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

You can condition on the value of $\lambda_t$. So $E[dN_t] = E[E[dN_t|\lambda_t]] = E[\lambda_t dt] = E[\lambda_t] dt$
  • 2,137
7 votes
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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, \...
  • 20.5k
7 votes

Ito's formula for Jump process

Write $X_t = A_t B_t$ with $A_t = e^{(\lambda - \eta)t}$ and $B_t = \left(\frac{\eta}{\lambda} \right)^{N_t}$. Then $dX_t = A_t dB_t + B_t dA_t$ by the product rule of calculus. There are no second ...
  • 1,780
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\...
  • 2,372
4 votes
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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 \...
  • 14.1k
3 votes

For Probability of Default in retail credit what is more popular logistic regression or GLM with Poisson distribution and why?

From http://rmi.nus.edu.sg/gcr/files/04%20GCR%20vol%201.pdf "One of the attractive features of the logistic function is the fact that it is bounded between 0 and 1, making it suitable to ...
  • 831
2 votes
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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 ...
  • 9,167
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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