# Tag Info

## Hot answers tagged poisson-process

### How can I learn stochastic process & stochastic calculus in two weeks?

This is impossible unless you are very intelligent with good memory-retention skills and already mathemathically proficient in the field of analysis and statistics (and no, a single course in basic ...
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### How do I learn the stochastic calculus of Poisson processes?

Summarizing the suggestions in comments: Nicolas Privault's chapter Stochastic Calculus of Jump Processes [available online] provides only a very brief overview. Chapter 11 of Shreve's II volume (...
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### Prove the jump times of a poisson process in a given interval are uniformly distributed

http://www.math.tau.ac.il/~uriy/Papers/encyc57.pdf - page 5 in this does that do it?
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### Second variation of a Brownian motion under jump-diffusion process

$$X_t = B_t 1_{t<0.5} + (x+ B_t) 1_{t\geq 0.5} = B_t + x1_{t\geq 0.5}$$ $$[X, X]_t = [B, B]_t + x^2 1_{t\geq 0.5} = t+ x^2 1_{t\geq 0.5}$$ (the author probably intended to use $0.5$ as jump size ...
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### Expected Value of Mean-Reverting Jump Process

First, we need to be careful about putting the condition at the right place: \begin{align} e^{kt}\mathbb{E}[\mu_t] -\mu_0 &= \mathbb{E}\bigg[\sum_{m=1}^{N_t} e^{k\tau_m}\eta_m\bigg]\\ &= \...
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### Separating jumps and diffusion

More some thoughts than a definite answer. Even if you want to allow some form of price divergence between the two markets (because arbitrage oportunities are hard to monetise) it is probabily ...
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### Ito Lemma for Poisson Process

The basic Ito formula for a Poisson process is $$dY_t = \mu_t dt + g_t dN_t$$ $$df(Y_t) = \mu_t f'(Y_t) dt + (f(Y_{t-}+g_t) - f(Y_{t-}))dN_t$$ (dropped $f$'s direct dependence on the time variable ...
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$N_t$ process comes with its own Poisson law (probability measure) $P$ defined via intensity $\lambda$. Under it, $N_t-\lambda t$ is a martingale wrt ${\cal F}_t =\sigma(N_u | u\in [0,t])$ (as $E^P[... • 5,028 1 vote ### Poisson process under equivalent martingale measure Consider a radon nikodym derivative, the Random variable:$Z(w)= 1_{N(T,w)=1}+1-Pr(N(T)=1)$. It is admissible since it is always positive and has an expectation 1. This will lead us to the formation ... • 1,662 1 vote Accepted ### Arbitrage free in a Black-Scholes/Poisson model This is easy to answer with the meta theorem given in the same chapter. Here you have two sources of randomness (W and N), and one risky asset. Q1: Arbitrage generally happens when you have more ... 1 vote ### Marked poisson process vs compounded The compound Poisson process isn't technically a marked process because we formulate the process with respect to$\sum_i D_i$instead of$(\tau_i, D_i)\$. However the compound process is constructed ...
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