I am reading Protter's book stochastic integration and differential equation. He proves the following theorem (Chp III thm 25):

If $A$ is a increasing process of locally integrable variation whose jumps occur at totally inaccessible time, then the compensator of $A$ is continuous.

I know this could be done by Doob-Meyer decomposition. But to use Doob-Meyer we need $A$ to be of class $D$ ($D$ for Doob), which means for any stopping time $T$, $A_{T\wedge t}$ is uniformly integrable. I think this could be relax a little bit to the class DL, which means for any bounded stopping time $T$, $A_{T\wedge t}$ is uniformly integrable. But I do not see how to prove A is of class DL here.

Here A is of locally integrable variation just means there exists a sequence of stopping time $T_n$ which goes to infinity a.s. and $E\left(A_{T_n}\right)$ is finite for all $n$.



Apply Doob-Meyer to each of the stopped processes $A^{T_n}_t:=A_{T_n\wedge t}$, which have only totally inaccessible jumps. This yields, for each $n$ a continuous increasing process $B^{(n)}$ (with $B^{(n)}_0=0$) such that $A^{T_n}-B^{(n)}$ is a local martingale. If $m<n$ then $A^{T_n}-A^{T_m}$ vanishes on $[0,T_m]$, so the stopped process $(B^{(n)}-B^{(m)})^{T_m}$ is a continuous local martingale with paths of finite variation, and is therefore a constant process. It follows that $B^{(n)}=B^{(m)}$ on $[0,T_m]$. This coincidence permits you to piece together the $B^{(n)}$ to obtain a continuous increasing process $B$ such that $B=B^{(n)}$ on $[0,T_n]$ for each $n$. This continuous process is the compensator of $A$.


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