Does anyone know the Ito's formula for jump-diffusions?
By jump-diffusion I mean this:
dX(t)=M(X(t))dt+S(X(t))dW(t) + dN(t)
where t is the time, X(t) is a scalar function of time, W(t) is a
standard Brownian motion and N(t) is a Poisson process with arrival
intensity L(X(t)) and jump-size distribution f(X(t)-X(t-))). If you
know anything about these, I am sure you understood what I meant.
For example, if there is no jump, that is, if the last term in the
above is zero, the Ito's formula for a function H(X(t)) is this:
dH(t) = {H'(t)M(X(t))+0.5H''(X(t))S(X(t))S(X(t))}dt+H'(X(t))S(X(t))dW(t)
The above is the so-called "differential" form of the Ito's formula
for diffusions. In the above, the "single prime" means the first
derivative and the "double prime" means the second derivative. I am
interested in seeing a clearly stated form of the corresponding
differential form of the Ito's formula for jump-diffusions. All the
books and papers I have been able to find so far are very confusing,
since most of the the authors of these books and papers seem to be
quite confused themselves. And I don't want to waste time to
rediscover the wheel myself. Someone must have figured it out.
I would appreciate any leads.
Best,
Sabri
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