Including random variables in differential equations
$begingroup$
I am having trouble finding information regarding the inclusion of random variables in differential equations. For example I have a simple model describing the growth of some population $X$ over time:
$$
dot{X} = X (r - alpha X)
$$
In this deterministic form we can easily solve to get the equilibrial and explicit solutions with regards to time
My question comes when I want to express random variations in parameter $alpha$, allowing it be represented as a random variable. This means that $X$ also becomes a random variable. Can we derive equations to describe the change in this distribution and its moments over time? Note that here i am not taking about a time varying stochastic process (like brownian walks) but rather static variation in the parameters. Additionally can we include these moments in the differential equation above so that they depend on the moments of the distribution?
$$
dot{X} = X (r - alpha X - E[X])
$$
calculus probability ordinary-differential-equations mathematical-modeling
edited Dec 13 '18 at 18:16
aghostinthefigures
1,2891217
asked Dec 12 '18 at 22:29
TomTom
436
$endgroup$
add a comment |
$begingroup$
I am having trouble finding information regarding the inclusion of random variables in differential equations. For example I have a simple model describing the growth of some population $X$ over time:
$$
dot{X} = X (r - alpha X)
$$
In this deterministic form we can easily solve to get the equilibrial and explicit solutions with regards to time
My question comes when I want to express random variations in parameter $alpha$, allowing it be represented as a random variable. This means that $X$ also becomes a random variable. Can we derive equations to describe the change in this distribution and its moments over time? Note that here i am not taking about a time varying stochastic process (like brownian walks) but rather static variation in the parameters. Additionally can we include these moments in the differential equation above so that they depend on the moments of the distribution?
$$
dot{X} = X (r - alpha X - E[X])
$$
calculus probability ordinary-differential-equations mathematical-modeling
edited Dec 13 '18 at 18:16
aghostinthefigures
1,2891217
asked Dec 12 '18 at 22:29
TomTom
436
$endgroup$
$begingroup$
If your $alpha$ is not varying with time, you can just solve your equation (considering $alpha$ a constant parameter) and, then, evaluate the moments of $X$ (that will be function of time), integrating over $alpha$ weighting by its pdf.
$endgroup$
– N74
Dec 17 '18 at 9:12
add a comment |
$begingroup$
I am having trouble finding information regarding the inclusion of random variables in differential equations. For example I have a simple model describing the growth of some population $X$ over time:
$$
dot{X} = X (r - alpha X)
$$
In this deterministic form we can easily solve to get the equilibrial and explicit solutions with regards to time
My question comes when I want to express random variations in parameter $alpha$, allowing it be represented as a random variable. This means that $X$ also becomes a random variable. Can we derive equations to describe the change in this distribution and its moments over time? Note that here i am not taking about a time varying stochastic process (like brownian walks) but rather static variation in the parameters. Additionally can we include these moments in the differential equation above so that they depend on the moments of the distribution?
$$
dot{X} = X (r - alpha X - E[X])
$$
calculus probability ordinary-differential-equations mathematical-modeling
edited Dec 13 '18 at 18:16
aghostinthefigures
1,2891217
asked Dec 12 '18 at 22:29
TomTom
436
$endgroup$
I am having trouble finding information regarding the inclusion of random variables in differential equations. For example I have a simple model describing the growth of some population $X$ over time:
$$
dot{X} = X (r - alpha X)
$$
In this deterministic form we can easily solve to get the equilibrial and explicit solutions with regards to time
My question comes when I want to express random variations in parameter $alpha$, allowing it be represented as a random variable. This means that $X$ also becomes a random variable. Can we derive equations to describe the change in this distribution and its moments over time? Note that here i am not taking about a time varying stochastic process (like brownian walks) but rather static variation in the parameters. Additionally can we include these moments in the differential equation above so that they depend on the moments of the distribution?
$$
dot{X} = X (r - alpha X - E[X])
$$
calculus probability ordinary-differential-equations mathematical-modeling
calculus probability ordinary-differential-equations mathematical-modeling
edited Dec 13 '18 at 18:16
aghostinthefigures
1,2891217
asked Dec 12 '18 at 22:29
TomTom
436
edited Dec 13 '18 at 18:16
aghostinthefigures
1,2891217
asked Dec 12 '18 at 22:29
TomTom
436
edited Dec 13 '18 at 18:16
aghostinthefigures
1,2891217
edited Dec 13 '18 at 18:16
aghostinthefigures
1,2891217
edited Dec 13 '18 at 18:16
aghostinthefigures
1,2891217
1,2891217
asked Dec 12 '18 at 22:29
TomTom
436
asked Dec 12 '18 at 22:29
TomTom
436
asked Dec 12 '18 at 22:29
TomTom
436
436
$begingroup$
If your $alpha$ is not varying with time, you can just solve your equation (considering $alpha$ a constant parameter) and, then, evaluate the moments of $X$ (that will be function of time), integrating over $alpha$ weighting by its pdf.
$endgroup$
– N74
Dec 17 '18 at 9:12
add a comment |
$begingroup$
If your $alpha$ is not varying with time, you can just solve your equation (considering $alpha$ a constant parameter) and, then, evaluate the moments of $X$ (that will be function of time), integrating over $alpha$ weighting by its pdf.
$endgroup$
– N74
Dec 17 '18 at 9:12
$begingroup$
If your $alpha$ is not varying with time, you can just solve your equation (considering $alpha$ a constant parameter) and, then, evaluate the moments of $X$ (that will be function of time), integrating over $alpha$ weighting by its pdf.
$endgroup$
– N74
Dec 17 '18 at 9:12
$begingroup$
If your $alpha$ is not varying with time, you can just solve your equation (considering $alpha$ a constant parameter) and, then, evaluate the moments of $X$ (that will be function of time), integrating over $alpha$ weighting by its pdf.
$endgroup$
– N74
Dec 17 '18 at 9:12
add a comment |
0
active
oldest
votes
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%2f3037304%2fincluding-random-variables-in-differential-equations%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3037304%2fincluding-random-variables-in-differential-equations%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
$begingroup$
If your $alpha$ is not varying with time, you can just solve your equation (considering $alpha$ a constant parameter) and, then, evaluate the moments of $X$ (that will be function of time), integrating over $alpha$ weighting by its pdf.
$endgroup$
– N74
Dec 17 '18 at 9:12