Terminology Filtered probability spaceSorting through “algebra of random variables,” vs. “probability space,” etcterminology of differencesThe difference between “identically distributed” and having “common probability space”Filtered Probability Space UnderstandingImpossible Events, Probability Zero Events, Change of Sample Space, Invariant, Canonical Sample Space?Probability measure, probability density function or probability event ? Are they different?How does the sample space remain constant in filteredIntuition behind a market uncertainty represented by a filtered complete probability space?A setup of filtered probability setup in canonical spaceWhat is the difference between a probability mass function and discrete probability distribution?
Are ETF trackers fundamentally better than individual stocks?
How to make healing in an exploration game interesting
If I am holding an item before I cast Blink, will it move with me through the Ethereal Plane?
I am confused as to how the inverse of a certain function is found.
Is it insecure to send a password in a `curl` command?
Why does overlay work only on the first tcolorbox?
Why do passenger jet manufacturers design their planes with stall prevention systems?
Brexit - No Deal Rejection
Print a physical multiplication table
A single argument pattern definition applies to multiple-argument patterns?
Is "upgrade" the right word to use in this context?
Did Ender ever learn that he killed Stilson and/or Bonzo?
As a new Ubuntu desktop 18.04 LTS user, do I need to use ufw for a firewall or is iptables sufficient?
Is there a symmetric-key algorithm which we can use for creating a signature?
What is "focus distance lower/upper" and how is it different from depth of field?
How to get the n-th line after a grepped one?
et qui - how do you really understand that kind of phraseology?
Aluminum electrolytic or ceramic capacitors for linear regulator input and output?
What's the meaning of a knight fighting a snail in medieval book illustrations?
How could a scammer know the apps on my phone / iTunes account?
Is it normal that my co-workers at a fitness company criticize my food choices?
What is the relationship between relativity and the Doppler effect?
Happy pi day, everyone!
What options are left, if Britain cannot decide?
Terminology Filtered probability space
Sorting through “algebra of random variables,” vs. “probability space,” etcterminology of differencesThe difference between “identically distributed” and having “common probability space”Filtered Probability Space UnderstandingImpossible Events, Probability Zero Events, Change of Sample Space, Invariant, Canonical Sample Space?Probability measure, probability density function or probability event ? Are they different?How does the sample space remain constant in filteredIntuition behind a market uncertainty represented by a filtered complete probability space?A setup of filtered probability setup in canonical spaceWhat is the difference between a probability mass function and discrete probability distribution?
$begingroup$
What is the difference between a "filtered probability space" and a "Stochastic base"? I see both terms used but it is not clear to be me if there is a difference. Maybe one is complete but the other need not be?
probability probability-theory stochastic-processes soft-question terminology
edited Mar 12 at 11:01
Bernard
123k741116
asked Mar 12 at 10:06
AIM_BLBAIM_BLB
2,5172819
$endgroup$
add a comment |
$begingroup$
What is the difference between a "filtered probability space" and a "Stochastic base"? I see both terms used but it is not clear to be me if there is a difference. Maybe one is complete but the other need not be?
probability probability-theory stochastic-processes soft-question terminology
edited Mar 12 at 11:01
Bernard
123k741116
asked Mar 12 at 10:06
AIM_BLBAIM_BLB
2,5172819
$endgroup$
add a comment |
$begingroup$
What is the difference between a "filtered probability space" and a "Stochastic base"? I see both terms used but it is not clear to be me if there is a difference. Maybe one is complete but the other need not be?
probability probability-theory stochastic-processes soft-question terminology
edited Mar 12 at 11:01
Bernard
123k741116
asked Mar 12 at 10:06
AIM_BLBAIM_BLB
2,5172819
$endgroup$
What is the difference between a "filtered probability space" and a "Stochastic base"? I see both terms used but it is not clear to be me if there is a difference. Maybe one is complete but the other need not be?
