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 ...
- 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
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[...
- 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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