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Questions tagged [poisson]

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3
votes
1answer
267 views

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

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 ...
3
votes
0answers
76 views

Marked poisson process vs compounded

I am a bit fuzzy about difference between compounded poisson process defined as $$\sum_{i=1}^{N_t} D_i $$ where $N_t$ is poisson process and $ D_i $ are iid random variables and marked poisson ...
5
votes
1answer
600 views

Black-Scholes formula for Poisson jumps

For underlying asset $$d S = r S dt + \sigma S d W + (J-1)Sd N$$ here $W$ is a Brownian motion, $N(t)$ is Poisson process with intensity $\lambda.$ Suppose $J$ is log-normal with standard deviation $\...
6
votes
1answer
5k views

How to simulate a jump-diffusion process?

I would like to price Asian and Digital options under Merton's jump-diffusion model. To that end, I will have to simulate from a jump diffusion process. In general, the stock price process is given ...
4
votes
1answer
160 views

Bond price under Poissonian model of interest rate

Working through an exercise in interest rate modelling and I have the following setup: $r_t = r_0 + \delta N_t$ where $\delta > 0$ and $\lambda > 0$ is the intensity of the Poisson pricess $N_t$...
2
votes
2answers
293 views

Does a Poisson process converge to an Ito process in long term?

I have heard that a Poisson process "converges" to an Ito (diffusion) process in long term. However I do not see how the characteristic function of the form morphs into that of the latter. In what ...
1
vote
0answers
50 views

Any idea of compound Poisson processes in betting? [closed]

Any suggestions on compound poisson processes in bets of a customer?
6
votes
2answers
2k views

Ito's formula for Jump process

Let $\{N_t\,|\,0\leq t\leq T\}$ be a Poisson process with intensity $\lambda>0$ defined on the probability space $(\Omega,\mathcal{F}_t,P)$ with respect to the filtration $\mathcal{F}_t$ and \...
7
votes
1answer
216 views

pdf of simple equation, compound Poisson noise

I would like to find the probability density function (at stationarity) of the random variable $X_t$, where: \begin{equation*} dX_t = -aX_t dt + d N_t, \end{equation*} $a$ is a constant and $N_t$ is a ...
1
vote
3answers
700 views

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

Trying to understand which regression model is more popular in retail credit card industry Logistic regression or GLM with Poisson distribution and why?