probability probability-theory stochastic-processes soft-question terminology
probability probability-theory stochastic-processes soft-question terminology
edited Mar 12 at 11:01
Bernard
123k741116
asked Mar 12 at 10:06
AIM_BLBAIM_BLB
2,5172819
edited Mar 12 at 11:01
Bernard
123k741116
asked Mar 12 at 10:06
AIM_BLBAIM_BLB
2,5172819
edited Mar 12 at 11:01
Bernard
123k741116
edited Mar 12 at 11:01
Bernard
123k741116
edited Mar 12 at 11:01
Bernard
123k741116
123k741116
asked Mar 12 at 10:06
AIM_BLBAIM_BLB
2,5172819
asked Mar 12 at 10:06
AIM_BLBAIM_BLB
2,5172819
asked Mar 12 at 10:06
AIM_BLBAIM_BLB
2,5172819
2,5172819
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
A filtered probability space is a probability space $(Omega,mathcal F,mathbb P)$ equipped with a filtration $(mathcal F_t)_tinmathbb R_+$.
Some authors define a stochastic basis as a filtered probability space whose filtration is right-continuous, that is
$$
forall tinmathbb R_+,quad F_t=cap_s>tmathcal F_s.
$$
(see for instance Jacod-Shiryaev:Limit theorems for stochastic processes, for a definition)
Other authors might define a stochastic basis as a filtered probability space whose filtration is right-continuous and complete, that is for all $tinmathbb R_+$, $F_t=cap_s>tmathcal F_s$ and $mathcal F_t$ contains all the $mathbb P$-null sets of $mathcal F$.
As far as I am concerned, I never use the term "stochastic basis". I rather say "probability space equipped with a right-continuous filtration" for instance. It's longer, yes, but at least it is unequivocal.
answered Mar 12 at 11:25
WillWill
1663
New contributor
Will is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
Awesome! I always wondered what it meant. Thanks :).
$endgroup$
– AIM_BLB
Mar 12 at 11:27
$begingroup$
Are you sure that there is such a strong consensus about the use of the term stochastic basis? You're right that Jacod and Shiryaev define it like that - but one should note that in their terminology a filtration is right-continuous by definition. I've also seen people use the term stochastic basis for filtered probability spaces with a right-continuous and complete filtration. My advice would be to always check the individual terminology of the pertaining author. Besides, I whole-heartedly agree with your suggestion not to use the term at all and instead be explicit and precise.
$endgroup$
– Mars Plastic
Mar 12 at 12:52
1
$begingroup$
You're right. People I discuss with mostly use the term "stochastic basis" for right-continuous filtration and I have already heard "complete stochastic basis" when they want the filtration to be complete. But indeed it does not seem to be a strong consensus. I'll edit my answer to highlight this ambiguity.
$endgroup$
– Will
Mar 12 at 13:01
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3144896%2fterminology-filtered-probability-space%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
A filtered probability space is a probability space $(Omega,mathcal F,mathbb P)$ equipped with a filtration $(mathcal F_t)_tinmathbb R_+$.
Some authors define a stochastic basis as a filtered probability space whose filtration is right-continuous, that is
$$
forall tinmathbb R_+,quad F_t=cap_s>tmathcal F_s.
$$
(see for instance Jacod-Shiryaev:Limit theorems for stochastic processes, for a definition)
Other authors might define a stochastic basis as a filtered probability space whose filtration is right-continuous and complete, that is for all $tinmathbb R_+$, $F_t=cap_s>tmathcal F_s$ and $mathcal F_t$ contains all the $mathbb P$-null sets of $mathcal F$.
As far as I am concerned, I never use the term "stochastic basis". I rather say "probability space equipped with a right-continuous filtration" for instance. It's longer, yes, but at least it is unequivocal.
answered Mar 12 at 11:25
WillWill
1663
New contributor
Will is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
Awesome! I always wondered what it meant. Thanks :).
$endgroup$
– AIM_BLB
Mar 12 at 11:27
$begingroup$
Are you sure that there is such a strong consensus about the use of the term stochastic basis? You're right that Jacod and Shiryaev define it like that - but one should note that in their terminology a filtration is right-continuous by definition. I've also seen people use the term stochastic basis for filtered probability spaces with a right-continuous and complete filtration. My advice would be to always check the individual terminology of the pertaining author. Besides, I whole-heartedly agree with your suggestion not to use the term at all and instead be explicit and precise.
