Calculating $mathbb E big( sin W(t) big)$, where $W$ is a Wiener process [closed]
I don't know how to calculate $$mathbb E big( sin W(t) big)$$ which is the expectation of Brownian motion. Is it right to use $sin w(t) = frac{e^{iw} - e^{-iw}}{2i}$?
stochastic-calculus brownian-motion expected-value
edited Dec 9 at 23:19
Rodrigo de Azevedo
12.8k41855
asked Dec 9 at 11:30
Hobong
11
closed as off-topic by Did, José Carlos Santos, Davide Giraudo, Cesareo, Leucippus Dec 10 at 1:01
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Did, José Carlos Santos, Davide Giraudo, Cesareo, Leucippus
If this question can be reworded to fit the rules in the help center, please edit the question.
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I don't know how to calculate $$mathbb E big( sin W(t) big)$$ which is the expectation of Brownian motion. Is it right to use $sin w(t) = frac{e^{iw} - e^{-iw}}{2i}$?
stochastic-calculus brownian-motion expected-value
edited Dec 9 at 23:19
Rodrigo de Azevedo
12.8k41855
asked Dec 9 at 11:30
Hobong
11
closed as off-topic by Did, José Carlos Santos, Davide Giraudo, Cesareo, Leucippus Dec 10 at 1:01
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Did, José Carlos Santos, Davide Giraudo, Cesareo, Leucippus
If this question can be reworded to fit the rules in the help center, please edit the question.
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Dec 9 at 11:38
1
Approach 1: Apply Itô's formula to $f(x)=sin(x)$. Approach 2: Write $mathbb{E}sin(W_t)$ in terms of the characteristic function of $W_t$. Approach 3: Use that the distribution of $W_t$ is symmetric and that $x mapsto sin(x)$ is an odd function. So many possibilities...
– saz
Dec 9 at 11:40
I'm sorry I don't know well of this calculus. So can you write down more specifically?
– Hobong
Dec 9 at 11:43
What do you mean by "this calculus"? For the third approach you only need to know the very definition of Brownian motion. Surely you can check that $W_t$ has the same distribution as $-W_t$.
– saz
Dec 9 at 12:15
3
I'm not here to solve your homework problems. If you just copy solutions from other people, you will never understand the mathematics behind it. Even if it may be hard at the begining, you have to try (hard) to solve the problems on your own.
– saz
Dec 9 at 13:24
|
show 4 more comments
I don't know how to calculate $$mathbb E big( sin W(t) big)$$ which is the expectation of Brownian motion. Is it right to use $sin w(t) = frac{e^{iw} - e^{-iw}}{2i}$?
stochastic-calculus brownian-motion expected-value
edited Dec 9 at 23:19
Rodrigo de Azevedo
12.8k41855
asked Dec 9 at 11:30
Hobong
11
I don't know how to calculate $$mathbb E big( sin W(t) big)$$ which is the expectation of Brownian motion. Is it right to use $sin w(t) = frac{e^{iw} - e^{-iw}}{2i}$?
stochastic-calculus brownian-motion expected-value
stochastic-calculus brownian-motion expected-value
edited Dec 9 at 23:19
Rodrigo de Azevedo
12.8k41855
asked Dec 9 at 11:30
Hobong
11
edited Dec 9 at 23:19
Rodrigo de Azevedo
12.8k41855
asked Dec 9 at 11:30
Hobong
11
edited Dec 9 at 23:19
Rodrigo de Azevedo
12.8k41855
edited Dec 9 at 23:19
Rodrigo de Azevedo
12.8k41855
edited Dec 9 at 23:19
Rodrigo de Azevedo
12.8k41855
12.8k41855
asked Dec 9 at 11:30
Hobong
11
asked Dec 9 at 11:30
Hobong
11
asked Dec 9 at 11:30
Hobong
11
11
closed as off-topic by Did, José Carlos Santos, Davide Giraudo, Cesareo, Leucippus Dec 10 at 1:01
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Did, José Carlos Santos, Davide Giraudo, Cesareo, Leucippus
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Did, José Carlos Santos, Davide Giraudo, Cesareo, Leucippus Dec 10 at 1:01
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Did, José Carlos Santos, Davide Giraudo, Cesareo, Leucippus
If this question can be reworded to fit the rules in the help center, please edit the question.
