resolve this paradox?
probability-theory stochastic-processes stochastic-calculus markov-process
asked Nov 28 at 9:57
user3503589
1,1961721
add a comment |
up vote
0
down vote
favorite
Included in the defintion of the Markov process is the following
$text{ for } x in mathbb{R}^d,s,t ge 0, Gamma in mathcal{B}(mathbb{R}^d)$
$$,
P^x[X_{t+s} in Gamma mid X_s=y]=P^y[X_t in Gamma],P^xX_s^{-1}text{- a.s.} y
$$
Now what is bothering me is that if $X_s$ has a continuous density this conditional probability is not well defined as the set ${X_s=y}$ has measure zero for all $y$. But for any nice stochastic process to be markov, it has to satisfy this condition(and some others which I havent written here). How do I resolve this paradox?
probability-theory stochastic-processes stochastic-calculus markov-process
asked Nov 28 at 9:57
user3503589
1,1961721
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Included in the defintion of the Markov process is the following
$text{ for } x in mathbb{R}^d,s,t ge 0, Gamma in mathcal{B}(mathbb{R}^d)$
$$,
P^x[X_{t+s} in Gamma mid X_s=y]=P^y[X_t in Gamma],P^xX_s^{-1}text{- a.s.} y
$$
Now what is bothering me is that if $X_s$ has a continuous density this conditional probability is not well defined as the set ${X_s=y}$ has measure zero for all $y$. But for any nice stochastic process to be markov, it has to satisfy this condition(and some others which I havent written here). How do I resolve this paradox?
probability-theory stochastic-processes stochastic-calculus markov-process
asked Nov 28 at 9:57
user3503589
1,1961721
Included in the defintion of the Markov process is the following
$text{ for } x in mathbb{R}^d,s,t ge 0, Gamma in mathcal{B}(mathbb{R}^d)$
$$,
P^x[X_{t+s} in Gamma mid X_s=y]=P^y[X_t in Gamma],P^xX_s^{-1}text{- a.s.} y
$$
Now what is bothering me is that if $X_s$ has a continuous density this conditional probability is not well defined as the set ${X_s=y}$ has measure zero for all $y$. But for any nice stochastic process to be markov, it has to satisfy this condition(and some others which I havent written here). How do I resolve this paradox?
probability-theory stochastic-processes stochastic-calculus markov-process
probability-theory stochastic-processes stochastic-calculus markov-process
asked Nov 28 at 9:57
user3503589
1,1961721
asked Nov 28 at 9:57
user3503589
1,1961721
asked Nov 28 at 9:57
user3503589
1,1961721
asked Nov 28 at 9:57
user3503589
1,1961721
asked Nov 28 at 9:57
user3503589
1,1961721
1,1961721
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
We can write $P^{}[X_{t+s}in Gamma |X_s]$ as $f(X_s)$ for some measurable function $f$. In Markov property LHS is interpreted as $f(y)$.
answered Nov 28 at 10:06
Kavi Rama Murthy
47.2k31854
Thank you for the quick answer. I was wondering if you have some advise on reading more about the makor and strong markov property in a fashion similar to Brownian Motion and stochastic calculus by Karatazas and Shreve. I apologize if this comment is inappropriate
– user3503589
Nov 28 at 10:15
1
Lectures from Markov Processes to Brownian Motion by KL Chung may be on interest to you.
– Kavi Rama Murthy
Nov 28 at 10:18
Thank you very much . I just downloaded it
– user3503589
Nov 28 at 10:25
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
We can write $P^{}[X_{t+s}in Gamma |X_s]$ as $f(X_s)$ for some measurable function $f$. In Markov property LHS is interpreted as $f(y)$.
answered Nov 28 at 10:06
Kavi Rama Murthy
47.2k31854
Thank you for the quick answer. I was wondering if you have some advise on reading more about the makor and strong markov property in a fashion similar to Brownian Motion and stochastic calculus by Karatazas and Shreve. I apologize if this comment is inappropriate
– user3503589
Nov 28 at 10:15
1
Lectures from Markov Processes to Brownian Motion by KL Chung may be on interest to you.