$endgroup$
– Mars Plastic
Mar 12 at 12:52
1
$begingroup$
You're right. People I discuss with mostly use the term "stochastic basis" for right-continuous filtration and I have already heard "complete stochastic basis" when they want the filtration to be complete. But indeed it does not seem to be a strong consensus. I'll edit my answer to highlight this ambiguity.
$endgroup$
– Will
Mar 12 at 13:01
add a comment |
$begingroup$
A filtered probability space is a probability space $(Omega,mathcal F,mathbb P)$ equipped with a filtration $(mathcal F_t)_tinmathbb R_+$.
Some authors define a stochastic basis as a filtered probability space whose filtration is right-continuous, that is
$$
forall tinmathbb R_+,quad F_t=cap_s>tmathcal F_s.
$$
(see for instance Jacod-Shiryaev:Limit theorems for stochastic processes, for a definition)
Other authors might define a stochastic basis as a filtered probability space whose filtration is right-continuous and complete, that is for all $tinmathbb R_+$, $F_t=cap_s>tmathcal F_s$ and $mathcal F_t$ contains all the $mathbb P$-null sets of $mathcal F$.
As far as I am concerned, I never use the term "stochastic basis". I rather say "probability space equipped with a right-continuous filtration" for instance. It's longer, yes, but at least it is unequivocal.
answered Mar 12 at 11:25
WillWill
1663
New contributor
Will is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
Awesome! I always wondered what it meant. Thanks :).
$endgroup$
– AIM_BLB
Mar 12 at 11:27
$begingroup$
Are you sure that there is such a strong consensus about the use of the term stochastic basis? You're right that Jacod and Shiryaev define it like that - but one should note that in their terminology a filtration is right-continuous by definition. I've also seen people use the term stochastic basis for filtered probability spaces with a right-continuous and complete filtration. My advice would be to always check the individual terminology of the pertaining author. Besides, I whole-heartedly agree with your suggestion not to use the term at all and instead be explicit and precise.
$endgroup$
– Mars Plastic
Mar 12 at 12:52
1
$begingroup$
You're right. People I discuss with mostly use the term "stochastic basis" for right-continuous filtration and I have already heard "complete stochastic basis" when they want the filtration to be complete. But indeed it does not seem to be a strong consensus. I'll edit my answer to highlight this ambiguity.
$endgroup$
– Will
Mar 12 at 13:01
add a comment |
$begingroup$
A filtered probability space is a probability space $(Omega,mathcal F,mathbb P)$ equipped with a filtration $(mathcal F_t)_tinmathbb R_+$.
Some authors define a stochastic basis as a filtered probability space whose filtration is right-continuous, that is
$$
forall tinmathbb R_+,quad F_t=cap_s>tmathcal F_s.
$$
(see for instance Jacod-Shiryaev:Limit theorems for stochastic processes, for a definition)
Other authors might define a stochastic basis as a filtered probability space whose filtration is right-continuous and complete, that is for all $tinmathbb R_+$, $F_t=cap_s>tmathcal F_s$ and $mathcal F_t$ contains all the $mathbb P$-null sets of $mathcal F$.
As far as I am concerned, I never use the term "stochastic basis". I rather say "probability space equipped with a right-continuous filtration" for instance. It's longer, yes, but at least it is unequivocal.
answered Mar 12 at 11:25
WillWill
1663
New contributor
Will is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
A filtered probability space is a probability space $(Omega,mathcal F,mathbb P)$ equipped with a filtration $(mathcal F_t)_tinmathbb R_+$.
Some authors define a stochastic basis as a filtered probability space whose filtration is right-continuous, that is
$$
forall tinmathbb R_+,quad F_t=cap_s>tmathcal F_s.
$$
(see for instance Jacod-Shiryaev:Limit theorems for stochastic processes, for a definition)
Other authors might define a stochastic basis as a filtered probability space whose filtration is right-continuous and complete, that is for all $tinmathbb R_+$, $F_t=cap_s>tmathcal F_s$ and $mathcal F_t$ contains all the $mathbb P$-null sets of $mathcal F$.
As far as I am concerned, I never use the term "stochastic basis". I rather say "probability space equipped with a right-continuous filtration" for instance. It's longer, yes, but at least it is unequivocal.
answered Mar 12 at 11:25
WillWill
1663
New contributor
Will is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Mar 12 at 13:05
answered Mar 12 at 11:25
WillWill
1663
New contributor
Will is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered Mar 12 at 11:25
WillWill
1663
answered Mar 12 at 11:25
WillWill
1663
1663
New contributor
Will is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Will is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Will is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
Awesome! I always wondered what it meant. Thanks :).
