if some one can help me, thank you in advance. My background knowledge is poor Please don't mock me.
It comes from a small paper this.
During the paper page 24 about the proof, it says:
$$M_{ij}(t)=\int_{(0,t]}I_i(s^-)dM^{j}(s)$$
$$=\sum_{o<s\leq t}I_i(s^-)I_j(s)-\int_{(0,t]}\sum_k\mu_{kj}(s)I_i(s^-)I_k(s)ds$$
where $M^{j}(t)=I_j(t)-I_j(0)-\int_0^t\sum_k\mu_{kj}(s)I_k(s)ds$ is a martingale and I is an indicator process: $1_{\{x(t)=i\}}=I_i(t)$ and $X(t)$ is continuous-time markov chain defined on page 23 from that paper. $\mu_{ij}$ is transition intensity by X(t) (see page 22).
For more details, please see page 22-24 about that paper, I think it is very basic for you.
My question:
- What is the meaning of ij in $M_{ij}$ ?
My trying: It can be described that at time s-, the state is i, and at time s, the state is j, during such (jump) moment, we denote the martingale as $M_{ij}$. or given the time is s-, which the state is i, we collect all the martingale $M^{j}$ by time, where the state comes to j in the time s.
Am I correct?
- What is the dimension of $M_{ij}$ ?
Is it mxm martingale?matrix? (if the state is finite up to m)
- Why it equals the second line formula?
My trying:
Since $$M^{j}(t)=I_j(t)-I_j(0)-\int_0^t\sum_k\mu_{kj}(s)I_k(s)ds,$$ then $$dM^{j}(t)=dI_j(t)-\sum_k\mu_{kj}(t)I_k(t)ds,$$ then $$\int_{(0,t]}I_i(s^-)dM^{j}(s)=**\int_{(0,t]}I_i(s^-)dI_j(t)**-\int_{(0,t]}I_i(s^-)\sum_k\mu_{kj}(t)I_k(t)ds$$ then I cannot get it through.
How does the first term on the RHS? From the proof, why it becomes the summation instead of integral? I totally don't understand. If you know or understand, please help me in detail.