Expectation in a stochastic differential equation The Next CEO of Stack OverflowWhat is Ito's lemma used for in quantitative finance?Question about the stochastic differential equation in the Merton modelComputation of ExpectationSquare of arithmetic brownian motion processBaxter & Rennie HJM: differentiating Ito integralSimple HJM model, differentiating the bond priceStochastic Leibniz ruleStochastic differential equation of a Brownian MotionHow to calculate the product of forward rates with different reset times using Ito's lemma?For an Ito Process, $dlnX neq fracdXX$ and $(dlnX)^2 = (fracdXX)^2$, but $dlnX neq pm fracdXX$
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Expectation in a stochastic differential equation
The Next CEO of Stack OverflowWhat is Ito's lemma used for in quantitative finance?Question about the stochastic differential equation in the Merton modelComputation of ExpectationSquare of arithmetic brownian motion processBaxter & Rennie HJM: differentiating Ito integralSimple HJM model, differentiating the bond priceStochastic Leibniz ruleStochastic differential equation of a Brownian MotionHow to calculate the product of forward rates with different reset times using Ito's lemma?For an Ito Process, $dlnX neq fracdXX$ and $(dlnX)^2 = (fracdXX)^2$, but $dlnX neq pm fracdXX$
$begingroup$
I'm new to stochastic calculus, I want to find the mean of $X_2$ with $X_t = exp(W_t)$, with $W_t$ a Wiener process.
I used Ito's Lemma is arrive at the SDE:
beginalign
d(X_t) = frac12X_t dt + X_t dW_t
endalign
But how can I get the mean of $X_2$?
itos-lemma sde
asked 2 hours ago
VictorVictor
614
$endgroup$
add a comment |
$begingroup$
I'm new to stochastic calculus, I want to find the mean of $X_2$ with $X_t = exp(W_t)$, with $W_t$ a Wiener process.
I used Ito's Lemma is arrive at the SDE:
beginalign
d(X_t) = frac12X_t dt + X_t dW_t
endalign
But how can I get the mean of $X_2$?
itos-lemma sde
asked 2 hours ago
VictorVictor
614
$endgroup$
add a comment |
$begingroup$
I'm new to stochastic calculus, I want to find the mean of $X_2$ with $X_t = exp(W_t)$, with $W_t$ a Wiener process.
I used Ito's Lemma is arrive at the SDE:
beginalign
d(X_t) = frac12X_t dt + X_t dW_t
endalign
But how can I get the mean of $X_2$?
itos-lemma sde
asked 2 hours ago
VictorVictor
614
$endgroup$
I'm new to stochastic calculus, I want to find the mean of $X_2$ with $X_t = exp(W_t)$, with $W_t$ a Wiener process.
I used Ito's Lemma is arrive at the SDE:
beginalign
d(X_t) = frac12X_t dt + X_t dW_t
endalign
But how can I get the mean of $X_2$?
itos-lemma sde
itos-lemma sde
asked 2 hours ago
VictorVictor
614
asked 2 hours ago
VictorVictor
614
edited 1 hour ago
Victor
asked 2 hours ago
VictorVictor
614
asked 2 hours ago
VictorVictor
614
asked 2 hours ago
VictorVictor
614
614
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Assuming you are talking about unconditional expectation, in general you have
$$
mathbbE[X_t] = mathbbE[e^W_t] = e^mathbbE[W_t] + frac12textVar(W_t)
$$
which yields
$$
mathbbE[X_t]= e^frac12 t
$$
Hence,
$$ mathbbE[X_2]= e $$
answered 1 hour ago
RafaelCRafaelC
1263
New contributor
RafaelC is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
I'm quite new to the theory. What is the name of the first equality and under which hypotheses is it true?
$endgroup$
– Victor
1 hour ago
1
$begingroup$
@Victor the first equality comes from the moment-generating function of a normal. Take a look here for more details. In general, $mathbbE[e^X] = e^mu + frac12 sigma^2$ holds whenever $X sim mathcalN(mu, sigma^2)$
$endgroup$
– RafaelC
1 hour ago
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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active
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votes
$begingroup$
Assuming you are talking about unconditional expectation, in general you have
$$
mathbbE[X_t] = mathbbE[e^W_t] = e^mathbbE[W_t] + frac12textVar(W_t)
$$
which yields
$$
mathbbE[X_t]= e^frac12 t
$$
Hence,
$$ mathbbE[X_2]= e $$
answered 1 hour ago
RafaelCRafaelC
1263
New contributor
RafaelC is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
I'm quite new to the theory. What is the name of the first equality and under which hypotheses is it true?