$endgroup$
– AIM_BLB
Mar 12 at 11:27
$begingroup$
Are you sure that there is such a strong consensus about the use of the term stochastic basis? You're right that Jacod and Shiryaev define it like that - but one should note that in their terminology a filtration is right-continuous by definition. I've also seen people use the term stochastic basis for filtered probability spaces with a right-continuous and complete filtration. My advice would be to always check the individual terminology of the pertaining author. Besides, I whole-heartedly agree with your suggestion not to use the term at all and instead be explicit and precise.
$endgroup$
– Mars Plastic
Mar 12 at 12:52
1
$begingroup$
You're right. People I discuss with mostly use the term "stochastic basis" for right-continuous filtration and I have already heard "complete stochastic basis" when they want the filtration to be complete. But indeed it does not seem to be a strong consensus. I'll edit my answer to highlight this ambiguity.
$endgroup$
– Will
Mar 12 at 13:01
add a comment |
$begingroup$
Awesome! I always wondered what it meant. Thanks :).
$endgroup$
– AIM_BLB
Mar 12 at 11:27
$begingroup$
Are you sure that there is such a strong consensus about the use of the term stochastic basis? You're right that Jacod and Shiryaev define it like that - but one should note that in their terminology a filtration is right-continuous by definition. I've also seen people use the term stochastic basis for filtered probability spaces with a right-continuous and complete filtration. My advice would be to always check the individual terminology of the pertaining author. Besides, I whole-heartedly agree with your suggestion not to use the term at all and instead be explicit and precise.
$endgroup$
– Mars Plastic
Mar 12 at 12:52
1
$begingroup$
You're right. People I discuss with mostly use the term "stochastic basis" for right-continuous filtration and I have already heard "complete stochastic basis" when they want the filtration to be complete. But indeed it does not seem to be a strong consensus. I'll edit my answer to highlight this ambiguity.
$endgroup$
– Will
Mar 12 at 13:01
$begingroup$
Awesome! I always wondered what it meant. Thanks :).
$endgroup$
– AIM_BLB
Mar 12 at 11:27
$begingroup$
Awesome! I always wondered what it meant. Thanks :).
$endgroup$
– AIM_BLB
Mar 12 at 11:27
$begingroup$
Are you sure that there is such a strong consensus about the use of the term stochastic basis? You're right that Jacod and Shiryaev define it like that - but one should note that in their terminology a filtration is right-continuous by definition. I've also seen people use the term stochastic basis for filtered probability spaces with a right-continuous and complete filtration. My advice would be to always check the individual terminology of the pertaining author. Besides, I whole-heartedly agree with your suggestion not to use the term at all and instead be explicit and precise.
$endgroup$
– Mars Plastic
Mar 12 at 12:52
$begingroup$
Are you sure that there is such a strong consensus about the use of the term stochastic basis? You're right that Jacod and Shiryaev define it like that - but one should note that in their terminology a filtration is right-continuous by definition. I've also seen people use the term stochastic basis for filtered probability spaces with a right-continuous and complete filtration. My advice would be to always check the individual terminology of the pertaining author. Besides, I whole-heartedly agree with your suggestion not to use the term at all and instead be explicit and precise.
$endgroup$
– Mars Plastic
Mar 12 at 12:52
1
1
$begingroup$
You're right. People I discuss with mostly use the term "stochastic basis" for right-continuous filtration and I have already heard "complete stochastic basis" when they want the filtration to be complete. But indeed it does not seem to be a strong consensus. I'll edit my answer to highlight this ambiguity.
$endgroup$
– Will
Mar 12 at 13:01
$begingroup$
You're right. People I discuss with mostly use the term "stochastic basis" for right-continuous filtration and I have already heard "complete stochastic basis" when they want the filtration to be complete. But indeed it does not seem to be a strong consensus. I'll edit my answer to highlight this ambiguity.
$endgroup$
– Will
Mar 12 at 13:01
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3144896%2fterminology-filtered-probability-space%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
