Counterexample in Kolmogorov theorem about existence of almost surely continuous modification
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I want to understand this Kolmogorov theorem about existence of almost surely continuous modification:
A process ${xi_t, in[0,T]}$ admits an almost surely continuous modification if there exist constants $a,b,c>0$ such that $$mathbb{E}[|xi_t-xi_s|^a]leq b|t-s|^{1+c}$$ for all $s,tin [0,T]$.
And I have such a question. What if we take a process ${xi_t=e^{w_t^3}, tin[0,T]}$, where $w_t$ is standard wiener process. Then will it be determined $mathbb{E}[|xi_t-xi_s|^a]$ ?
stochastic-processes examples-counterexamples expected-value
edited Dec 3 '18 at 15:08
GNUSupporter 8964民主女神 地下教會
12.8k72445
asked Dec 3 '18 at 14:58
EmeraldEmerald
378
$endgroup$
|
show 1 more comment
$begingroup$
I want to understand this Kolmogorov theorem about existence of almost surely continuous modification:
A process ${xi_t, in[0,T]}$ admits an almost surely continuous modification if there exist constants $a,b,c>0$ such that $$mathbb{E}[|xi_t-xi_s|^a]leq b|t-s|^{1+c}$$ for all $s,tin [0,T]$.
And I have such a question. What if we take a process ${xi_t=e^{w_t^3}, tin[0,T]}$, where $w_t$ is standard wiener process. Then will it be determined $mathbb{E}[|xi_t-xi_s|^a]$ ?
stochastic-processes examples-counterexamples expected-value
edited Dec 3 '18 at 15:08
GNUSupporter 8964民主女神 地下教會
12.8k72445
asked Dec 3 '18 at 14:58
EmeraldEmerald
378
$endgroup$
1
$begingroup$
You are right, this will not be integrable. However, the Kolmogorov theorem it also true with $|cdot|^a$ replaced by $max{|cdot|^a,1}$.
$endgroup$
– zhoraster
Dec 3 '18 at 15:12
$begingroup$
@zhoraster Thank you very much for your answer.
$endgroup$
– Emerald
Dec 3 '18 at 15:14
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@zhoraster The problem is that I still can't understand why we conclude about non-integrability. Why is it true $forall a$?
$endgroup$
– Emerald
Dec 5 '18 at 17:25
$begingroup$
The function $e^{a |x|^3 - b x^2}$ is non-integrable for any $a>0$, $binmathbb{R}$.
$endgroup$
– zhoraster
Dec 5 '18 at 20:57
$begingroup$
@zhoraster Sorry, but how did we move from this $e^{w_t^3}$ to this $e^{a|x|^3−bx^2}$ ?
$endgroup$
– Emerald
Dec 10 '18 at 17:08
|
show 1 more comment
$begingroup$
I want to understand this Kolmogorov theorem about existence of almost surely continuous modification:
A process ${xi_t, in[0,T]}$ admits an almost surely continuous modification if there exist constants $a,b,c>0$ such that $$mathbb{E}[|xi_t-xi_s|^a]leq b|t-s|^{1+c}$$ for all $s,tin [0,T]$.
And I have such a question. What if we take a process ${xi_t=e^{w_t^3}, tin[0,T]}$, where $w_t$ is standard wiener process. Then will it be determined $mathbb{E}[|xi_t-xi_s|^a]$ ?
stochastic-processes examples-counterexamples expected-value
edited Dec 3 '18 at 15:08
GNUSupporter 8964民主女神 地下教會
12.8k72445
asked Dec 3 '18 at 14:58
EmeraldEmerald
378
$endgroup$
I want to understand this Kolmogorov theorem about existence of almost surely continuous modification:
A process ${xi_t, in[0,T]}$ admits an almost surely continuous modification if there exist constants $a,b,c>0$ such that $$mathbb{E}[|xi_t-xi_s|^a]leq b|t-s|^{1+c}$$ for all $s,tin [0,T]$.
And I have such a question. What if we take a process ${xi_t=e^{w_t^3}, tin[0,T]}$, where $w_t$ is standard wiener process. Then will it be determined $mathbb{E}[|xi_t-xi_s|^a]$ ?
stochastic-processes examples-counterexamples expected-value
stochastic-processes examples-counterexamples expected-value
edited Dec 3 '18 at 15:08
GNUSupporter 8964民主女神 地下教會
12.8k72445
asked Dec 3 '18 at 14:58
EmeraldEmerald
378
edited Dec 3 '18 at 15:08
GNUSupporter 8964民主女神 地下教會
12.8k72445
asked Dec 3 '18 at 14:58
EmeraldEmerald
378
edited Dec 3 '18 at 15:08
GNUSupporter 8964民主女神 地下教會
12.8k72445
edited Dec 3 '18 at 15:08
GNUSupporter 8964民主女神 地下教會
12.8k72445
edited Dec 3 '18 at 15:08
GNUSupporter 8964民主女神 地下教會
12.8k72445
12.8k72445
asked Dec 3 '18 at 14:58
EmeraldEmerald
378
asked Dec 3 '18 at 14:58
EmeraldEmerald
378
asked Dec 3 '18 at 14:58
EmeraldEmerald
378
378
1
$begingroup$
You are right, this will not be integrable. However, the Kolmogorov theorem it also true with $|cdot|^a$ replaced by $max{|cdot|^a,1}$.