– Kavi Rama Murthy
Nov 28 at 10:18
Thank you very much . I just downloaded it
– user3503589
Nov 28 at 10:25
add a comment |
up vote
1
down vote
accepted
We can write $P^{}[X_{t+s}in Gamma |X_s]$ as $f(X_s)$ for some measurable function $f$. In Markov property LHS is interpreted as $f(y)$.
answered Nov 28 at 10:06
Kavi Rama Murthy
47.2k31854
Thank you for the quick answer. I was wondering if you have some advise on reading more about the makor and strong markov property in a fashion similar to Brownian Motion and stochastic calculus by Karatazas and Shreve. I apologize if this comment is inappropriate
– user3503589
Nov 28 at 10:15
1
Lectures from Markov Processes to Brownian Motion by KL Chung may be on interest to you.
– Kavi Rama Murthy
Nov 28 at 10:18
Thank you very much . I just downloaded it
– user3503589
Nov 28 at 10:25
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
We can write $P^{}[X_{t+s}in Gamma |X_s]$ as $f(X_s)$ for some measurable function $f$. In Markov property LHS is interpreted as $f(y)$.
answered Nov 28 at 10:06
Kavi Rama Murthy
47.2k31854
We can write $P^{}[X_{t+s}in Gamma |X_s]$ as $f(X_s)$ for some measurable function $f$. In Markov property LHS is interpreted as $f(y)$.
answered Nov 28 at 10:06
Kavi Rama Murthy
47.2k31854
answered Nov 28 at 10:06
Kavi Rama Murthy
47.2k31854
answered Nov 28 at 10:06
Kavi Rama Murthy
47.2k31854
answered Nov 28 at 10:06
Kavi Rama Murthy
47.2k31854
47.2k31854
Thank you for the quick answer. I was wondering if you have some advise on reading more about the makor and strong markov property in a fashion similar to Brownian Motion and stochastic calculus by Karatazas and Shreve. I apologize if this comment is inappropriate
– user3503589
Nov 28 at 10:15
1
Lectures from Markov Processes to Brownian Motion by KL Chung may be on interest to you.
– Kavi Rama Murthy
Nov 28 at 10:18
Thank you very much . I just downloaded it
– user3503589
Nov 28 at 10:25
add a comment |
Thank you for the quick answer. I was wondering if you have some advise on reading more about the makor and strong markov property in a fashion similar to Brownian Motion and stochastic calculus by Karatazas and Shreve. I apologize if this comment is inappropriate
– user3503589
Nov 28 at 10:15
1
Lectures from Markov Processes to Brownian Motion by KL Chung may be on interest to you.
– Kavi Rama Murthy
Nov 28 at 10:18
Thank you very much . I just downloaded it
– user3503589
Nov 28 at 10:25
Thank you for the quick answer. I was wondering if you have some advise on reading more about the makor and strong markov property in a fashion similar to Brownian Motion and stochastic calculus by Karatazas and Shreve. I apologize if this comment is inappropriate
– user3503589
Nov 28 at 10:15
Thank you for the quick answer. I was wondering if you have some advise on reading more about the makor and strong markov property in a fashion similar to Brownian Motion and stochastic calculus by Karatazas and Shreve. I apologize if this comment is inappropriate
– user3503589
Nov 28 at 10:15
1
1
Lectures from Markov Processes to Brownian Motion by KL Chung may be on interest to you.
– Kavi Rama Murthy
Nov 28 at 10:18
Lectures from Markov Processes to Brownian Motion by KL Chung may be on interest to you.
– Kavi Rama Murthy
Nov 28 at 10:18
Thank you very much . I just downloaded it
– user3503589
Nov 28 at 10:25
Thank you very much . I just downloaded it
– user3503589
Nov 28 at 10:25
add a comment |
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