How to compute $mathbb{E}[X_{s}^{2}e^{lambda X_{s}}]$ where $X_s$ is a Brownian motion with drift $s$?
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I'm working on a problem and at a certain point I ran into the problem as described in the title. We have that ${W_t,tgeq 0}$ is a Brownian motion and $mathscr{F}_t$ is the corresponding filtration. We have that $mu>0$ is given in the process ${X_t,tgeq 0}$ defined via $X_t:=mu t+W_t$.
I don't want to post the full problem I was solving yet, rather I'd like to know if what I ended up with is even solvable, because if not, I'll know I'm definitely wrong.
As posted in the title, I came at a point where I was left to compute the expectation:
$mathbb{E}[X_{s}^{2}e^{lambda X_{s}}]$
Earlier in the exercise (it consisted of multiple parts) I used the moment generating function for the normal distribution. However, as far as I know, I cannot take the $X_{s}^{2}$ out of the expectation, stopping me from applying the moment generating function.
Is this expectation solvable in a relatively easy way? If not, I'll know I'm wrong and start over.
probability-theory stochastic-processes brownian-motion expected-value
asked yesterday
S. Crim
13212
add a comment |
I'm working on a problem and at a certain point I ran into the problem as described in the title. We have that ${W_t,tgeq 0}$ is a Brownian motion and $mathscr{F}_t$ is the corresponding filtration. We have that $mu>0$ is given in the process ${X_t,tgeq 0}$ defined via $X_t:=mu t+W_t$.
I don't want to post the full problem I was solving yet, rather I'd like to know if what I ended up with is even solvable, because if not, I'll know I'm definitely wrong.
As posted in the title, I came at a point where I was left to compute the expectation:
$mathbb{E}[X_{s}^{2}e^{lambda X_{s}}]$
Earlier in the exercise (it consisted of multiple parts) I used the moment generating function for the normal distribution. However, as far as I know, I cannot take the $X_{s}^{2}$ out of the expectation, stopping me from applying the moment generating function.
Is this expectation solvable in a relatively easy way? If not, I'll know I'm wrong and start over.
probability-theory stochastic-processes brownian-motion expected-value
asked yesterday
S. Crim
13212
6
Differentiate twice $E(e^{lambda X_t})$ with respect to $lambda$.
– Did
yesterday
@Did That works. Thank you very much!
– S. Crim
20 hours ago
add a comment |
I'm working on a problem and at a certain point I ran into the problem as described in the title. We have that ${W_t,tgeq 0}$ is a Brownian motion and $mathscr{F}_t$ is the corresponding filtration. We have that $mu>0$ is given in the process ${X_t,tgeq 0}$ defined via $X_t:=mu t+W_t$.
I don't want to post the full problem I was solving yet, rather I'd like to know if what I ended up with is even solvable, because if not, I'll know I'm definitely wrong.
As posted in the title, I came at a point where I was left to compute the expectation:
$mathbb{E}[X_{s}^{2}e^{lambda X_{s}}]$
Earlier in the exercise (it consisted of multiple parts) I used the moment generating function for the normal distribution. However, as far as I know, I cannot take the $X_{s}^{2}$ out of the expectation, stopping me from applying the moment generating function.
Is this expectation solvable in a relatively easy way? If not, I'll know I'm wrong and start over.
probability-theory stochastic-processes brownian-motion expected-value
asked yesterday
S. Crim
13212
I'm working on a problem and at a certain point I ran into the problem as described in the title. We have that ${W_t,tgeq 0}$ is a Brownian motion and $mathscr{F}_t$ is the corresponding filtration. We have that $mu>0$ is given in the process ${X_t,tgeq 0}$ defined via $X_t:=mu t+W_t$.
I don't want to post the full problem I was solving yet, rather I'd like to know if what I ended up with is even solvable, because if not, I'll know I'm definitely wrong.
As posted in the title, I came at a point where I was left to compute the expectation:
$mathbb{E}[X_{s}^{2}e^{lambda X_{s}}]$
Earlier in the exercise (it consisted of multiple parts) I used the moment generating function for the normal distribution. However, as far as I know, I cannot take the $X_{s}^{2}$ out of the expectation, stopping me from applying the moment generating function.
Is this expectation solvable in a relatively easy way? If not, I'll know I'm wrong and start over.
probability-theory stochastic-processes brownian-motion expected-value
probability-theory stochastic-processes brownian-motion expected-value
asked yesterday
S. Crim
13212
asked yesterday
S. Crim
13212
asked yesterday
S. Crim
13212
asked yesterday
S. Crim
13212
asked yesterday
S. Crim
13212
13212
6
Differentiate twice $E(e^{lambda X_t})$ with respect to $lambda$.
– Did
yesterday
@Did That works. Thank you very much!
– S. Crim
20 hours ago
add a comment |
6
Differentiate twice $E(e^{lambda X_t})$ with respect to $lambda$.
– Did
yesterday
@Did That works. Thank you very much!
– S. Crim
20 hours ago
6
6
Differentiate twice $E(e^{lambda X_t})$ with respect to $lambda$.
– Did
yesterday
Differentiate twice $E(e^{lambda X_t})$ with respect to $lambda$.
– Did
yesterday
@Did That works. Thank you very much!
– S. Crim
20 hours ago
@Did That works. Thank you very much!
– S. Crim
20 hours ago
add a comment |
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6
Differentiate twice $E(e^{lambda X_t})$ with respect to $lambda$.
– Did
yesterday
@Did That works. Thank you very much!
– S. Crim
20 hours ago