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Brownian motion question for $mathbbEleft[W_s W_t^2 right]$ and $mathbbEleft[W_s^2 W_t^2 right]$

0

$begingroup$

Let $W_t:t ge 0$ be a Brownian motion. Find for all $0 le s < t$:

• $mathbbEleft[W_s W_t right]$


• $mathbbEleft[W_s W_t^2 right]$


• $mathbbEleft[W_s^2 W_t^2 right]$


• $mathbbEleft[left. e^2W_t right| mathcalF_sright]$

Hint: Recall that the kurtosis for $Z sim mathcalN(0,1)$ is $mathbbEleft[Z^4right] = 3$.

I am confused. I think based on the theory, $mathbbEleft[W_s W_t right]$ should be $s$, and then I do not understand what I should do for $mathbbEleft[W_s W_t^2 right]$.

stochastic-processes brownian-motion expected-value

share|cite|improve this question

edited Apr 1 at 17:43

gt6989b

36k22557

asked Apr 1 at 17:37

leeswimleeswim

1

$endgroup$

add a comment | 

0

$begingroup$

Let $W_t:t ge 0$ be a Brownian motion. Find for all $0 le s < t$:

• $mathbbEleft[W_s W_t right]$


• $mathbbEleft[W_s W_t^2 right]$


• $mathbbEleft[W_s^2 W_t^2 right]$


• $mathbbEleft[left. e^2W_t right| mathcalF_sright]$

Hint: Recall that the kurtosis for $Z sim mathcalN(0,1)$ is $mathbbEleft[Z^4right] = 3$.

I am confused. I think based on the theory, $mathbbEleft[W_s W_t right]$ should be $s$, and then I do not understand what I should do for $mathbbEleft[W_s W_t^2 right]$.

stochastic-processes brownian-motion expected-value

share|cite|improve this question

edited Apr 1 at 17:43

gt6989b

36k22557

asked Apr 1 at 17:37

leeswimleeswim

1

$endgroup$

add a comment | 

$begingroup$

Let $W_t:t ge 0$ be a Brownian motion. Find for all $0 le s < t$:

• $mathbbEleft[W_s W_t right]$


• $mathbbEleft[W_s W_t^2 right]$


• $mathbbEleft[W_s^2 W_t^2 right]$


• $mathbbEleft[left. e^2W_t right| mathcalF_sright]$

Hint: Recall that the kurtosis for $Z sim mathcalN(0,1)$ is $mathbbEleft[Z^4right] = 3$.

I am confused. I think based on the theory, $mathbbEleft[W_s W_t right]$ should be $s$, and then I do not understand what I should do for $mathbbEleft[W_s W_t^2 right]$.

stochastic-processes brownian-motion expected-value

share|cite|improve this question

edited Apr 1 at 17:43

gt6989b

36k22557

asked Apr 1 at 17:37

leeswimleeswim

1

$endgroup$

share|cite|improve this question

edited Apr 1 at 17:43

gt6989b

36k22557

asked Apr 1 at 17:37

leeswimleeswim

1

share|cite|improve this question

edited Apr 1 at 17:43

gt6989b

36k22557

asked Apr 1 at 17:37

leeswimleeswim

1

edited Apr 1 at 17:43

gt6989b

36k22557

edited Apr 1 at 17:43

gt6989b

36k22557

asked Apr 1 at 17:37

leeswimleeswim

1

asked Apr 1 at 17:37

leeswimleeswim

1

1 Answer
1

active

oldest

votes

0

$begingroup$

HINT

Recall that you want to use independent increments, so
$$
mathbbE[W_s W_t]
= mathbbE[W_s (W_s + W_s,t)]
= mathbbE[W_s^2] + mathbbE[W_s W_s,t]
= s+mathbbE[W_s] mathbbE[ W_s,t]
= s,
$$
where $W_s,t$ is the process starting at $s$ and evolving until $t$.

Can you use a similar argument to solve others?

share|cite|improve this answer

answered Apr 1 at 17:42

gt6989bgt6989b

36k22557

$endgroup$

add a comment | 

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0

$begingroup$

HINT

Recall that you want to use independent increments, so
$$
mathbbE[W_s W_t]
= mathbbE[W_s (W_s + W_s,t)]
= mathbbE[W_s^2] + mathbbE[W_s W_s,t]
= s+mathbbE[W_s] mathbbE[ W_s,t]
= s,
$$
where $W_s,t$ is the process starting at $s$ and evolving until $t$.

Can you use a similar argument to solve others?

share|cite|improve this answer

answered Apr 1 at 17:42

gt6989bgt6989b

36k22557

$endgroup$

add a comment | 

0

$begingroup$

HINT

Recall that you want to use independent increments, so
$$
mathbbE[W_s W_t]
= mathbbE[W_s (W_s + W_s,t)]
= mathbbE[W_s^2] + mathbbE[W_s W_s,t]
= s+mathbbE[W_s] mathbbE[ W_s,t]
= s,
$$
where $W_s,t$ is the process starting at $s$ and evolving until $t$.

Can you use a similar argument to solve others?

share|cite|improve this answer

answered Apr 1 at 17:42

gt6989bgt6989b

36k22557

$endgroup$

add a comment | 

$begingroup$

HINT

Recall that you want to use independent increments, so
$$
mathbbE[W_s W_t]
= mathbbE[W_s (W_s + W_s,t)]
= mathbbE[W_s^2] + mathbbE[W_s W_s,t]
= s+mathbbE[W_s] mathbbE[ W_s,t]
= s,
$$
where $W_s,t$ is the process starting at $s$ and evolving until $t$.

Can you use a similar argument to solve others?

share|cite|improve this answer

answered Apr 1 at 17:42

gt6989bgt6989b

36k22557

$endgroup$

share|cite|improve this answer

answered Apr 1 at 17:42

gt6989bgt6989b

36k22557

answered Apr 1 at 17:42

gt6989bgt6989b

36k22557

answered Apr 1 at 17:42

gt6989bgt6989b

36k22557