Find values of $a$ and $lambda$ for which $Z_{0}e^{at+bW_{t}}-lambda t$ is a martingaleFor what value of...
OP Amp not amplifying audio signal
How can saying a song's name be a copyright violation?
What is required to make GPS signals available indoors?
How seriously should I take size and weight limits of hand luggage?
Why was Sir Cadogan fired?
Where would I need my direct neural interface to be implanted?
Are British MPs missing the point, with these 'Indicative Votes'?
Using "tail" to follow a file without displaying the most recent lines
Notepad++ delete until colon for every line with replace all
Is it a bad idea to plug the other end of ESD strap to wall ground?
What reasons are there for a Capitalist to oppose a 100% inheritance tax?
Solving an equation with constraints
How can a day be of 24 hours?
How many wives did king shaul have
How to show a landlord what we have in savings?
Is it "common practice in Fourier transform spectroscopy to multiply the measured interferogram by an apodizing function"? If so, why?
how do we prove that a sum of two periods is still a period?
Why is the sentence "Das ist eine Nase" correct?
What is the opposite of "eschatology"?
How does a dynamic QR code work?
How to compactly explain secondary and tertiary characters without resorting to stereotypes?
How to prevent "they're falling in love" trope
Knowledge-based authentication using Domain-driven Design in C#
Finding the reason behind the value of the integral.
Find values of $a$ and $lambda$ for which $Z_{0}e^{at+bW_{t}}-lambda t$ is a martingale
For what value of $alpha$ is $W_t^{alpha}$ a martingale?d-dimensional Brownian motion and martingalesWhy for a continous local martingale ,on an enlarged probability space, it possibly holds that $M_t=B_{langle M rangle _t}$Is $X_t = tWleft(frac{1}{t}right)$ a Martingale？If not, how could it be a Brownian Motion?Laplace transform of the square of Brownian motion hitting timeFor Brownian Motion $B_t$, and stop time $nu = inf{t: B_t = r}$, how to show $E(e^{-alphanu}) = e^{-|r|sqrt{2alpha}}$?Brownian motion with Lévy’s CharacterizationProof that the exponential martingale is a Brownian MotionExpectation stopped Brownian motion with driftChecking for martingale and Itô's formula
$begingroup$
Find values of $a$ and $lambda$ for which $Z(t)=Z_{0}e^{at+bW_{t}}-lambda t$ is a martingale. In here $W_{t}$ is a Brownian motion and $a,binmathbb{R}$ can be positive as well as negative, since $b$ is derived by substracting one variance from another.
I cannot find a form for which this is a martingale except the form of the exponential martingale $$Ccdot e^{alpha W_{t}-frac{alpha^{2}t}{2}}$$ with $C$ a constant. This would imply that only for the values $$a=frac{-b^{2}}{2}$$ and $$lambda=0$$ $Z(t)$ is a martingale. Is this correct or is there another form for which $Z(t)$ is a martingle, since I have to find a solution for $lambdainmathbb{R}^{+}$, but this is quite an ambiguous expression.
exponential-function brownian-motion martingales expected-value filtrations
asked Mar 18 at 13:53
rs4rs35rs4rs35
447
$endgroup$
add a comment |
$begingroup$
Find values of $a$ and $lambda$ for which $Z(t)=Z_{0}e^{at+bW_{t}}-lambda t$ is a martingale. In here $W_{t}$ is a Brownian motion and $a,binmathbb{R}$ can be positive as well as negative, since $b$ is derived by substracting one variance from another.
I cannot find a form for which this is a martingale except the form of the exponential martingale $$Ccdot e^{alpha W_{t}-frac{alpha^{2}t}{2}}$$ with $C$ a constant. This would imply that only for the values $$a=frac{-b^{2}}{2}$$ and $$lambda=0$$ $Z(t)$ is a martingale. Is this correct or is there another form for which $Z(t)$ is a martingle, since I have to find a solution for $lambdainmathbb{R}^{+}$, but this is quite an ambiguous expression.
exponential-function brownian-motion martingales expected-value filtrations
asked Mar 18 at 13:53
rs4rs35rs4rs35
447
$endgroup$
add a comment |
$begingroup$
Find values of $a$ and $lambda$ for which $Z(t)=Z_{0}e^{at+bW_{t}}-lambda t$ is a martingale. In here $W_{t}$ is a Brownian motion and $a,binmathbb{R}$ can be positive as well as negative, since $b$ is derived by substracting one variance from another.
