Let $T<\infty$ and $Y$ be some cadlag semimartingale such that : $$ dY_t = r_tdt + \sigma_t dW_t + \beta_tdN_t,\ t\in [0,T] $$
Where $W$ is a $1$-dimensional Brownian Motion, $N$ be a jump process and $r,\sigma,\beta$ are assumed to be bounded.
My questions are:
- How to develop $((Y_t)^+)_{t\in[0,T]}$? Do the processes $r,\sigma,\beta$ bounded is enough ?
- Does the Itô-Tanaka's formula works, since we have a jump process ?
- How could we express the Local time term in this case ?
- Is there a reference for this specific problem (article, book ...) ? it could be enough for me.
My idea is maybe to develop Itô-Tanaka formula on every $[\tau_k,\tau_{k+1}[$.
I think we can write $Y$ such that $$ (Y_t)^+=\sum^{\infty}_{k=0}1_{\tau_{k}\leq t<\tau_{k+1}}(X_t+C_t)^+ $$ where $X=A+M$ with $A$ is a finite variation term (i.e. $dt$), $M$ be the continuous (Brownian) martingale part, $C$ be the cadlag part.
For a reference there is Chapter IV, Section 7 in Ph. Protter. Stochastic Integration and Differential Equations. Springer, Berlin, Second edition, 2005.
If someone can give me more ideas thank you.