I was reading the beginning of chapter 3 of Karatzas, Shreve about stochastic integrals. I already had a brief introduction to stochastic integration but never clearly understood the meaning of $\mathrm{d}A_s$ in an integral. The goal is to understand later the introduction of stochastic integration of this chapter.
In chapter 1 of Karatzas, Shreve they introduce a new notion without precise explanation:
4.5 Definition. An increasing process $A$ is called natural if for every bounded, right-continuous martingale $(M_t,\mathscr{F}_t)_t$ we have $$ \tag{1} \mathbb{E}[\int_{(0;t]}M_s\mathrm{d}A_s]=\mathbb{E}[\int_{(0;t]}M_{s^-}\mathrm{d}A_s] $$ 4.6 Remark. If $A$ is increasing and $X$ measurable process, then for $\omega\in\Omega$ fixed, the sample path $(X_t(\omega))_t$ is a measurable function from $[0;\infty)$ to $\mathbb{R}$. It follows that the Lebesgue-Stieltjes integrals $$ \tag{2} I^{\pm}_t(\omega):=\int_{(0;t]}X_s^{\pm}(\omega)\mathrm{d}A_s(\omega) $$ are well defined. If $X$ is progressively measurable (i.e. right-continuous and adapted), and if $I_t=I_t^+-I_t^-$ is well defined and finite for all $t\geq0$, then $I$ is right-continuous and progessively measurable.
I am talking about the notion $\int_{(0;t]}M_s\mathrm{d}A_s$ or $\int_{(0;t]}X^{\pm}_s\mathrm{d}A_s$.
Questions
- What is the definition of $\int_{(0;t]}M_s\mathrm{d}A_s$?
- Why do they write $(0;t]$ instead of $[0;t]$?
My Approach
I ignore the distinction between $(0;t]$ and $[0;t]$ and simply write $(Y_t)_t=(\int_0^tM_s\mathrm{d}A_s)_t$. For almost every $\omega\in\Omega$ I have then $$ \tag{3} Y_t(\omega)=\int_{\mu_\omega^t}M_s(\omega)\mathrm{d}s $$ where the measure $\mu_\omega^t$ is given by $$ \tag{4} \forall B\in B(\mathbb{R})\cap 2^{[0;t]}: \mu_\omega(B)=\lambda(f_{\omega}^{-1}(B)) $$ where $f_\omega:[0;t]\ni s\mapsto A_t(\omega)$ and $\lambda$ is the Lebesgue on $B(\mathbb{R})$. Since $A$ an increasing process, $\mu_\omega^t$ defines a measure.
My Concerns
- Somehow the measure $\mu_\omega^t$ does not really include all of the proces $A$, since the image of $A$ is almost surely a subset of $[0;\infty)$ and not necessarily of $[0;t]$. Hence I would prefer to definie $\mu_\omega^t(B)=\lambda(f_\omega(B))$, but this is in general probably not possible. How can I define the measure $\mu_\omega^t$ correctly in this manner?
- In the definition of an adapted process we also have that $A_0=0$ almost surely, but I didn't use this in my interpretation of this integral in (1). Can we drop this requirement to define this integral in (1) accordingly? Similarly I didin't use right-continuity of $A$ almost surely. In fact I only used that $f_\omega$ is measurable.