7
votes
Accepted
How to calculate the expectation of Poisson process when its intensity is also stochastic
You can condition on the value of $\lambda_t$. So
$E[dN_t] = E[E[dN_t|\lambda_t]] = E[\lambda_t dt] = E[\lambda_t] dt$
- 2,187
7
votes
Accepted
Black-Scholes formula for Poisson jumps
We assume that the process $\{J_t, \, t\ge 0\}$ is defined at the jump times of the Poisson process $\{N_t, \, t \ge 0\}$, and all the jump sizes are independent and identically distributed. That is,
\...
- 21k
4
votes
Probability density function of simple equation, compound Poisson noise
I don't think you can have an explicit form.
Let $Y_t= e^{at}X_t$ then :
$$ Y_t -Y_0 =\sum_{i=1}^{N_t}e^{aT_i} $$ where $(T_i)_{i=1...N_t}$ are the jump times of your poisson process.
then $$P(Y_t\...
- 2,412
4
votes
Accepted
Bond price under Poissonian model of interest rate
Let $$Y_t = \int_0^t N_u du$$
where $(N_t)_{t \geq 0}$ figures a Poisson process with intensity $\lambda$.
Using the stochastic Fubini theorem we have that:
\begin{align}
Y_T &= \int_0^T N_t dt \...
- 14.5k
2
votes
Accepted
Does a Poisson process converge to an Ito process in long term?
There are, as for random variables, different types of convergences for stochastic processes. Probably you mean convergence in the Skorokhod topology $J_1$. This is one convergence concept for $d$-...
- 136
1
vote
Marked poisson process vs compounded
The compound Poisson process isn't technically a marked process because we formulate the process with respect to $\sum_i D_i$ instead of $(\tau_i, D_i)$. However the compound process is constructed ...
- 111
1
vote
Does a Poisson process converge to an Ito process in long term?
If $X_t$ is a Poisson Counting Process with intensity $\lambda$ then the Martingale $M_t=X_t−\lambda t$ is called a Compensated Poisson Process. As $\lambda$ becomes large $M_t$ does converge to a ...
- 9,332
1
vote
For Probability of Default in retail credit what is more popular logistic regression or GLM with Poisson distribution and why?
http://jgscott.github.io/SDS325H_Spring2015/files/logit_poisson_cox.pdf
Based on my understanding from reading the above document, I think it could be because Poisson is used for count data and ...
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