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Dec 9 at 11:38
1
Approach 1: Apply Itô's formula to $f(x)=sin(x)$. Approach 2: Write $mathbb{E}sin(W_t)$ in terms of the characteristic function of $W_t$. Approach 3: Use that the distribution of $W_t$ is symmetric and that $x mapsto sin(x)$ is an odd function. So many possibilities...
– saz
Dec 9 at 11:40
I'm sorry I don't know well of this calculus. So can you write down more specifically?
– Hobong
Dec 9 at 11:43
What do you mean by "this calculus"? For the third approach you only need to know the very definition of Brownian motion. Surely you can check that $W_t$ has the same distribution as $-W_t$.
– saz
Dec 9 at 12:15
3
I'm not here to solve your homework problems. If you just copy solutions from other people, you will never understand the mathematics behind it. Even if it may be hard at the begining, you have to try (hard) to solve the problems on your own.
– saz
Dec 9 at 13:24
|
show 4 more comments
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Dec 9 at 11:38
1
Approach 1: Apply Itô's formula to $f(x)=sin(x)$. Approach 2: Write $mathbb{E}sin(W_t)$ in terms of the characteristic function of $W_t$. Approach 3: Use that the distribution of $W_t$ is symmetric and that $x mapsto sin(x)$ is an odd function. So many possibilities...
– saz
Dec 9 at 11:40
I'm sorry I don't know well of this calculus. So can you write down more specifically?
– Hobong
Dec 9 at 11:43
What do you mean by "this calculus"? For the third approach you only need to know the very definition of Brownian motion. Surely you can check that $W_t$ has the same distribution as $-W_t$.
– saz
Dec 9 at 12:15
3
I'm not here to solve your homework problems. If you just copy solutions from other people, you will never understand the mathematics behind it. Even if it may be hard at the begining, you have to try (hard) to solve the problems on your own.
– saz
Dec 9 at 13:24
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Dec 9 at 11:38
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Dec 9 at 11:38
1
1
Approach 1: Apply Itô's formula to $f(x)=sin(x)$. Approach 2: Write $mathbb{E}sin(W_t)$ in terms of the characteristic function of $W_t$. Approach 3: Use that the distribution of $W_t$ is symmetric and that $x mapsto sin(x)$ is an odd function. So many possibilities...
– saz
Dec 9 at 11:40
Approach 1: Apply Itô's formula to $f(x)=sin(x)$. Approach 2: Write $mathbb{E}sin(W_t)$ in terms of the characteristic function of $W_t$. Approach 3: Use that the distribution of $W_t$ is symmetric and that $x mapsto sin(x)$ is an odd function. So many possibilities...
– saz
Dec 9 at 11:40
I'm sorry I don't know well of this calculus. So can you write down more specifically?
– Hobong
Dec 9 at 11:43
I'm sorry I don't know well of this calculus. So can you write down more specifically?
– Hobong
Dec 9 at 11:43
What do you mean by "this calculus"? For the third approach you only need to know the very definition of Brownian motion. Surely you can check that $W_t$ has the same distribution as $-W_t$.
– saz
Dec 9 at 12:15
What do you mean by "this calculus"? For the third approach you only need to know the very definition of Brownian motion. Surely you can check that $W_t$ has the same distribution as $-W_t$.
– saz
Dec 9 at 12:15
3
3
I'm not here to solve your homework problems. If you just copy solutions from other people, you will never understand the mathematics behind it. Even if it may be hard at the begining, you have to try (hard) to solve the problems on your own.
– saz
Dec 9 at 13:24
I'm not here to solve your homework problems. If you just copy solutions from other people, you will never understand the mathematics behind it. Even if it may be hard at the begining, you have to try (hard) to solve the problems on your own.