$endgroup$
– zhoraster
Dec 3 '18 at 15:12
$begingroup$
@zhoraster Thank you very much for your answer.
$endgroup$
– Emerald
Dec 3 '18 at 15:14
$begingroup$
@zhoraster The problem is that I still can't understand why we conclude about non-integrability. Why is it true $forall a$?
$endgroup$
– Emerald
Dec 5 '18 at 17:25
$begingroup$
The function $e^{a |x|^3 - b x^2}$ is non-integrable for any $a>0$, $binmathbb{R}$.
$endgroup$
– zhoraster
Dec 5 '18 at 20:57
$begingroup$
@zhoraster Sorry, but how did we move from this $e^{w_t^3}$ to this $e^{a|x|^3−bx^2}$ ?
$endgroup$
– Emerald
Dec 10 '18 at 17:08
|
show 1 more comment
1
$begingroup$
You are right, this will not be integrable. However, the Kolmogorov theorem it also true with $|cdot|^a$ replaced by $max{|cdot|^a,1}$.
$endgroup$
– zhoraster
Dec 3 '18 at 15:12
$begingroup$
@zhoraster Thank you very much for your answer.
$endgroup$
– Emerald
Dec 3 '18 at 15:14
$begingroup$
@zhoraster The problem is that I still can't understand why we conclude about non-integrability. Why is it true $forall a$?
$endgroup$
– Emerald
Dec 5 '18 at 17:25
$begingroup$
The function $e^{a |x|^3 - b x^2}$ is non-integrable for any $a>0$, $binmathbb{R}$.
$endgroup$
– zhoraster
Dec 5 '18 at 20:57
$begingroup$
@zhoraster Sorry, but how did we move from this $e^{w_t^3}$ to this $e^{a|x|^3−bx^2}$ ?
$endgroup$
– Emerald
Dec 10 '18 at 17:08
1
1
$begingroup$
You are right, this will not be integrable. However, the Kolmogorov theorem it also true with $|cdot|^a$ replaced by $max{|cdot|^a,1}$.
$endgroup$
– zhoraster
Dec 3 '18 at 15:12
$begingroup$
You are right, this will not be integrable. However, the Kolmogorov theorem it also true with $|cdot|^a$ replaced by $max{|cdot|^a,1}$.
$endgroup$
– zhoraster
Dec 3 '18 at 15:12
$begingroup$
@zhoraster Thank you very much for your answer.
$endgroup$
– Emerald
Dec 3 '18 at 15:14
$begingroup$
@zhoraster Thank you very much for your answer.
$endgroup$
– Emerald
Dec 3 '18 at 15:14
$begingroup$
@zhoraster The problem is that I still can't understand why we conclude about non-integrability. Why is it true $forall a$?
$endgroup$
– Emerald
Dec 5 '18 at 17:25
$begingroup$
@zhoraster The problem is that I still can't understand why we conclude about non-integrability. Why is it true $forall a$?
$endgroup$
– Emerald
Dec 5 '18 at 17:25
$begingroup$
The function $e^{a |x|^3 - b x^2}$ is non-integrable for any $a>0$, $binmathbb{R}$.
$endgroup$
– zhoraster
Dec 5 '18 at 20:57
$begingroup$
The function $e^{a |x|^3 - b x^2}$ is non-integrable for any $a>0$, $binmathbb{R}$.
$endgroup$
– zhoraster
Dec 5 '18 at 20:57
$begingroup$
@zhoraster Sorry, but how did we move from this $e^{w_t^3}$ to this $e^{a|x|^3−bx^2}$ ?
$endgroup$
– Emerald
Dec 10 '18 at 17:08
$begingroup$
@zhoraster Sorry, but how did we move from this $e^{w_t^3}$ to this $e^{a|x|^3−bx^2}$ ?
$endgroup$
– Emerald
Dec 10 '18 at 17:08
|
show 1 more comment
0
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$begingroup$
You are right, this will not be integrable. However, the Kolmogorov theorem it also true with $|cdot|^a$ replaced by $max{|cdot|^a,1}$.
$endgroup$
– zhoraster
Dec 3 '18 at 15:12
$begingroup$
@zhoraster Thank you very much for your answer.
$endgroup$
– Emerald
Dec 3 '18 at 15:14
$begingroup$
@zhoraster The problem is that I still can't understand why we conclude about non-integrability. Why is it true $forall a$?
$endgroup$
– Emerald
Dec 5 '18 at 17:25
$begingroup$
The function $e^{a |x|^3 - b x^2}$ is non-integrable for any $a>0$, $binmathbb{R}$.
$endgroup$
– zhoraster
Dec 5 '18 at 20:57
$begingroup$
@zhoraster Sorry, but how did we move from this $e^{w_t^3}$ to this $e^{a|x|^3−bx^2}$ ?
$endgroup$
– Emerald
Dec 10 '18 at 17:08