Semimartingale characteristics for stochastic integral? I'm recently reading Limit Theorems for Stochastic Processes. A question came to my mind when going through the theory of Characteristics of Semimartingales in Ch. 2. How to figure out the characteristics for a general stochastic integral? To be specific, 

Let $X$ be a $d$-dimensional semimartingale, with characteristics $(B,C,\nu)$ relative to a truncation function $h$, $H$ be a locally bounded predictable
  processes. Then it's well-known that, the stochastic integral $H\cdot X=\int_0^\cdot H_s dX_s$ is a semimartingale. The question is, what the characteristics of $H\cdot X$ look like?

I cannot find it out within this book. Could anyone give some reference or comments? Appreciate!
 A: I post an answer of myself here. Denote $Y=H\cdot X$, and the characteristics of $Y$ by $(B'',C'',\nu'')$.
If we want to find out the characteristics of $Y$, we'd better make an assumption for the integrand $H$:

Assumption The $d\times d$ matrix-valued process $H$ is a.s. nondegenerate, that is,
  \begin{equation}
\mathbf P\{\text{det}(H_t)\ne0, \forall t\ge0\}=1.
\end{equation}

The assumption is to insure that 
\begin{equation}\tag{*}
\Delta Y\ne0 \quad\text{if and only if}\quad \Delta X \ne0,
\end{equation}
since $\Delta Y=H\Delta X$. One may make other assumptions to insure it. Thus if $\mu^X$ and $\mu^Y$ denote the random measures associated with the jumps of $X$ and $Y$, then
\begin{equation*}
\begin{split}
\mu^Y (dt,dy) & = \sum_s\mathbf 1_{\{\Delta Y_s\ne 0\}} \delta_{(s,\Delta Y_s)}(dt,dy) \\
& = \sum_s\mathbf 1_{\{\Delta X_s\ne 0\}} \delta_{(s,H_s\Delta X_s)}(dt,dy).
\end{split}
\end{equation*}
Thus, for any $G\in\mathcal B(\mathbb R_+)\otimes\mathcal  B(\mathbb R^d)$,
\begin{equation*}
\begin{split}
(\mathbf 1_G*\mu^Y)_t & = \mu^Y(([0,t]\times\mathbb R^d)\cap G) \\
& = \sum_s\mathbf 1_{\{\Delta X_s\ne 0\}} \delta_{(s,H_s\Delta X_s)}(([0,t]\times\mathbb R^d)\cap G)\\
& = \sum_s\mathbf 1_{\{\Delta X_s\ne 0\}} \delta_{(s,\Delta X_s)}(([0,t]\times\mathbb R^d)\cap\{(r,x):(r,H_r x))\in G\})\\
& = \int_0^t\int_{\mathbb R^d}\mathbf 1_G(r,H_r x))\mu^X(dr,dx).
\end{split}
\end{equation*}
Once we get this, all the rest of procedure are the same as the proof of Proposition IX.5.3 in the book "Limit Theorems for Stochastic Processes". Finally, we obtain the characteristics
\begin{equation*}
\begin{split}
B''&=\{B'_i;d+1\le i\le d+m\} = H\cdot B + (h'(Hx)-Hh(x))*\nu, \\
C''&=\{C'_{ij};d+1\le i\le d+m,d+1\le j\le d+m\} = \{(H^k_i H^l_j)\cdot C_{kl}\}, \\
(\mathbf 1_G*\nu'')_t & =\int_0^t\int_{\mathbb R^d}\mathbf 1_G(r,H_r x))\nu(dr,dx).
\end{split}
\end{equation*}
Remark In fact, the condition (*) can be weakened to that
\begin{equation}\tag{**}
\Delta Y\ne0 \Rightarrow \Delta X \ne0,
\end{equation}
and the conclusion still holds. That is, the Assumption is not necessary at all.
