I consider a Lévy process of the form $$ X_t = X_0 e^{(\mu-\frac{\sigma^2}{2})t+\sigma B_t-\sum_{i=1}^{N_t} Y_i}, $$ where $\mu$ and $\sigma$ are constants, $B_t$ is a Brownian motion, $N_t$ is a Poisson process and $Y_i$ are i.i.d. random variables indepentent with $N_t$.

In different papers I noted that the Lévy process can be described as a solution of $$ dX_t = \mu(t, X_t)dt + \sigma(t, X_t)dB_t+ \int_{\mathbb{R}}\gamma(t, X_t, z)\tilde{v}(dt, dz), $$ where $\tilde{v}(dt, dz)$ is a compensated Poisson measure.

My question is: What SDE satisfies process $X_t$ and how can I derive it?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.