$endgroup$
– Victor
1 hour ago
1
$begingroup$
@Victor the first equality comes from the moment-generating function of a normal. Take a look here for more details. In general, $mathbbE[e^X] = e^mu + frac12 sigma^2$ holds whenever $X sim mathcalN(mu, sigma^2)$
$endgroup$
– RafaelC
1 hour ago
add a comment |
$begingroup$
Assuming you are talking about unconditional expectation, in general you have
$$
mathbbE[X_t] = mathbbE[e^W_t] = e^mathbbE[W_t] + frac12textVar(W_t)
$$
which yields
$$
mathbbE[X_t]= e^frac12 t
$$
Hence,
$$ mathbbE[X_2]= e $$
answered 1 hour ago
RafaelCRafaelC
1263
New contributor
RafaelC is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
I'm quite new to the theory. What is the name of the first equality and under which hypotheses is it true?
$endgroup$
– Victor
1 hour ago
1
$begingroup$
@Victor the first equality comes from the moment-generating function of a normal. Take a look here for more details. In general, $mathbbE[e^X] = e^mu + frac12 sigma^2$ holds whenever $X sim mathcalN(mu, sigma^2)$
$endgroup$
– RafaelC
1 hour ago
add a comment |
$begingroup$
Assuming you are talking about unconditional expectation, in general you have
$$
mathbbE[X_t] = mathbbE[e^W_t] = e^mathbbE[W_t] + frac12textVar(W_t)
$$
which yields
$$
mathbbE[X_t]= e^frac12 t
$$
Hence,
$$ mathbbE[X_2]= e $$
answered 1 hour ago
RafaelCRafaelC
1263
New contributor
RafaelC is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Assuming you are talking about unconditional expectation, in general you have
$$
mathbbE[X_t] = mathbbE[e^W_t] = e^mathbbE[W_t] + frac12textVar(W_t)
$$
which yields
$$
mathbbE[X_t]= e^frac12 t
$$
Hence,
$$ mathbbE[X_2]= e $$
answered 1 hour ago
RafaelCRafaelC
1263
New contributor
RafaelC is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 1 hour ago
RafaelCRafaelC
1263
New contributor
RafaelC is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 1 hour ago
RafaelCRafaelC
1263
answered 1 hour ago
RafaelCRafaelC
1263
1263
New contributor
RafaelC is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
RafaelC is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
RafaelC is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
I'm quite new to the theory. What is the name of the first equality and under which hypotheses is it true?
$endgroup$
– Victor
1 hour ago
1
$begingroup$
@Victor the first equality comes from the moment-generating function of a normal. Take a look here for more details. In general, $mathbbE[e^X] = e^mu + frac12 sigma^2$ holds whenever $X sim mathcalN(mu, sigma^2)$
$endgroup$
– RafaelC
1 hour ago
add a comment |
$begingroup$
I'm quite new to the theory. What is the name of the first equality and under which hypotheses is it true?
$endgroup$
– Victor
1 hour ago
1
$begingroup$
@Victor the first equality comes from the moment-generating function of a normal. Take a look here for more details. In general, $mathbbE[e^X] = e^mu + frac12 sigma^2$ holds whenever $X sim mathcalN(mu, sigma^2)$
$endgroup$
– RafaelC
1 hour ago
$begingroup$
I'm quite new to the theory. What is the name of the first equality and under which hypotheses is it true?
$endgroup$
– Victor
1 hour ago
$begingroup$
I'm quite new to the theory. What is the name of the first equality and under which hypotheses is it true?
$endgroup$
– Victor
1 hour ago
1
1
$begingroup$
@Victor the first equality comes from the moment-generating function of a normal. Take a look here for more details. In general, $mathbbE[e^X] = e^mu + frac12 sigma^2$ holds whenever $X sim mathcalN(mu, sigma^2)$
$endgroup$
– RafaelC
1 hour ago
$begingroup$
@Victor the first equality comes from the moment-generating function of a normal. Take a look here for more details. In general, $mathbbE[e^X] = e^mu + frac12 sigma^2$ holds whenever $X sim mathcalN(mu, sigma^2)$
$endgroup$
– RafaelC
1 hour ago
add a comment |
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