# How to transform a stochastic jump diffusion equation to a Levy stochastic differential equation?

If I have this type of stochastic differential equation : $$dX(t) = A(X(t),t)\ dt +B(X(t),t)\ dW(t) + C(X(t),t)\ dP(t)$$ With $$\begin{align} dW(t)& : \text{A wiener process}\\ dP(t)& : \text{A Poisson process with parameter }\Lambda\\ A,B,C& : \text{Smooth functions} \end{align}$$ and I want to transform it to this type of stochastic differential equations: $$dX(t) = F(X(t),t)\ dt + G(X(t),t)\ dL(t)\\$$ with $$\begin{align} L(t)& : \text{An }\alpha\text{-stable Levy process} \end{align}$$ I was trying to identify the relations between the coefficients and parameters. Could some one tell me if I can use the development of a Levy process to a Brownian motion, a drift and a poisson process to find a relation between those two equations ?

I wanted to use it in the case $F$ and $G$ are constants.

Thank you.

Kind regards

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@ Samatix : When possible I think that the best way to do it is to use Lévy-Khintchine decomposition of the process and identifying terms. Best regards – TheBridge Aug 7 '12 at 15:53
Hi TheBridge. I found the equation using google but it is too complicated to apply. Could you, please, write down the Lévy-Khintchine formula ? Best regards – Samatix Aug 7 '12 at 17:03
Do you know where I can find the formula you talked about ? – Samatix Aug 8 '12 at 10:42
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