– saz
Dec 9 at 13:24
|
show 4 more comments
1 Answer
1
active
oldest
votes
Since $W_t$ equals in distribution $-W_t$ for each $t geq 0$, we have
$$mathbb{E}(sin(W_t)) = mathbb{E}(sin(-W_t)).$$
As $x mapsto sin(x)$ is an odd function, i.e. $sin(-x)=-sin(x)$, we get
$$mathbb{E}(sin(W_t)) = - mathbb{E}(sin(W_t)),$$
and so $mathbb{E}(sin(W_t))=0$ for all $t geq 0$.
answered Dec 9 at 13:21
saz
78.1k758122
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Since $W_t$ equals in distribution $-W_t$ for each $t geq 0$, we have
$$mathbb{E}(sin(W_t)) = mathbb{E}(sin(-W_t)).$$
As $x mapsto sin(x)$ is an odd function, i.e. $sin(-x)=-sin(x)$, we get
$$mathbb{E}(sin(W_t)) = - mathbb{E}(sin(W_t)),$$
and so $mathbb{E}(sin(W_t))=0$ for all $t geq 0$.
answered Dec 9 at 13:21
saz
78.1k758122
add a comment |
Since $W_t$ equals in distribution $-W_t$ for each $t geq 0$, we have
$$mathbb{E}(sin(W_t)) = mathbb{E}(sin(-W_t)).$$
As $x mapsto sin(x)$ is an odd function, i.e. $sin(-x)=-sin(x)$, we get
$$mathbb{E}(sin(W_t)) = - mathbb{E}(sin(W_t)),$$
and so $mathbb{E}(sin(W_t))=0$ for all $t geq 0$.
answered Dec 9 at 13:21
saz
78.1k758122
add a comment |
Since $W_t$ equals in distribution $-W_t$ for each $t geq 0$, we have
$$mathbb{E}(sin(W_t)) = mathbb{E}(sin(-W_t)).$$
As $x mapsto sin(x)$ is an odd function, i.e. $sin(-x)=-sin(x)$, we get
$$mathbb{E}(sin(W_t)) = - mathbb{E}(sin(W_t)),$$
and so $mathbb{E}(sin(W_t))=0$ for all $t geq 0$.
answered Dec 9 at 13:21
saz
78.1k758122
Since $W_t$ equals in distribution $-W_t$ for each $t geq 0$, we have
$$mathbb{E}(sin(W_t)) = mathbb{E}(sin(-W_t)).$$
As $x mapsto sin(x)$ is an odd function, i.e. $sin(-x)=-sin(x)$, we get
$$mathbb{E}(sin(W_t)) = - mathbb{E}(sin(W_t)),$$
and so $mathbb{E}(sin(W_t))=0$ for all $t geq 0$.
answered Dec 9 at 13:21
saz
78.1k758122
answered Dec 9 at 13:21
saz
78.1k758122
answered Dec 9 at 13:21
saz
78.1k758122
answered Dec 9 at 13:21
saz
78.1k758122
78.1k758122
add a comment |
add a comment |
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Dec 9 at 11:38
1
Approach 1: Apply Itô's formula to $f(x)=sin(x)$. Approach 2: Write $mathbb{E}sin(W_t)$ in terms of the characteristic function of $W_t$. Approach 3: Use that the distribution of $W_t$ is symmetric and that $x mapsto sin(x)$ is an odd function. So many possibilities...
– saz
Dec 9 at 11:40
I'm sorry I don't know well of this calculus. So can you write down more specifically?
– Hobong
Dec 9 at 11:43
What do you mean by "this calculus"? For the third approach you only need to know the very definition of Brownian motion. Surely you can check that $W_t$ has the same distribution as $-W_t$.
– saz
Dec 9 at 12:15
3
I'm not here to solve your homework problems. If you just copy solutions from other people, you will never understand the mathematics behind it. Even if it may be hard at the begining, you have to try (hard) to solve the problems on your own.
– saz
Dec 9 at 13:24