I have just learned Ito fomula for jump processes but I have still not understood it well. Assume that I have

$dS_t=S_{t-}\mu+S_{t-}\sigma dB_t +S_{t-}\int_{\mathbb{R}^+}(y-1)N(dt,dy), \;\; 0\leq t\leq T$

where $N$ is a counting measure whose mean measure is $\nu(dy)dt$.

Define $q_t=\frac{1}{rT}(1-e^{-r(T-t)})$, and $X_0=q_0S_0-e^{-rT}K$ where $0<r, K$ are given real numbers, let $X_t$ be a process that satisfies

$ dX_t=q_tdS_t+r(X_{t-}-q_tS_{t-})dt\;\;0\leq t\leq T$. Define $Z_t=\frac{X_t}{S_t}$

1) I would like to know what is $dZ_t$ ??

2) What is the intuitive meaning of the mean measure $\nu(dy)dt$??

I know that I need to use Ito's product rule but I am still confused about jump processes. Could someone help me with this problem? Thank you so much.


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.