How to calculate the expectation of Poisson process when its intensity is also stochastic

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How to calculate the expectation of Poisson process $N_t$ when its intensity is also stochastic? Since when intensity $lambda_t$ is non-random, then we have
$$E[dN_t] = lambda_tdt.$$
But how about the stochastic $lambda_t?$ I have no idea to calculate it. You can simply assume $lambda_t$ is a Gaussian process.

poisson

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user6703592

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up vote
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down vote

favorite

How to calculate the expectation of Poisson process $N_t$ when its intensity is also stochastic? Since when intensity $lambda_t$ is non-random, then we have
$$E[dN_t] = lambda_tdt.$$
But how about the stochastic $lambda_t?$ I have no idea to calculate it. You can simply assume $lambda_t$ is a Gaussian process.

poisson

share|improve this question

asked 3 hours ago

user6703592

896

add a comment | 

up vote
2
down vote

favorite

up vote
2
down vote

favorite

How to calculate the expectation of Poisson process $N_t$ when its intensity is also stochastic? Since when intensity $lambda_t$ is non-random, then we have
$$E[dN_t] = lambda_tdt.$$
But how about the stochastic $lambda_t?$ I have no idea to calculate it. You can simply assume $lambda_t$ is a Gaussian process.

poisson

share|improve this question

asked 3 hours ago

user6703592

896

How to calculate the expectation of Poisson process $N_t$ when its intensity is also stochastic? Since when intensity $lambda_t$ is non-random, then we have
$$E[dN_t] = lambda_tdt.$$
But how about the stochastic $lambda_t?$ I have no idea to calculate it. You can simply assume $lambda_t$ is a Gaussian process.

poisson

poisson

share|improve this question

asked 3 hours ago

user6703592

896

share|improve this question

asked 3 hours ago

user6703592

896

share|improve this question

share|improve this question

asked 3 hours ago

user6703592

896

asked 3 hours ago

user6703592

896

asked 3 hours ago

user6703592

896

896

add a comment | 

add a comment | 

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You can condition on the value of $lambda_t$. So

$E[dN_t] = E[E[dN_t|lambda_t]] = E[lambda_t dt] = E[lambda_t] dt$

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answered 2 hours ago

Ezy

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
3
down vote



accepted

You can condition on the value of $lambda_t$. So

$E[dN_t] = E[E[dN_t|lambda_t]] = E[lambda_t dt] = E[lambda_t] dt$

share|improve this answer

answered 2 hours ago

Ezy

31616

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Check out our Code of Conduct.

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up vote
3
down vote



accepted

You can condition on the value of $lambda_t$. So

$E[dN_t] = E[E[dN_t|lambda_t]] = E[lambda_t dt] = E[lambda_t] dt$

share|improve this answer

answered 2 hours ago

Ezy

31616

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Ezy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

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up vote
3
down vote



accepted

up vote
3
down vote



accepted

You can condition on the value of $lambda_t$. So

$E[dN_t] = E[E[dN_t|lambda_t]] = E[lambda_t dt] = E[lambda_t] dt$

share|improve this answer

answered 2 hours ago

Ezy

31616

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Ezy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

You can condition on the value of $lambda_t$. So

$E[dN_t] = E[E[dN_t|lambda_t]] = E[lambda_t dt] = E[lambda_t] dt$

share|improve this answer

answered 2 hours ago

Ezy

31616

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Ezy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|improve this answer

share|improve this answer

answered 2 hours ago

Ezy

31616

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answered 2 hours ago

Ezy

31616

answered 2 hours ago

Ezy

31616

31616

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Ezy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

New contributor

Ezy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

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