# Tag Info

Accepted

### 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 ...
• 4,963
1 vote
Accepted

### 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 ...
• 1,419
1 vote
Accepted

### 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 ...
• 4,963
1 vote

$N_t$ 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 $E^P[... • 4,963 1 vote ### Poisson process under equivalent martingale measure Consider a radon nikodym derivative, the Random variable:$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

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