I cannot find a form for which this is a martingale except the form of the exponential martingale $$Ccdot e^{alpha W_{t}-frac{alpha^{2}t}{2}}$$ with $C$ a constant. This would imply that only for the values $$a=frac{-b^{2}}{2}$$ and $$lambda=0$$ $Z(t)$ is a martingale. Is this correct or is there another form for which $Z(t)$ is a martingle, since I have to find a solution for $lambdainmathbb{R}^{+}$, but this is quite an ambiguous expression.
exponential-function brownian-motion martingales expected-value filtrations
asked Mar 18 at 13:53
rs4rs35rs4rs35
447
$endgroup$
Find values of $a$ and $lambda$ for which $Z(t)=Z_{0}e^{at+bW_{t}}-lambda t$ is a martingale. In here $W_{t}$ is a Brownian motion and $a,binmathbb{R}$ can be positive as well as negative, since $b$ is derived by substracting one variance from another.
I cannot find a form for which this is a martingale except the form of the exponential martingale $$Ccdot e^{alpha W_{t}-frac{alpha^{2}t}{2}}$$ with $C$ a constant. This would imply that only for the values $$a=frac{-b^{2}}{2}$$ and $$lambda=0$$ $Z(t)$ is a martingale. Is this correct or is there another form for which $Z(t)$ is a martingle, since I have to find a solution for $lambdainmathbb{R}^{+}$, but this is quite an ambiguous expression.
exponential-function brownian-motion martingales expected-value filtrations
exponential-function brownian-motion martingales expected-value filtrations
asked Mar 18 at 13:53
rs4rs35rs4rs35
447
asked Mar 18 at 13:53
rs4rs35rs4rs35
447
edited Mar 18 at 14:06
rs4rs35
asked Mar 18 at 13:53
rs4rs35rs4rs35
447
asked Mar 18 at 13:53
rs4rs35rs4rs35
447
asked Mar 18 at 13:53
rs4rs35rs4rs35
447
447
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The mean of $Z(t)$ is $E[Z_0]cdot e^{at+b^2t/2}-lambda t$, which is constant (in $t$) if and only if $a+b^2/2=0=lambda$. This means that the sufficient condition you have found is also necessary, for $Z(t)$ to be a martingale.
answered Mar 18 at 16:54
John DawkinsJohn Dawkins
13.3k11017
$endgroup$
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%2f3152819%2ffind-values-of-a-and-lambda-for-which-z-0eatbw-t-lambda-t-is-a-m%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
$begingroup$
The mean of $Z(t)$ is $E[Z_0]cdot e^{at+b^2t/2}-lambda t$, which is constant (in $t$) if and only if $a+b^2/2=0=lambda$. This means that the sufficient condition you have found is also necessary, for $Z(t)$ to be a martingale.
answered Mar 18 at 16:54
John DawkinsJohn Dawkins
13.3k11017
$endgroup$
add a comment |
$begingroup$
The mean of $Z(t)$ is $E[Z_0]cdot e^{at+b^2t/2}-lambda t$, which is constant (in $t$) if and only if $a+b^2/2=0=lambda$. This means that the sufficient condition you have found is also necessary, for $Z(t)$ to be a martingale.
answered Mar 18 at 16:54
John DawkinsJohn Dawkins
13.3k11017
$endgroup$
add a comment |
$begingroup$
The mean of $Z(t)$ is $E[Z_0]cdot e^{at+b^2t/2}-lambda t$, which is constant (in $t$) if and only if $a+b^2/2=0=lambda$. This means that the sufficient condition you have found is also necessary, for $Z(t)$ to be a martingale.
answered Mar 18 at 16:54
John DawkinsJohn Dawkins
13.3k11017
$endgroup$
The mean of $Z(t)$ is $E[Z_0]cdot e^{at+b^2t/2}-lambda t$, which is constant (in $t$) if and only if $a+b^2/2=0=lambda$. This means that the sufficient condition you have found is also necessary, for $Z(t)$ to be a martingale.
answered Mar 18 at 16:54
John DawkinsJohn Dawkins
13.3k11017
answered Mar 18 at 16:54
John DawkinsJohn Dawkins
13.3k11017
answered Mar 18 at 16:54
John DawkinsJohn Dawkins
13.3k11017
answered Mar 18 at 16:54
John DawkinsJohn Dawkins
13.3k11017
13.3k11017
add a comment |
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%2f3152819%2ffind-values-of-a-and-lambda-for-which-z-0eatbw-t-lambda-t-is-a-m%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
