3 votes
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 ...
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  • 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 ...
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  • 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 ...
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  • 4,963
1 vote

Poisson process under equivalent martingale measure

$N_t$ 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[...
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  • 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 ...
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  • 1,662

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